TeX sample extracted from math.SE tutorial
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| <!DOCTYPE html> | |
| <html> | |
| <head> | |
| <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> | |
| <meta name="viewport" content="width=device-width, initial-scale=1"> | |
| <title>Math.SE extracted Samples</title> | |
| </head> | |
| <body> | |
| <p>Licensed under <a href="https://creativecommons.org/licenses/by-sa/3.0/" rel="license">cc by-sa 3.0</a>. Original source: <a href="http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference">meta.math.stackexchange</a> (cc by-sa 3.0), with <a href="http://blog.stackoverflow.com/2009/06/attribution-required/" rel="license">attribution required</a>.</p> | |
| <div>\(\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}\)</div> | |
| <div>\[\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}\]</div> | |
| <div>\(\alpha, \beta, … \omega\)</div> | |
| <div>\(\Gamma, \Delta, …, \Omega\)</div> | |
| <div>\(x_i^2\)</div> | |
| <div>\(\log_2 x\)</div> | |
| <div>\(10^10\)</div> | |
| <div>\(10^{10}\)</div> | |
| <div>\({x^y}^z\)</div> | |
| <div>\(x^{y^z}\)</div> | |
| <div>\(x_i^2\)</div> | |
| <div>\(x_{i^2}\)</div> | |
| <div>\((2+3)[4+4]\)</div> | |
| <div>\(\{\}\)</div> | |
| <div>\((\frac{\sqrt x}{y^3})\)</div> | |
| <div>\(\left(\frac{\sqrt x}{y^3}\right)\)</div> | |
| <div>\((x)\)</div> | |
| <div>\([x]\)</div> | |
| <div>\(\{ x \}\)</div> | |
| <div>\(|x|\)</div> | |
| <div>\(\vert x \vert\)</div> | |
| <div>\(\Vert x \Vert\)</div> | |
| <div>\(\langle x \rangle\)</div> | |
| <div>\(\lceil x \rceil\)</div> | |
| <div>\(\lfloor x \rfloor\)</div> | |
| <div>\(\left.\frac12\right\rbrace\)</div> | |
| <div>\(\Biggl(\biggl(\Bigl(\bigl((x)\bigr)\Bigr)\biggr)\Biggr)\)</div> | |
| <div>\(\sum_1^n\)</div> | |
| <div>\(\sum_{i=0}^\infty i^2\)</div> | |
| <div>\(\prod\)</div> | |
| <div>\(\int\)</div> | |
| <div>\(\bigcup\)</div> | |
| <div>\(\bigcap\)</div> | |
| <div>\(\iint\)</div> | |
| <div>\(\iiint\)</div> | |
| <div>\(\frac ab\)</div> | |
| <div>\(\frac{a+1}{b+1}\)</div> | |
| <div>\({a+1\over b+1}\)</div> | |
| <div>\(\cfrac{a}{b}\)</div> | |
| <div>\(\mathbb{CHNQRZ}\)</div> | |
| <div>\(\mathbf{ABCDEFGHIJKLMNOPQRSTUVWXYZ}\)</div> | |
| <div>\(\mathbf{abcdefghijklmnopqrstuvwxyz}\)</div> | |
| <div>\(\mathtt{ABCDEFGHIJKLMNOPQRSTUVWXYZ}\)</div> | |
| <div>\(\mathtt{abcdefghijklmnopqrstuvwxyz}\)</div> | |
| <div>\(\mathrm{ABCDEFGHIJKLMNOPQRSTUVWXYZ}\)</div> | |
| <div>\(\mathrm{abcdefghijklmnopqrstuvwxyz}\)</div> | |
| <div>\(\mathsf{ABCDEFGHIJKLMNOPQRSTUVWXYZ}\)</div> | |
| <div>\(\mathsf{abcdefghijklmnopqrstuvwxyz}\)</div> | |
| <div>\(\mathcal{ ABCDEFGHIJKLMNOPQRSTUVWXYZ}\)</div> | |
| <div>\(\mathscr{ABCDEFGHIJKLMNOPQRSTUVWXYZ}\)</div> | |
| <div>\(\mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZ} \mathfrak{abcdefghijklmnopqrstuvwxyz}\)</div> | |
| <div>\(\sqrt{x^3}\)</div> | |
| <div>\(\sqrt[3]{\frac xy}\)</div> | |
| <div>\(\sin x\)</div> | |
| <div>\(sin x\)</div> | |
| <div>\[\lim_{x\to 0}\]</div> | |
| <div>\(\lt\, \gt\, \le\, \leq\, \leqq\, \leqslant\, \ge\, \geq\, \geqq\, \geqslant\, \neq\)</div> | |
| <div>\(\not\lt\)</div> | |
| <div>\(\times\, \div\, \pm\, \mp\)</div> | |
| <div>\(x\cdot y\)</div> | |
| <div>\(\cup\, \cap\, \setminus\, \subset\, \subseteq \,\subsetneq \,\supset\, \in\, \notin\, \emptyset\, \varnothing\)</div> | |
| <div>\({n+1 \choose 2k}\)</div> | |
| <div>\(\to\, \rightarrow\, \leftarrow\, \Rightarrow\, \Leftarrow\, \mapsto\)</div> | |
| <div>\(\land\, \lor\, \lnot\, \forall\, \exists\, \top\, \bot\, \vdash\, \vDash\)</div> | |
| <div>\(\star\, \ast\, \oplus\, \circ\, \bullet\)</div> | |
| <div>\(\approx\, \sim \, \simeq\, \cong\, \equiv\, \prec, \lhd\)</div> | |
| <div>\(\infty\, \aleph_0\)</div> | |
| <div>\(\nabla\, \partial\)</div> | |
| <div>\(\Im\, \Re\)</div> | |
| <div>\(a\equiv b\pmod n\)</div> | |
| <div>\(a_1, a_2, \ldots ,a_n\)</div> | |
| <div>\(a_1+a_2+\cdots+a_n\)</div> | |
| <div>\(\epsilon\, \varepsilon\)</div> | |
| <div>\(\phi\, \varphi\)</div> | |
| <div>\(\ell\)</div> | |
| <div>\(\TeX\)</div> | |
| <div>\(\LaTeX\)</div> | |
| <div>\(\TeX\)</div> | |
| <div>\(a b\)</div> | |
| <div>\(a\,b\)</div> | |
| <div>\(a\;b\)</div> | |
| <div>\(a\quad b\)</div> | |
| <div>\(a\qquad b\)</div> | |
| <div>\(\{x\in s\mid x\text{ is extra large}\}\)</div> | |
| <div>\(\hat x\)</div> | |
| <div>\(\widehat{xy}\)</div> | |
| <div>\(\bar x\)</div> | |
| <div>\(\overline{xyz}\)</div> | |
| <div>\(\vec x\)</div> | |
| <div>\(\overrightarrow{xy}\)</div> | |
| <div>\(\overleftrightarrow{xy}\)</div> | |
| <div>\(\frac d{dx}x\dot x = \dot x^2 + x\ddot x\)</div> | |
| <div>\(\$\)</div> | |
| <div>\(\{\)</div> | |
| <div>\(\_\)</div> | |
| <div>\(\backslash\)</div> | |
| <div>\(\LaTeX\)</div> | |
| <div>\(\mathrm{B}\)</div> | |
| <div>\(\sin\)</div> | |
| <div>\(\operatorname{Spec} A\)</div> | |
| <div>\(_5C_3\)</div> | |
| <div>\(\TeX\)</div> | |
| <div>\(\TeX\)</div> | |
| <div>\(\LaTeX\)</div> | |
| <div>\(\mathrm{d}x\)</div> | |
| <div>\[ | |
| \begin{matrix} | |
| 1 & x & x^2 \\ | |
| 1 & y & y^2 \\ | |
| 1 & z & z^2 \\ | |
| \end{matrix} | |
| \] | |
| </div> | |
| <div>\(\begin{pmatrix}1&2\\3&4\\ \end{pmatrix}\)</div> | |
| <div>\(\begin{bmatrix}1&2\\3&4\\ \end{bmatrix}\)</div> | |
| <div>\(\begin{Bmatrix}1&2\\3&4\\ \end{Bmatrix}\)</div> | |
| <div>\(\begin{vmatrix}1&2\\3&4\\ \end{vmatrix}\)</div> | |
| <div>\(\begin{Vmatrix}1&2\\3&4\\ \end{Vmatrix}\)</div> | |
| <div>\(\cdots\)</div> | |
| <div>\(\ddots\)</div> | |
| <div>\(\vdots\)</div> | |
| <div>\begin{pmatrix} | |
| 1 & a_1 & a_1^2 & \cdots & a_1^n \\ | |
| 1 & a_2 & a_2^2 & \cdots & a_2^n \\ | |
| \vdots & \vdots& \vdots & \ddots & \vdots \\ | |
| 1 & a_m & a_m^2 & \cdots & a_m^n | |
| \end{pmatrix}</div> | |
| <div>\[ | |
| \left[\begin{array}{cc|c} | |
| 1&2&3\\ | |
| 4&5&6 | |
| \end{array}\right] | |
| \] | |
| </div> | |
| <div>\[ | |
| \begin{pmatrix} | |
| a & b \\ | |
| c & d\\ | |
| \hline | |
| 1 & 0\\ | |
| 0 & 1 | |
| \end{pmatrix} | |
| \] | |
| </div> | |
| <div>\(\bigl( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \bigr)\)</div> | |
| <div>\begin{align} | |
| \sqrt{37} & = \sqrt{\frac{73^2-1}{12^2}} \\ | |
| & = \sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\ | |
| & = \sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\ | |
| & = \frac{73}{12}\sqrt{1 - \frac{1}{73^2}} \\ | |
| & \approx \frac{73}{12}\left(1 - \frac{1}{2\cdot73^2}\right) | |
| \end{align}</div> | |
| <div>\[ | |
| \begin{align} f(x)&=\left(x^3\right)+\left(x^3+x^2+x^1\right)+\left(x^3+x^2\right)\\ f'(x)&=\left(\left(3x^2+2x+1\right)+\left(3x^2+2x\right)\right)\\ f''(x)&=\left(6x+2\right)\\ \end{align} | |
| \]</div> | |
| <div>\({}{}{}{}{}{}{}{}\)</div> | |
| <div>\(\Psi\)</div> | |
| <div>\(\delta\)</div> | |
| <div>\(\zeta\)</div> | |
| <div>\(\ge\)</div> | |
| <div>\(\subseteq\)</div> | |
| <div>\(\LaTeX\)</div> | |
| <div>\(\LaTeX\)</div> | |
| <div>f(n) = | |
| \begin{cases} | |
| n/2, & \text{if $n$ is even} \\ | |
| 3n+1, & \text{if $n$ is odd} | |
| \end{cases}</div> | |
| <div>\[ | |
| \left. | |
| \begin{array}{l} | |
| \text{if $n$ is even:}&n/2\\ | |
| \text{if $n$ is odd:}&3n+1 | |
| \end{array} | |
| \right\} | |
| =f(n) | |
| \] | |
| </div> | |
| <div>f(n) = | |
| \begin{cases} | |
| \frac{n}{2}, & \text{if $n$ is even} \\[2ex] | |
| 3n+1, & \text{if $n$ is odd} | |
| \end{cases}</div> | |
| <div>\begin{array}{c|lcr} | |
| n & \text{Left} & \text{Center} & \text{Right} \\ | |
| \hline | |
| 1 & 0.24 & 1 & 125 \\ | |
| 2 & -1 & 189 & -8 \\ | |
| 3 & -20 & 2000 & 1+10i | |
| \end{array} | |
| </div> | |
| <div>\[ | |
| % outer vertical array of arrays | |
| \begin{array}{c} | |
| % inner horizontal array of arrays | |
| \begin{array}{cc} | |
| % inner array of minimum values | |
| \begin{array}{c|cccc} | |
| \text{min} & 0 & 1 & 2 & 3\\ | |
| \hline | |
| 0 & 0 & 0 & 0 & 0\\ | |
| 1 & 0 & 1 & 1 & 1\\ | |
| 2 & 0 & 1 & 2 & 2\\ | |
| 3 & 0 & 1 & 2 & 3 | |
| \end{array} | |
| & | |
| % inner array of maximum values | |
| \begin{array}{c|cccc} | |
| \text{max}&0&1&2&3\\ | |
| \hline | |
| 0 & 0 & 1 & 2 & 3\\ | |
| 1 & 1 & 1 & 2 & 3\\ | |
| 2 & 2 & 2 & 2 & 3\\ | |
| 3 & 3 & 3 & 3 & 3 | |
| \end{array} | |
| \end{array} | |
| \\ | |
| % inner array of delta values | |
| \begin{array}{c|cccc} | |
| \Delta&0&1&2&3\\ | |
| \hline | |
| 0 & 0 & 1 & 2 & 3\\ | |
| 1 & 1 & 0 & 1 & 2\\ | |
| 2 & 2 & 1 & 0 & 1\\ | |
| 3 & 3 & 2 & 1 & 0 | |
| \end{array} | |
| \end{array} | |
| \] | |
| </div> | |
| <div>\(\mathsf{Show\ Math\ As\ }\blacktriangleright\mathsf{\ TeX\ Commands}\)</div> | |
| <div>\(all italics, weird-looking spacing, an' odd apostrophes\)</div> | |
| <div>\(\LaTeX\)</div> | |
| <div>\[ | |
| \begin{array}{ll} \hfill\mathrm{Bad}\hfill & \hfill\mathrm{Better}\hfill \\ \hline \\ e^{i\frac{\pi}2} \quad e^{\frac{i\pi}2}& e^{i\pi/2} \\ \int_{-\frac\pi2}^\frac\pi2 \sin x\,dx & \int_{-\pi/2}^{\pi/2}\sin x\,dx \\ \end{array} | |
| \]</div> | |
| <div>\begin{array}{cc} | |
| \mathrm{Bad} & \mathrm{Better} \\ | |
| \hline \\ | |
| e^{i\frac{\pi}2} \quad e^{\frac{i\pi}2}& e^{i\pi/2} \\ | |
| \int_{-\frac\pi2}^\frac\pi2 \sin x\,dx & \int_{-\pi/2}^{\pi/2}\sin x\,dx \\ | |
| \end{array}</div> | |
| <div>\begin{array}{cc} | |
| \mathrm{Bad} & \mathrm{Better} \\ | |
| \hline \\ | |
| \{x|x^2\in\Bbb Z\} & \{x\mid x^2\in\Bbb Z\} \\ | |
| \end{array}</div> | |
| <div>\begin{array}{cc} | |
| \mathrm{Bad} & \mathrm{Better} \\ | |
| \hline \\ | |
| \int\int_S f(x)\,dy\,dx & \iint_S f(x)\,dy\,dx \\ | |
| \int\int\int_V f(x)\,dz\,dy\,dx & \iiint_V f(x)\,dz\,dy\,dx | |
| \end{array}</div> | |
| <div>\(\TeX\)</div> | |
| <div>\begin{array}{cc} | |
| \mathrm{Bad} & \mathrm{Better} \\ | |
| \hline \\ | |
| \iiint_V f(x)dz dy dx & \iiint_V f(x)\,dz\,dy\,dx | |
| \end{array}</div> | |
| <div>\(\left\{x\middle | \frac{x^2}{2} \in \mathbb{z}\right\}\)</div> | |
| <div>\[ | |
| \require{cancel}\begin{array}{rl} | |
| \verb|y+\cancel{x}| & y+\cancel{x}\\ | |
| \verb|\cancel{y+x}| & \cancel{y+x}\\ | |
| \verb|y+\bcancel{x}| & y+\bcancel{x}\\ | |
| \verb|y+\xcancel{x}| & y+\xcancel{x}\\ | |
| \verb|y+\cancelto{0}{x}| & y+\cancelto{0}{x}\\ | |
| \verb+\frac{1\cancel9}{\cancel95} = \frac15+& \frac{1\cancel9}{\cancel95} = \frac15 \\ | |
| \end{array} | |
| \] | |
| </div> | |
| <div>\[ | |
| \require{enclose}\begin{array}{rl} | |
| \verb|\enclose{horizontalstrike}{x+y}| & \enclose{horizontalstrike}{x+y}\\ | |
| \verb|\enclose{verticalstrike}{\frac xy}| & \enclose{verticalstrike}{\frac xy}\\ | |
| \verb|\enclose{updiagonalstrike}{x+y}| & \enclose{updiagonalstrike}{x+y}\\ | |
| \verb|\enclose{downdiagonalstrike}{x+y}| & \enclose{downdiagonalstrike}{x+y}\\ | |
| \verb|\enclose{horizontalstrike,updiagonalstrike}{x+y}| & \enclose{horizontalstrike,updiagonalstrike}{x+y}\\ | |
| \end{array} | |
| \] | |
| </div> | |
| <div>\(19/95 = 1/5\)</div> | |
| <div>\(\cancelto{\cancelto{\cancelto{x^{2+x}}{\cancelto{x^2}{x}+4}}4}0\)</div> | |
| <div>\(\LaTeX\)</div> | |
| <div> | |
| \[ | |
| \left\{ | |
| \begin{array}{c} | |
| a_1x+b_1y+c_1z=d_1 \\ | |
| a_2x+b_2y+c_2z=d_2 \\ | |
| a_3x+b_3y+c_3z=d_3 | |
| \end{array} | |
| \right. | |
| \] | |
| </div> | |
| <div> | |
| \[ | |
| \begin{cases} | |
| a_1x+b_1y+c_1z=d_1 \\ | |
| a_2x+b_2y+c_2z=d_2 \\ | |
| a_3x+b_3y+c_3z=d_3 | |
| \end{cases} | |
| \] | |
| </div> | |
| <div>\[\left\{\begin{aligned} a_1x+b_1y+c_1z&=d_1+e_1 \\ | |
| a_2x+b_2y&=d_2 \\ | |
| a_3x+b_3y+c_3z&=d_3 \end{aligned} \right. \]</div> | |
| <div>\[ | |
| \left\{\begin{array}{ll}a_1x+b_1y+c_1z &=d_1+e_1 \\ a_2x+b_2y &=d_2 \\ a_3x+b_3y+c_3z &=d_3 \end{array} \right. | |
| \]</div> | |
| <div> | |
| \[\begin{cases} | |
| a_1x+b_1y+c_1z=\frac{p_1}{q_1} \\[2ex] | |
| a_2x+b_2y+c_2z=\frac{p_2}{q_2} \\[2ex] | |
| a_3x+b_3y+c_3z=\frac{p_3}{q_3} | |
| \end{cases} | |
| \] | |
| </div> | |
| <div> | |
| \[ | |
| \begin{cases} | |
| a_1x+b_1y+c_1z=\frac{p_1}{q_1} \\ | |
| a_2x+b_2y+c_2z=\frac{p_2}{q_2} \\ | |
| a_3x+b_3y+c_3z=\frac{p_3}{q_3} | |
| \end{cases} | |
| \] | |
| </div> | |
| <div>\[ | |
| \left\{ \begin{array}{l} 0 = c_x-a_{x0}-d_{x0}\dfrac{(c_x-a_{x0})\cdot d_{x0}}{\|d_{x0}\|^2} + c_x-a_{x1}-d_{x1}\dfrac{(c_x-a_{x1})\cdot d_{x1}}{\|d_{x1}\|^2} \\[2ex] 0 = c_y-a_{y0}-d_{y0}\dfrac{(c_y-a_{y0})\cdot d_{y0}}{\|d_{y0}\|^2} + c_y-a_{y1}-d_{y1}\dfrac{(c_y-a_{y1})\cdot d_{y1}}{\|d_{y1}\|^2} \end{array} \right. | |
| \] | |
| </div> | |
| <div>\(\longrightarrow\)</div> | |
| <div> | |
| \[ | |
| \begin{array}{|rc|} | |
| \hline | |
| \verb+\color{black}{text}+ & \color{black}{text} \\ | |
| \verb+\color{gray}{text}+ & \color{gray}{text} \\ | |
| \verb+\color{silver}{text}+ & \color{silver}{text} \\ | |
| \verb+\color{white}{text}+ & \color{white}{text} \\ | |
| \hline | |
| \verb+\color{maroon}{text}+ & \color{maroon}{text} \\ | |
| \verb+\color{red}{text}+ & \color{red}{text} \\ | |
| \verb+\color{yellow}{text}+ & \color{yellow}{text} \\ | |
| \verb+\color{lime}{text}+ & \color{lime}{text} \\ | |
| \verb+\color{olive}{text}+ & \color{olive}{text} \\ | |
| \verb+\color{green}{text}+ & \color{green}{text} \\ | |
| \verb+\color{teal}{text}+ & \color{teal}{text} \\ | |
| \verb+\color{aqua}{text}+ & \color{aqua}{text} \\ | |
| \verb+\color{blue}{text}+ & \color{blue}{text} \\ | |
| \verb+\color{navy}{text}+ & \color{navy}{text} \\ | |
| \verb+\color{purple}{text}+ & \color{purple}{text} \\ | |
| \verb+\color{fuchsia}{text}+ & \color{magenta}{text} \\ | |
| \hline | |
| \end{array} | |
| \]</div> | |
| <div>\(r, g, b\)</div> | |
| <div>\(0–15\)</div> | |
| <div> | |
| \[ | |
| \begin{array}{|rrrrrrrr|}\hline | |
| \verb+#000+ & \color{#000}{text} & & & | |
| \verb+#00F+ & \color{#00F}{text} & & \\ | |
| & & \verb+#0F0+ & \color{#0F0}{text} & | |
| & & \verb+#0FF+ & \color{#0FF}{text}\\ | |
| \verb+#F00+ & \color{#F00}{text} & & & | |
| \verb+#F0F+ & \color{#F0F}{text} & & \\ | |
| & & \verb+#FF0+ & \color{#FF0}{text} & | |
| & & \verb+#FFF+ & \color{#FFF}{text}\\ | |
| \hline | |
| \end{array} | |
| \] | |
| </div> | |
| <div>\begin{array}{|rrrrrrrr|} | |
| \hline | |
| \verb+#000+ & \color{#000}{text} & \verb+#005+ & \color{#005}{text} & \verb+#00A+ & \color{#00A}{text} & \verb+#00F+ & \color{#00F}{text} \\ | |
| \verb+#500+ & \color{#500}{text} & \verb+#505+ & \color{#505}{text} & \verb+#50A+ & \color{#50A}{text} & \verb+#50F+ & \color{#50F}{text} \\ | |
| \verb+#A00+ & \color{#A00}{text} & \verb+#A05+ & \color{#A05}{text} & \verb+#A0A+ & \color{#A0A}{text} & \verb+#A0F+ & \color{#A0F}{text} \\ | |
| \verb+#F00+ & \color{#F00}{text} & \verb+#F05+ & \color{#F05}{text} & \verb+#F0A+ & \color{#F0A}{text} & \verb+#F0F+ & \color{#F0F}{text} \\ | |
| \hline | |
| \verb+#080+ & \color{#080}{text} & \verb+#085+ & \color{#085}{text} & \verb+#08A+ & \color{#08A}{text} & \verb+#08F+ & \color{#08F}{text} \\ | |
| \verb+#580+ & \color{#580}{text} & \verb+#585+ & \color{#585}{text} & \verb+#58A+ & \color{#58A}{text} & \verb+#58F+ & \color{#58F}{text} \\ | |
| \verb+#A80+ & \color{#A80}{text} & \verb+#A85+ & \color{#A85}{text} & \verb+#A8A+ & \color{#A8A}{text} & \verb+#A8F+ & \color{#A8F}{text} \\ | |
| \verb+#F80+ & \color{#F80}{text} & \verb+#F85+ & \color{#F85}{text} & \verb+#F8A+ & \color{#F8A}{text} & \verb+#F8F+ & \color{#F8F}{text} \\ | |
| \hline | |
| \verb+#0F0+ & \color{#0F0}{text} & \verb+#0F5+ & \color{#0F5}{text} & \verb+#0FA+ & \color{#0FA}{text} & \verb+#0FF+ & \color{#0FF}{text} \\ | |
| \verb+#5F0+ & \color{#5F0}{text} & \verb+#5F5+ & \color{#5F5}{text} & \verb+#5FA+ & \color{#5FA}{text} & \verb+#5FF+ & \color{#5FF}{text} \\ | |
| \verb+#AF0+ & \color{#AF0}{text} & \verb+#AF5+ & \color{#AF5}{text} & \verb+#AFA+ & \color{#AFA}{text} & \verb+#AFF+ & \color{#AFF}{text} \\ | |
| \verb+#FF0+ & \color{#FF0}{text} & \verb+#FF5+ & \color{#FF5}{text} & \verb+#FFA+ & \color{#FFA}{text} & \verb+#FFF+ & \color{#FFF}{text} \\ | |
| \hline | |
| \end{array}</div> | |
| <div>\(\def\demo#1#2{#1{#2}\ #1{#2#2}\ #1{#2#2#2}}\)</div> | |
| <div>\(\demo\overline A\)</div> | |
| <div>\(\demo\underline B\)</div> | |
| <div>\(\demo\widetilde C\)</div> | |
| <div>\(\demo\widehat D\)</div> | |
| <div>\(\demo\fbox {$E$}\)</div> | |
| <div>\(\demo\underleftarrow{F}\qquad\)</div> | |
| <div>\(\xleftarrow{abc}\)</div> | |
| <div>\(\demo\underrightarrow{G}\qquad\)</div> | |
| <div>\(\xrightarrow{abc}\)</div> | |
| <div>\(\demo\underleftrightarrow{H}\)</div> | |
| <div>\(\overbrace{(n - 2) + \overbrace{(n - 1) + n + (n + 1)} + (n + 2)}\)</div> | |
| <div>\((n \underbrace{- 2) + (n \underbrace{- 1) + n + (n +} 1) + (n +} 2)\)</div> | |
| <div>\[\underbrace{a\cdot a\cdots a}_{b\text{ times}}\]</div> | |
| <div>\(\varliminf\)</div> | |
| <div>\(\varlimsup\)</div> | |
| <div>\(\check{I}\)</div> | |
| <div>\(\acute{J}\)</div> | |
| <div>\(\grave{K}\)</div> | |
| <div>\(\vec u\ \vec{AB}\)</div> | |
| <div>\(\bar z\)</div> | |
| <div>\(\hat x\)</div> | |
| <div>\(\tilde x\)</div> | |
| <div>\(\dot x,\ddot x,\dddot x\)</div> | |
| <div>\(\mathring A\)</div> | |
| <div>\(\overset{@}{ABC}\ \overset{x^2}{\longmapsto}\ \overset{\bullet\circ\circ\bullet}{T}\)</div> | |
| <div> \[ | |
| x = a_0 + \cfrac{1^2}{a_1 | |
| + \cfrac{2^2}{a_2 | |
| + \cfrac{3^2}{a_3 + \cfrac{4^4}{a_4 + \cdots}}}} | |
| \] | |
| </div> | |
| <div> \[ | |
| x = a_0 + \frac{1^2}{a_1 | |
| + \frac{2^2}{a_2 | |
| + \frac{3^2}{a_3 + \frac{4^4}{a_4 + \cdots}}}} | |
| \] | |
| </div> | |
| <div> \[ | |
| x = a_0 + \frac{1^2}{a_1+} | |
| \frac{2^2}{a_2+} | |
| \frac{3^2}{a_3 +} \frac{4^4}{a_4 +} \cdots | |
| \] | |
| </div> | |
| <div>\([a_0; a_1, a_2, a_3, \ldots]\)</div> | |
| <div>\[ \cfrac{a_{1}}{b_{1}+\cfrac{a_{2}}{b_{2}+\cfrac{a_{3}}{b_{3}+\ddots }}}= {\genfrac{}{}{}{}{a_1}{b_1}} {\genfrac{}{}{0pt}{}{}{+}} {\genfrac{}{}{}{}{a_2}{b_2}} {\genfrac{}{}{0pt}{}{}{+}} {\genfrac{}{}{}{}{a_3}{b_3}} {\genfrac{}{}{0pt}{}{}{+\dots}} \]</div> | |
| <div>\(\LaTeX\)</div> | |
| <div>\[\underset{j=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{a_j}{b_j}=\cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+\ddots}}}.\]</div> | |
| <div>\[\mathop{\LARGE\mathrm K}_{i=1}^\infty \frac{a_i}{b_i}\]</div> | |
| <div> | |
| \[ | |
| \require{AMScd} | |
| \begin{CD} | |
| A @>a>> B\\ | |
| @V b V V= @VV c V\\ | |
| C @>>d> D | |
| \end{CD} | |
| \] | |
| </div> | |
| <div> | |
| \[ | |
| \begin{CD} | |
| A @>>> B @>{\text{very long label}}>> C \\ | |
| @. @AAA @| \\ | |
| D @= E @<<< F | |
| \end{CD} | |
| \] | |
| </div> | |
| <div> | |
| \[ | |
| \require{AMScd} | |
| \begin{CD} | |
| RCOHR'SO_3Na @>{\text{Hydrolysis,$\Delta, Dil.HCl$}}>> (RCOR')+NaCl+SO_2+ H_2O | |
| \end{CD} | |
| \] | |
| </div> | |
| <div>\( \newcommand{\SES}[3]{ 0 \to #1 \to #2 \to #3 \to 0 }\)</div> | |
| <div>\[ \SES{A}{B}{C} \]</div> | |
| <div>\(\implies\)</div> | |
| <div>\(\Rightarrow\)</div> | |
| <div>\(\iff\)</div> | |
| <div>\(\impliedby\)</div> | |
| <div>\(\to\)</div> | |
| <div>\(f\colon A \to B\)</div> | |
| <div>\(\gets\)</div> | |
| <div>\(T:\mathbb R\to \mathbb R,\; x\mapsto x+1\)</div> | |
| <div>\(p\land((q\lor r)\to s)\)</div> | |
| <div>\[x+2=4-x\implies x=1.\]</div> | |
| <div>\[ a := x^2-y^3 \tag{*}\label{*} \]</div> | |
| <div>\[ a+y^3 \stackrel{\eqref{*}}= x^2 \]</div> | |
| <div>\(\eqref{*}\)</div> | |
| <div>\(\ref{*}\)</div> | |
| <div>\(\eqref{*}\)</div> | |
| <div>\[(1)\qquad\qquad\sum\limits_{j}k\tag*{}\]</div> | |
| <div>\((1)\)</div> | |
| <div> | |
| \[ | |
| f\left(\left[ \frac{1+\left\{x,y\right\}}{\left(\frac{x}{y}+\frac{y}{x}\right)\left(u+1\right)}+a\right]^{3/2}\right). | |
| \] | |
| </div> | |
| <div> | |
| \[ | |
| \begin{aligned} | |
| a=&\left(1+2+3+ \cdots \right. \\ | |
| & \cdots+ \left. \infty-2+\infty-1+\infty\right). | |
| \end{aligned} | |
| \] | |
| </div> | |
| <div> | |
| \[ | |
| \left\langle q\middle\|\frac{\frac{x}{y}}{\frac{u}{v}} \middle| p \right\rangle. | |
| \] | |
| </div> | |
| <div>\(\Big(\dots\Big)\)</div> | |
| <div>\[\operatorname{arsinh}(x)\]</div> | |
| <div>\(arsinh(x)\)</div> | |
| <div> | |
| \[ | |
| \operatorname*{Res}_{z=1}\left(\frac1{z^2-z}\right)=1 | |
| \] | |
| </div> | |
| <div>\(\verb*{\rm ...}*\)</div> | |
| <div>\(\verb*{\rm arsinh}*\)</div> | |
| <div>\({\rm arsinh}\)</div> | |
| <div>\(\operatorname{arsinh}x\)</div> | |
| <div>\({\rm arsinh}x\)</div> | |
| <div>\({\tt operatorname}\)</div> | |
| <div>\(\text{arsinh }x\)</div> | |
| <div>\(\operatorname{arsinh}x\)</div> | |
| <div>\(\lim \limits_{x \to 1} \frac{x^2-1}{x-1}\)</div> | |
| <div>\(\lim\)</div> | |
| <div>\(lim\)</div> | |
| <div>\(\lim \limits_{x \to 1}\)</div> | |
| <div>\(\to\)</div> | |
| <div>\(x \to 1\)</div> | |
| <div>\(\lim\)</div> | |
| <div>\(x \to 1\)</div> | |
| <div>\(\lim \limits_{x \to 1} \frac{x^2-1}{x-1}\)</div> | |
| <div>\[\lim_{x\to 1}?\]</div> | |
| <div>\[\mathop{+}\limits_{i=1}^k\text{ instead of }+_{i=1}^k\]</div> | |
| <div>\(\lim\limits_{x\to 1}\)</div> | |
| <div>\(\lim_{x\to 1}\)</div> | |
| <div>\(\lim_{x\to 1}\)</div> | |
| <div>\(\lim\limits_{x\to 1}\)</div> | |
| <div>\(x\to 1\)</div> | |
| <div>\(\lim \limits_{x \to 1}\)</div> | |
| <div>\(\lim\limits_{x\mapsto 1}\dfrac1x\)</div> | |
| <div>\(\TeX\)</div> | |
| <div>\(\lim_{x\mapsto 1}\frac1x\)</div> | |
| <div>\(\TeX\)</div> | |
| <div>\(\lvert x\rvert\)</div> | |
| <div>\(\lVert v\rVert\)</div> | |
| <div>\[|x|, ||v|| \quad\longrightarrow\quad \lvert x\rvert, \lVert v\rVert\]</div> | |
| <div>\(\|x\|\)</div> | |
| <div>\(\lVert x \rVert\)</div> | |
| <div> | |
| \[ | |
| \bbox[yellow] | |
| { | |
| e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n | |
| \qquad (1) | |
| } | |
| \] | |
| </div> | |
| <div> | |
| \[ | |
| \bbox[yellow,5px] | |
| { | |
| e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n | |
| \qquad (1) | |
| } | |
| \] | |
| </div> | |
| <div> | |
| \[ | |
| \bbox[5px,border:2px solid red] | |
| { | |
| e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n | |
| \qquad (2) | |
| } | |
| \] | |
| </div> | |
| <div> | |
| \[ | |
| \bbox[yellow,5px,border:2px solid red] | |
| { | |
| e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n | |
| \qquad (1) | |
| } | |
| \] | |
| </div> | |
| <div>\begin{align} | |
| v + w & = 0 &&\text{Given} \tag 1\\ | |
| -w & = -w + 0 && \text{additive identity} \tag 2\\ | |
| -w + 0 & = -w + (v + w) && \text{equations $(1)$ and $(2)$} | |
| \end{align}</div> | |
| <div>\[\spadesuit\quad\heartsuit\quad\diamondsuit\quad\clubsuit\]</div> | |
| <div>\[\color{red}{\heartsuit}\quad\color{red}{\diamondsuit}\]</div> | |
| <div>\[♠\quad♡\quad♢\quad♣\\ | |
| ♤\quad♥\quad♦\quad♧\] | |
| </div> | |
| <div>\(\Rightarrow\)</div> | |
| <div>\(\Leftarrow\)</div> | |
| <div>\(\Leftrightarrow\)</div> | |
| <div>\(\implies\)</div> | |
| <div>\(\impliedby\)</div> | |
| <div>\(\iff\)</div> | |
| <div> | |
| \[ | |
| \require{enclose} | |
| \begin{array}{r} | |
| 13 \\[-3pt] | |
| 4 \enclose{longdiv}{52} \\[-3pt] | |
| \underline{4}\phantom{2} \\[-3pt] | |
| 12 \\[-3pt] | |
| \underline{12} | |
| \end{array} | |
| \] | |
| </div> | |
| <div>\[x^3−6x^2+11x−6=(x−{\color{red}1})(x^2−5x+6)+{\color{blue}0}\]</div> | |
| <div>\[\begin{array}{c|rrrr}& x^3 & x^2 & x^1 & x^0\\ & 1 & -6 & 11 & -6\\ {\color{red}1} & \downarrow & 1 & -5 & 6\\ \hline & 1 & -5 & 6 & |\phantom{-} {\color{blue}0} \end{array}\]</div> | |
| <div>\(90°\)</div> | |
| <div>\(90^\circ\)</div> | |
| <div>\[\sum_{n=1}^\infty \frac{1}{n^2} \to | |
| \textstyle \sum_{n=1}^\infty \frac{1}{n^2} \to | |
| \displaystyle \sum_{n=1}^\infty \frac{1}{n^2}\]</div> | |
| <div>\(\displaystyle \lim_{t \to 0} \int_t^1 f(t)\, dt\)</div> | |
| <div>\(\lim_{t \to 0} \int_t^1 f(t)\, dt\)</div> | |
| <div>\(\overline a+\overline b=\overline {a\cdot b}\)</div> | |
| <div>\(\sqrt{a}-\sqrt{b}\)</div> | |
| <div>\(\sqrt{\mathstrut a} - \sqrt{\mathstrut b}\)</div> | |
| <div>\(\sqrt{\vphantom{b} a} - \sqrt{b}\)</div> | |
| <div>\(\dfrac{1}{\sqrt{\vphantom{b} a} - \sqrt{b}}\)</div> | |
| <div>\(\smash{\dfrac{1}{\sqrt{\vphantom{b} a} - \sqrt{b}}}\)</div> | |
| <div>\(n\)</div> | |
| <div>\(T_1,\ldots,T_n\)</div> | |
| <div>\(T_1\gets 1\)</div> | |
| <div>\(k\)</div> | |
| <div>\(2\)</div> | |
| <div>\(n\)</div> | |
| <div>\(T_k\gets (k−1)T_{k−1}\)</div> | |
| <div>\(k\)</div> | |
| <div>\(2\)</div> | |
| <div>\(n\)</div> | |
| <div>\(j\)</div> | |
| <div>\(k\)</div> | |
| <div>\(n\)</div> | |
| <div>\(T_j\gets (j −k)T_{j−1} + (j −k+2)T_j\)</div> | |
| <div>\(\;T_1,T_2,\ldots,T_n\)</div> | |
| <div>\(n\)</div> | |
| <div>\(T_1,\ldots,T_n\)</div> | |
| <div>\(T_1\gets 1\)</div> | |
| <div>\(\;k\;\)</div> | |
| <div>\(2\;\)</div> | |
| <div>\(\;n\)</div> | |
| <div>\(\phantom{{}++{}}\)</div> | |
| <div>\(T_k\gets (k−1)T_{k−1}\)</div> | |
| <div>\(\;k\;\)</div> | |
| <div>\(2\;\)</div> | |
| <div>\(\;n\)</div> | |
| <div>\(\phantom{{}++{}}\)</div> | |
| <div>\(\;j\;\)</div> | |
| <div>\(\;k\;\)</div> | |
| <div>\(\;n\)</div> | |
| <div>\(\phantom{{}++{}}\)</div> | |
| <div>\(\phantom{{}++{}}\)</div> | |
| <div>\(T_j\gets (j −k)T_{j−1} + (j −k+2)T_j\)</div> | |
| <div>\(\;T_1,T_2,\ldots,T_n\)</div> | |
| <div>\(\LaTeX\)</div> | |
| <div>\[e=mc^2 \tag{1}\label{eq1}\]</div> | |
| <div>\(\eqref{eq1}\)</div> | |
| <div>\(\eqref{eq1}\)</div> | |
| <div> | |
| \[ | |
| \begin{equation}\begin{aligned} | |
| a &= b + c \\ | |
| &= d + e + f + g \\ | |
| &= h + i | |
| \end{aligned}\end{equation}\tag{2}\label{eq2} | |
| \] | |
| </div> | |
| <div>\(\eqref{eq2}\)</div> | |
| <div>\(\eqref{eq2}\)</div> | |
| <div>\begin{align} | |
| a &= b + c \tag{3}\label{eq3} \\ | |
| x &= yz \tag{4}\label{eq4}\\ | |
| l &= m - n \tag{5}\label{eq5} | |
| \end{align}</div> | |
| <div>\(\eqref{eq3}\)</div> | |
| <div>\(\eqref{eq4}\)</div> | |
| <div>\(\eqref{eq5}\)</div> | |
| <div>\begin{array}{ll} | |
| \text{maximize} & c^T x \\ | |
| \text{subject to}& d^T x = \alpha \\ | |
| &0 \le x \le 1. | |
| \end{array}</div> | |
| <div>\begin{alignat}{5} | |
| \max \quad & z = & x_1 & + & 12 x_2 & & & && \\ | |
| \mbox{s.t.} \quad & & 13 x_1 & + & x_2 & + & 12x_3 & \geq 5 && \tag{constraint 1} \\ | |
| & & x_1 & & & + & x_3 & \leq 16 && \tag{constraint 2} \\ | |
| & & 15 x_1 & + & 201 x_2 & & & = 14 && \tag{constraint 3} \\ | |
| & & \rlap{x_i \ge 0, i = 1, 2, 3} | |
| \end{alignat}</div> | |
| <div>\(\max\)</div> | |
| <div>\(z\)</div> | |
| <div>\(\pm\)</div> | |
| <div>\(\rm\LaTeX\)</div> | |
| <div>\(\rm\LaTeX\)</div> | |
| <div> | |
| \[ | |
| \begin{bmatrix} | |
| 1 & 2 & 2 \\ | |
| 2 & 3 & 4 \\ | |
| 4 & 4 & 2 | |
| \end{bmatrix}. | |
| \] | |
| </div> | |
| <div>\(z\)</div> | |
| <div>\(z\)</div> | |
| <div>\begin{array}{rrrrrr|r} | |
| & x_1 & x_2 & s_1 & s_2 & s_3 & \\ \hline | |
| s_1 & 0 & 1 & 1 & 0 & 0 & 8 \\ | |
| s_2 & 1 & -1 & 0 & 1 & 0 & 4 \\ | |
| s_3 & 1 & 1 & 0 & 0 & 1 & 12 \\ \hline | |
| & -1 & -1 & 0 & 0 & 0 & 0 | |
| \end{array}</div> | |
| <div>\begin{array}{rrrrrrr|rr} | |
| & x_1 & x_2 & s_1 & s_2 & s_3 & w & & \text{ratio} \\ \hline | |
| s_1 & 0 & 1 & 1 & 0 & 0 & 0 & 8 & - \\ | |
| w & 1^* & -1 & 0 & -1 & 0 & 1 & 4 & 4 \\ | |
| s_3 & 1 & 1 & 0 & 0 & 1 & 0 & 12 & 12 \\ \hdashline | |
| & 1 & -1 & 0 & -1 & 0 & 0 & 4 & \\ \hline | |
| s_1 & 0 & 1 & 1 & 0 & 0 & 0 & 8 & \\ | |
| x_1 & 1 & -1 & 0 & -1 & 0 & 1 & 4 & \\ | |
| s_3 & 0 & 2 & 0 & 2 & 1 & -1 & 8 & \\ \hdashline | |
| & 0 & 0 & 0 & 0 & 0 & -1 & 0 & | |
| \end{array}</div> | |
| <div>\begin{array}{rrrrrrrr|r} | |
| & x_1 & x_2 & x_3 & x_4 & x_5 & x_6 & x_7 & \\ \hline | |
| x_4 & 0 & -3 & 7 & 1 & 0 & 0 & 2 & 2M -4 \\ | |
| x_5 & 0 & -9 & 0 & 0 & 1 & 0 & -1 & -M -3 \\ | |
| x_6 & 0 & 6 & -1 & 0 & 0 & 1 & -4^* & -4M +8 \\ | |
| x_1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & M \\ \hline | |
| & 0 & 1 & 1 & 0 & 0 & 0 & 2 & 2M \\ | |
| \text{ratio} & & & 1 & & & & 1/2 & | |
| \end{array}</div> | |
| <div>\begin{array}{rrrrrrr|r} | |
| & x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & \\ \hline | |
| s_1 & -2 & 0 & -2 & 1 & 0 & 0 & -60 \\ | |
| s_2 & -2 & -4^* & -5 & 0 & 1 & 0 & -70 \\ | |
| s_3 & 0 & -3 & -1 & 0 & 0 & 1 & -27 \\ \hdashline | |
| & 8 & 10 & 25 & 0 & 0 & 0 & 0 \\ | |
| \text{ratio} & -4 & -5/2 & -5 & & & & \\ \hline | |
| s_1 & -2^* & 0 & -2 & 1 & 0 & 0 & -60 \\ | |
| x_2 & 1/2 & 1 & 5/4 & 0 & -1/4 & 0 & 35/2 \\ | |
| s_3 & 3/2 & 0 & 11/4 & 0 & -3/4 & 1 & 51/2 \\ \hdashline | |
| & 3 & 0 & 25/2 & 0 & 5/2 & 0 & -175 \\ | |
| \text{ratio} & -3/2 & & 25/4 & & & & \\ \hline | |
| x_1 & 1 & 0 & 1 & -1/2 & 0 & 0 & 30 \\ | |
| x_2 & 0 & 1 & 3/4 & 1/4 & -1/4 & 0 & 5/2 \\ | |
| s_3 & 0 & 0 & 5/4 & 3/4 & -3/4^* & 1 & -39/2 \\ \hdashline | |
| & 0 & 0 & 19/2 & 3/2 & 5/2 & 0 & -265 \\ | |
| \text{ratio} & & & & & \dots & & \\ \hline | |
| x_1 & 1 & 0 & 1 & -1/2 & 0 & 0 & 30 \\ | |
| x_2 & 0 & 1 & 1/3 & 0 & 0 & -1/3 & 9 \\ | |
| s_2 & 0 & 0 & -5/3 & -1 & 1 & -4/3 & 26 \\ \hdashline | |
| & 0 & 0 & 41/3 & 4 & 0 & 10/3 & -330 | |
| \end{array}</div> | |
| <div>\[ | |
| \require{extpfeil} % produce extensible horizontal arrows | |
| \begin{array}{ccc} % arrange LPPs | |
| % first row | |
| % first LPP | |
| \begin{array}{ll} | |
| \max & z = c^T x \\ | |
| \text{s.t.} & A x \le b \\ | |
| & x \ge 0 | |
| \end{array} | |
| & \xtofrom{\text{duality}} & | |
| % second LPP | |
| \begin{array}{ll} | |
| \min & v = b^T y \\ | |
| \text{s.t.} & A^T y \ge c \\ | |
| & y \ge 0 | |
| \end{array} \\ | |
| ({\cal PC}) & & ({\cal DC}) \\ | |
| \text{add } {\Large \downharpoonleft} \text{slack var} & & \text{minus } {\Large \downharpoonright} \text{surplus var}\\ % Change to your favorite arrow style | |
| % | |
| % second row | |
| % third LPP | |
| \begin{array}{ll} | |
| \max & z = c^T x \\ | |
| \text{s.t.} & A x + s = b \\ | |
| & x,s \ge 0 | |
| \end{array} | |
| & \xtofrom[\text{some steps skipped}]{\text{duality}} & | |
| % fourth LPP | |
| \begin{array}{ll} | |
| \min & v = b^T y \\ | |
| \text{s.t.} & A^T y - t = c \\ | |
| & y,t \ge 0 | |
| \end{array} \\ | |
| ({\cal PS}) & & ({\cal DS}) | |
| % | |
| \end{array} | |
| \] | |
| </div> | |
| <div>\(\sum_{i=n}^\mathbb{N}\)</div> | |
| <div>\(\sum\)</div> | |
| </body> | |
| </html> |
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| <!DOCTYPE html><html><head> | |
| <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> | |
| <title>Math.SE extracted Samples</title> | |
| <style type="text/css">.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0} | |
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| .mjx-denominator {display: block; text-align: center} | |
| .MJXc-stacked {height: 0; position: relative} | |
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| .mjx-over {display: block} | |
| .mjx-over > * {padding-left: 0px!important; padding-right: 0px!important} | |
| .mjx-under > * {padding-left: 0px!important; padding-right: 0px!important} | |
| .mjx-stack > .mjx-sup {display: block} | |
| .mjx-stack > .mjx-sub {display: block} | |
| .mjx-prestack > .mjx-presup {display: block} | |
| .mjx-prestack > .mjx-presub {display: block} | |
| .mjx-delim-h > .mjx-char {display: inline-block} | |
| .mjx-surd {vertical-align: top} | |
| .mjx-mphantom * {visibility: hidden} | |
| .mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%} | |
| .mjx-annotation-xml {line-height: normal} | |
| .mjx-menclose > svg {fill: none; stroke: currentColor} | |
| .mjx-mtr {display: table-row} | |
| .mjx-mlabeledtr {display: table-row} | |
| .mjx-mtd {display: table-cell; text-align: center} | |
| .mjx-label {display: table-row} | |
| .mjx-box {display: inline-block} | |
| .mjx-block {display: block} | |
| .mjx-span {display: inline} | |
| .mjx-char {display: block; white-space: pre} | |
| .mjx-itable {display: inline-table; width: auto} | |
| .mjx-row {display: table-row} | |
| .mjx-cell {display: table-cell} | |
| .mjx-table {display: table; width: 100%} | |
| .mjx-line {display: block; height: 0} | |
| .mjx-strut {width: 0; padding-top: 1em} | |
| .mjx-vsize {width: 0} | |
| .MJXc-space1 {margin-left: .167em} | |
| .MJXc-space2 {margin-left: .222em} | |
| .MJXc-space3 {margin-left: .278em} | |
| .mjx-ex-box-test {position: absolute; overflow: hidden; width: 1px; height: 60ex} | |
| .mjx-line-box-test {display: table!important} | |
| .mjx-line-box-test span {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0} | |
| .MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal} | |
| .MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal} | |
| .MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold} | |
| .MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold} | |
| .MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw} | |
| .MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw} | |
| .MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw} | |
| .MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw} | |
| .MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw} | |
| .MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw} | |
| .MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw} | |
| .MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw} | |
| .MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw} | |
| .MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw} | |
| .MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw} | |
| .MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw} | |
| .MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw} | |
| .MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw} | |
| .MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw} | |
| .MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw} | |
| .MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw} | |
| .MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw} | |
| .MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw} | |
| .MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw} | |
| .MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw} | |
| /* @font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax_AMS'), local('MathJax_AMS-Regular')} */ | |
| @font-face {font-family: MJXc-TeX-ams-Rw; src 1: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_AMS-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_AMS-Regular.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax_Caligraphic Bold'), local('MathJax_Caligraphic-Bold')} */ | |
| /* @font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax_Caligraphic'); font-weight: bold} */ | |
| @font-face {font-family: MJXc-TeX-cal-Bw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Caligraphic-Bold.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Caligraphic-Bold.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax_Fraktur'), local('MathJax_Fraktur-Regular')} */ | |
| @font-face {font-family: MJXc-TeX-frak-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Fraktur-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Fraktur-Regular.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax_Fraktur Bold'), local('MathJax_Fraktur-Bold')} */ | |
| /* @font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax_Fraktur'); font-weight: bold} */ | |
| @font-face {font-family: MJXc-TeX-frak-Bw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Fraktur-Bold.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Fraktur-Bold.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax_Math BoldItalic'), local('MathJax_Math-BoldItalic')} */ | |
| /* @font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax_Math'); font-weight: bold; font-style: italic} */ | |
| @font-face {font-family: MJXc-TeX-math-BIw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Math-BoldItalic.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Math-BoldItalic.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax_SansSerif'), local('MathJax_SansSerif-Regular')} */ | |
| @font-face {font-family: MJXc-TeX-sans-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_SansSerif-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_SansSerif-Regular.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax_SansSerif Bold'), local('MathJax_SansSerif-Bold')} */ | |
| /* @font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax_SansSerif'); font-weight: bold} */ | |
| @font-face {font-family: MJXc-TeX-sans-Bw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_SansSerif-Bold.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_SansSerif-Bold.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax_SansSerif Italic'), local('MathJax_SansSerif-Italic')} */ | |
| /* @font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax_SansSerif'); font-style: italic} */ | |
| @font-face {font-family: MJXc-TeX-sans-Iw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_SansSerif-Italic.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_SansSerif-Italic.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-script-R; src: local('MathJax_Script'), local('MathJax_Script-Regular')} */ | |
| @font-face {font-family: MJXc-TeX-script-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Script-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Script-Regular.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-type-R; src: local('MathJax_Typewriter'), local('MathJax_Typewriter-Regular')} */ | |
| @font-face {font-family: MJXc-TeX-type-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Typewriter-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Typewriter-Regular.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax_Caligraphic'), local('MathJax_Caligraphic-Regular')} */ | |
| @font-face {font-family: MJXc-TeX-cal-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Caligraphic-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Caligraphic-Regular.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-main-B; src: local('MathJax_Main Bold'), local('MathJax_Main-Bold')} */ | |
| /* @font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax_Main'); font-weight: bold} */ | |
| @font-face {font-family: MJXc-TeX-main-Bw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Main-Bold.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Main-Bold.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-main-I; src: local('MathJax_Main Italic'), local('MathJax_Main-Italic')} */ | |
| /* @font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax_Main'); font-style: italic} */ | |
| @font-face {font-family: MJXc-TeX-main-Iw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Main-Italic.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Main-Italic.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-main-R; src: local('MathJax_Main'), local('MathJax_Main-Regular')} */ | |
| @font-face {font-family: MJXc-TeX-main-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Main-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Main-Regular.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-math-I; src: local('MathJax_Math Italic'), local('MathJax_Math-Italic')} */ | |
| /* @font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax_Math'); font-style: italic} */ | |
| @font-face {font-family: MJXc-TeX-math-Iw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Math-Italic.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Math-Italic.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax_Size1'), local('MathJax_Size1-Regular')} */ | |
| @font-face {font-family: MJXc-TeX-size1-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Size1-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Size1-Regular.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax_Size2'), local('MathJax_Size2-Regular')} */ | |
| @font-face {font-family: MJXc-TeX-size2-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Size2-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Size2-Regular.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax_Size3'), local('MathJax_Size3-Regular')} */ | |
| @font-face {font-family: MJXc-TeX-size3-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Size3-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Size3-Regular.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax_Size4'), local('MathJax_Size4-Regular')} */ | |
| @font-face {font-family: MJXc-TeX-size4-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Size4-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Size4-Regular.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax_Vector'), local('MathJax_Vector-Regular')} */ | |
| @font-face {font-family: MJXc-TeX-vec-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Vector-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Vector-Regular.otf') format('opentype')} | |
| /* @font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax_Vector Bold'), local('MathJax_Vector-Bold')} */ | |
| /* @font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax_Vector'); font-weight: bold} */ | |
| @font-face {font-family: MJXc-TeX-vec-Bw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Vector-Bold.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Vector-Bold.otf') format('opentype')} | |
| </style></head> | |
| <body> | |
| <p>Licensed under <a href="https://creativecommons.org/licenses/by-sa/3.0/" rel="license">cc by-sa 3.0</a>. Original source: <a href="http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference">meta.math.stackexchange</a> (cc by-sa 3.0), with <a href="http://blog.stackoverflow.com/2009/06/attribution-required/" rel="license">attribution required</a>.</p> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-munderover"><span class="mjx-base"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">∑</span></span></span><span class="mjx-stack" style="vertical-align: -0.31em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.422em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span></span><span class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.646em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 6.57em; top: -1.87em;"><span class="mjx-mrow" style=""><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span><span class="mjx-sup" style="font-size: 83.3%; vertical-align: 0.435em; padding-left: 0px; padding-right: 0.06em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 6.57em; bottom: -0.687em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">6</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 4.646em;" class="mjx-line"></span></span><span style="height: 1.808em; vertical-align: -0.486em;" class="mjx-vsize"></span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-munderover"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; padding-bottom: 0.247em; padding-top: 0.141em; padding-left: 0.721em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span><span class="mjx-op"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.74em; padding-bottom: 0.74em;">∑</span></span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em; padding-left: 0.21em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span></span></span><span class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 7.39em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 7.39em; top: -1.667em;"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span class="mjx-denominator" style="width: 7.39em; bottom: -0.778em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">6</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 7.39em;" class="mjx-line"></span></span><span style="height: 2.445em; vertical-align: -0.778em;" class="mjx-vsize"></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\alpha, \beta, … \omega"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">α</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.446em; padding-right: 0.007em;">β</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.372em;">…</span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">ω</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\Gamma, \Delta, …, \Omega"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">Γ</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">Δ</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.372em;">…</span></span><span class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">Ω</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="x_i^2"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-stack" style="vertical-align: -0.311em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\log_2 x"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.519em;">log</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.377em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo"><span class="mjx-char"></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="10^10"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">10</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="10^{10}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">10</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">10</span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="{x^y}^z"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span></span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.594em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="x^{y^z}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.006em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sup" style="font-size: 83.3%; vertical-align: 0.435em; padding-left: 0.069em; padding-right: 0.06em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="x_i^2"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-stack" style="vertical-align: -0.311em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="x_{i^2}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.365em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span></span><span class="mjx-sup" style="font-size: 83.3%; vertical-align: 0.347em; padding-left: 0px; padding-right: 0.06em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="(2+3)[4+4]"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">[</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">4</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">4</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">]</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\{\}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">{</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">}</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="(\frac{\sqrt x}{y^3})"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.135em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.605em; top: -1.766em;"><span class="mjx-msqrt" style=""><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">√</span></span><span class="mjx-box" style="padding-top: 0.236em; border-top: 1px solid;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.605em; bottom: -1.043em;"><span class="mjx-msubsup" style=""><span class="mjx-base" style="margin-right: -0.006em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sup" style="font-size: 83.3%; vertical-align: 0.347em; padding-left: 0.069em; padding-right: 0.06em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.135em;" class="mjx-line"></span></span><span style="height: 1.987em; vertical-align: -0.738em;" class="mjx-vsize"></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\left(\frac{\sqrt x}{y^3}\right)"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">(</span></span><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.135em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.605em; top: -1.766em;"><span class="mjx-msqrt" style=""><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">√</span></span><span class="mjx-box" style="padding-top: 0.236em; border-top: 1px solid;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.605em; bottom: -1.043em;"><span class="mjx-msubsup" style=""><span class="mjx-base" style="margin-right: -0.006em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sup" style="font-size: 83.3%; vertical-align: 0.347em; padding-left: 0.069em; padding-right: 0.06em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.135em;" class="mjx-line"></span></span><span style="height: 1.987em; vertical-align: -0.738em;" class="mjx-vsize"></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">)</span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="(x)"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="[x]"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">[</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">]</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\{ x \}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">{</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">}</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="|x|"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\vert x \vert"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\Vert x \Vert"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">∥</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">∥</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\langle x \rangle"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">⟨</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">⟩</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\lceil x \rceil"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">⌈</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">⌉</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\lfloor x \rfloor"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">⌊</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">⌋</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\left.\frac12\right\rbrace"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo" style="width: 0.12em;"></span><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.372em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.665em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 0.495em;" class="mjx-line"></span></span><span style="height: 1.441em; vertical-align: -0.47em;" class="mjx-vsize"></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">}</span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\Biggl(\biggl(\Bigl(\bigl((x)\bigr)\Bigr)\biggr)\Biggr)"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.551em; padding-bottom: 1.551em;">(</span></span></span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">(</span></span></span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">(</span></span></span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">(</span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">)</span></span></span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">)</span></span></span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">)</span></span></span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.551em; padding-bottom: 1.551em;">)</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\sum_1^n"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-munderover"><span class="mjx-base"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">∑</span></span></span><span class="mjx-stack" style="vertical-align: -0.31em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.422em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\sum_{i=0}^\infty i^2"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-munderover"><span class="mjx-base"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">∑</span></span></span><span class="mjx-stack" style="vertical-align: -0.31em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.422em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">∞</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span></span><span class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\prod"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">∏</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\int"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∫</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\bigcup"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">⋃</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\bigcap"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">⋂</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\iint"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∬</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\iiint"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∭</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\frac ab"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.515em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.729em; top: -1.157em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.729em; bottom: -0.704em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 0.515em;" class="mjx-line"></span></span><span style="height: 1.316em; vertical-align: -0.498em;" class="mjx-vsize"></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\frac{a+1}{b+1}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.419em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.007em; top: -1.454em;"><span class="mjx-mrow" style=""><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.007em; bottom: -0.775em;"><span class="mjx-mrow" style=""><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.419em;" class="mjx-line"></span></span><span style="height: 1.577em; vertical-align: -0.548em;" class="mjx-vsize"></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="{a+1\over b+1}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.419em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.007em; top: -1.454em;"><span class="mjx-mrow" style=""><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.007em; bottom: -0.775em;"><span class="mjx-mrow" style=""><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.419em;" class="mjx-line"></span></span><span style="height: 1.577em; vertical-align: -0.548em;" class="mjx-vsize"></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\cfrac{a}{b}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.67em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.948em; top: -1.816em;"><span class="mjx-mrow" style=""><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.14em; padding: 0px 0px 0.3em 0px;"><span class="mjx-box" style="width: 0px; margin: 0px 0px 0px;"><span class="mjx-mrow"><span class="mjx-mrow"></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mstyle" style="font-size: 141.4%;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span></span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.948em; bottom: -1.266em;"><span class="mjx-mrow" style=""><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.14em; padding: 0px 0px 0.3em 0px;"><span class="mjx-box" style="width: 0px; margin: 0px 0px 0px;"><span class="mjx-mrow"><span class="mjx-mrow"></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mstyle" style="font-size: 141.4%;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span></span></span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 0.67em;" class="mjx-line"></span></span><span style="height: 2.18em; vertical-align: -0.895em;" class="mjx-vsize"></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\mathbb{CHNQRZ}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">C</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">H</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">N</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.446em; padding-bottom: 0.446em;">Q</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">Z</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\mathbf{ABCDEFGHIJKLMNOPQRSTUVWXYZ}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">A</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">C</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">D</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">E</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">F</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">G</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">H</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">I</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">J</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">K</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">L</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">M</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">N</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">O</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">P</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.593em;">Q</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">R</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">S</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">T</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">U</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">V</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">W</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">X</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">Y</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">Z</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\mathbf{abcdefghijklmnopqrstuvwxyz}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.372em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">b</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.372em;">c</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.372em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em; padding-right: 0.101em;">f</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.593em;">g</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">h</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.593em;">j</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">k</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.372em; padding-bottom: 0.372em;">l</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.372em;">m</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.372em;">n</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.372em;">o</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.593em;">p</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.593em; padding-right: 0.002em;">q</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.372em;">r</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.372em;">s</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.298em; padding-bottom: 0.372em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.372em;">u</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.372em;">v</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.372em;">w</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.372em;">x</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.593em;">y</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-B" style="padding-top: 0.151em; padding-bottom: 0.372em;">z</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\mathtt{ABCDEFGHIJKLMNOPQRSTUVWXYZ}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.446em; padding-bottom: 0.225em;">A</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">C</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">D</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">E</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">F</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">G</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">H</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">I</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">J</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">K</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">L</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">M</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">N</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">O</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">P</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">Q</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">R</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">S</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">T</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.298em; padding-right: 0.003em;">U</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">V</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">W</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">X</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">Y</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">Z</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\mathtt{abcdefghijklmnopqrstuvwxyz}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">b</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.446em; padding-bottom: 0.225em;">f</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.519em;">g</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">h</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.519em;">j</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">k</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.225em;">l</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.225em; padding-right: 0.011em;">m</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.225em;">n</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">o</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.519em;">p</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.02em;">q</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.225em;">r</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">u</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">v</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">w</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.225em;">x</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.519em;">y</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.225em; padding-bottom: 0.225em;">z</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\mathrm{ABCDEFGHIJKLMNOPQRSTUVWXYZ}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">A</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">C</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">D</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">E</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">F</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">G</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">H</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">I</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">J</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">K</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">L</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">M</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">N</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">O</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">P</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.519em;">Q</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">R</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">S</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">T</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">U</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">V</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">W</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">X</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">Y</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">Z</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\mathrm{abcdefghijklmnopqrstuvwxyz}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">b</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">c</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em; padding-right: 0.066em;">f</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.519em;">g</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">h</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.519em;">j</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">k</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">l</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">m</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">n</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">o</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.519em;">p</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.519em; padding-right: 0.007em;">q</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">r</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">s</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">u</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">v</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">w</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">x</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.519em;">y</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">z</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\mathsf{ABCDEFGHIJKLMNOPQRSTUVWXYZ}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">A</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.519em; padding-bottom: 0.298em;">C</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">D</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">E</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">F</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.519em; padding-bottom: 0.298em;">G</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">H</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">I</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">J</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">K</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">L</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">M</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">N</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.519em; padding-bottom: 0.298em;">O</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">P</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.519em; padding-bottom: 0.372em;">Q</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.519em; padding-bottom: 0.298em;">S</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">T</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">U</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">V</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">W</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">X</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">Y</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">Z</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\mathsf{abcdefghijklmnopqrstuvwxyz}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.041em;">f</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.519em;">g</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">h</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.519em;">j</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">k</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">l</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">o</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.446em;">q</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">r</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">u</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">v</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">w</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.519em;">y</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">z</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\mathcal{ ABCDEFGHIJKLMNOPQRSTUVWXYZ}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em; padding-right: 0.021em;">A</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em; padding-right: 0.007em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em; padding-right: 0.006em;">C</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">D</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em; padding-right: 0.036em;">E</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em; padding-right: 0.11em;">F</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.446em; padding-right: 0.004em;">G</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">H</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.097em;">I</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.446em; padding-right: 0.161em;">J</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">K</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">L</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">M</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.519em; padding-bottom: 0.372em; padding-right: 0.159em;">N</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">O</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em; padding-right: 0.037em;">P</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.446em;">Q</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">R</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em; padding-right: 0.036em;">S</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em; padding-right: 0.288em;">T</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em; padding-right: 0.061em;">U</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em; padding-right: 0.045em;">V</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.372em; padding-right: 0.046em;">W</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.094em;">X</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.446em; padding-right: 0.046em;">Y</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-cal-R" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.042em;">Z</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\mathscr{ABCDEFGHIJKLMNOPQRSTUVWXYZ}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.225em; padding-right: 0.213em;">A</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.02em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.153em;">C</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.081em;">D</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.225em; padding-right: 0.156em;">E</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.095em;">F</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.128em;">G</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.272em;">H</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.225em; padding-right: 0.137em;">I</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.081em;">J</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.29em;">K</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.225em; padding-right: 0.161em;">L</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.136em;">M</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.306em;">N</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.225em; padding-right: 0.067em;">O</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.018em;">P</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.225em; padding-right: 0.002em;">Q</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.225em; padding-right: 0.037em;">R</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.148em;">S</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.249em;">T</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.225em; padding-right: 0.16em;">U</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.225em; padding-right: 0.228em;">V</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.225em; padding-right: 0.221em;">W</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.225em; padding-right: 0.187em;">X</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.225em; padding-right: 0.249em;">Y</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-script-R" style="padding-top: 0.519em; padding-bottom: 0.225em; padding-right: 0.211em;">Z</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\mathfrak{ABCDEFGHIJKLMNOPQRSTUVWXYZ} \mathfrak{abcdefghijklmnopqrstuvwxyz}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">A</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">C</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">D</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">E</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.446em; padding-right: 0.001em;">F</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">G</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.446em;">H</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">I</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.446em;">J</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.014em;">K</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">L</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">M</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">N</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.519em; padding-bottom: 0.298em;">O</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.519em;">P</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.519em; padding-bottom: 0.372em;">Q</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">S</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.007em;">T</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.019em;">U</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">V</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.008em;">W</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">X</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.519em;">Y</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.519em; padding-bottom: 0.446em;">Z</span></span></span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.519em;">f</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.225em; padding-bottom: 0.519em;">g</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.519em;">h</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.519em;">j</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">k</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">l</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">o</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.298em; padding-bottom: 0.519em;">p</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.225em; padding-bottom: 0.519em;">q</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">r</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.015em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">u</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.298em; padding-bottom: 0.298em;">v</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.298em; padding-bottom: 0.298em;">w</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.225em; padding-bottom: 0.446em;">x</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.298em; padding-bottom: 0.519em;">y</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.225em; padding-bottom: 0.519em;">z</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\sqrt{x^3}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">√</span></span><span class="mjx-box" style="padding-top: 0.081em; border-top: 1px solid;"><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\sqrt[3]{\frac xy}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mroot"><span class="mjx-root" style="font-size: 58.9%; vertical-align: 0.477em; width: 0px;"><span class="mjx-mn" style="padding-left: 0.561em;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">√</span></span><span class="mjx-box" style="padding-top: 0.224em; border-top: 1px solid;"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.546em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.772em; top: -1.159em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.772em; bottom: -0.718em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 0.546em;" class="mjx-line"></span></span><span style="height: 1.327em; vertical-align: -0.507em;" class="mjx-vsize"></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\sin x"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">sin</span></span><span class="mjx-mo"><span class="mjx-char"></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="sin x"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="\lim_{x\to 0}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-munderover"><span class="mjx-itable"><span class="mjx-row"><span class="mjx-cell"><span class="mjx-op" style="padding-left: 0.038em;"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">lim</span></span></span></span></span><span class="mjx-row"><span class="mjx-under" style="font-size: 70.7%; padding-top: 0.236em; padding-bottom: 0.141em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">→</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\lt\, \gt\, \le\, \leq\, \leqq\, \leqslant\, \ge\, \geq\, \geqq\, \geqslant\, \neq"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;"><</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">></span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.446em;">≤</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.446em;">≤</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.519em; padding-bottom: 0.446em;">≦</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.372em; padding-bottom: 0.446em;">⩽</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.446em;">≥</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.446em;">≥</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.519em; padding-bottom: 0.446em;">≧</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.372em; padding-bottom: 0.446em;">⩾</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">≠</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\not\lt"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.446em; padding-bottom: 0.519em;">≮</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\times\, \div\, \pm\, \mp"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">×</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">÷</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">±</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.519em;">∓</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="x\cdot y"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.004em; padding-bottom: 0.298em;">⋅</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\cup\, \cap\, \setminus\, \subset\, \subseteq \,\subsetneq \,\supset\, \in\, \notin\, \emptyset\, \varnothing"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">∪</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">∩</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">∖</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">⊂</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.446em;">⊆</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.372em; padding-bottom: 0.519em;">⊊</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">⊃</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">∈</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">∉</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.519em; padding-bottom: 0.446em;">∅</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">∅</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="{n+1 \choose 2k}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mrow"><span class="mjx-TeXmathchoice"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">(</span></span></span></span></span><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.328em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.878em; top: -1.319em;"><span class="mjx-mrow" style=""><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.878em; bottom: -0.524em;"><span class="mjx-mrow" style=""><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">k</span></span></span></span></span><span style="height: 1.303em; vertical-align: -0.37em;" class="mjx-vsize"></span></span><span class="mjx-TeXmathchoice"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">)</span></span></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\to\, \rightarrow\, \leftarrow\, \Rightarrow\, \Leftarrow\, \mapsto"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">→</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">→</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">←</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">⇒</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">⇐</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">↦</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\land\, \lor\, \lnot\, \forall\, \exists\, \top\, \bot\, \vdash\, \vDash"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">∧</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">∨</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">¬</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">∀</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">∃</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">⊤</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">⊥</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">⊢</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">⊨</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\star\, \ast\, \oplus\, \circ\, \bullet"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">⋆</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.298em;">∗</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">⊕</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.298em;">∘</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.298em;">∙</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\approx\, \sim \, \simeq\, \cong\, \equiv\, \prec, \lhd"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">≈</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">∼</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.298em;">≃</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char" style="padding-top: 0.298em; padding-bottom: 0.004em;"><span class="mjx-charbox MJXc-TeX-main-R" style="padding-bottom: 0.314em; margin-right: -0.222em;">≅</span></span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.298em;">≡</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">≺</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">⊲</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\infty\, \aleph_0"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">∞</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">ℵ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\nabla\, \partial"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">∇</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em; padding-right: 0.035em;">∂</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\Im\, \Re"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">I</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-frak-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="a\equiv b\pmod n"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.298em;">≡</span></span><span class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span><span class="mjx-TeXmathchoice"><span class="mjx-mspace" style="width: 0.444em; height: 0px;"></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">mod</span></span><span class="mjx-mspace" style="width: 0.333em; height: 0px;"></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="a_1, a_2, \ldots ,a_n"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.372em;">…</span></span><span class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="a_1+a_2+\cdots+a_n"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.004em; padding-bottom: 0.298em;">⋯</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\epsilon\, \varepsilon"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">ϵ</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">ε</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\phi\, \varphi"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.519em;">ϕ</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em;">φ</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\ell"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">ℓ</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\TeX"><span class="mjx-mrow" style="width: 2.065em;" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.12em;">T</span></span><span class="mjx-mspace" style="margin-right: -0.14em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.516em; padding: 0px 0.764em 0.246em 0px;"><span class="mjx-box" style="width: 0px; margin: -0.705em 0px -0.025em; position: relative; top: 0.221em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.115em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.024em;">X</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\LaTeX"><span class="mjx-mrow" style="width: 2.72em;" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">L</span></span><span class="mjx-mspace" style="margin-right: -0.325em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.327em; padding: 0px 0.469em 0px 0px;"><span class="mjx-box" style="width: 0px; margin: -0.463em 0px -0.016em; position: relative; top: -0.21em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mstyle" style="font-size: 80%;"><span class="mjx-mrow" style="font-size: 88.4%;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.298em;">A</span></span></span></span></span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.17em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.12em;">T</span></span><span class="mjx-mspace" style="margin-right: -0.14em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.516em; padding: 0px 0.764em 0.246em 0px;"><span class="mjx-box" style="width: 0px; margin: -0.705em 0px -0.025em; position: relative; top: 0.221em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.115em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.024em;">X</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\TeX"><span class="mjx-mrow" style="width: 2.065em;" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.12em;">T</span></span><span class="mjx-mspace" style="margin-right: -0.14em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.516em; padding: 0px 0.764em 0.246em 0px;"><span class="mjx-box" style="width: 0px; margin: -0.705em 0px -0.025em; position: relative; top: 0.221em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.115em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.024em;">X</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="a b"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="a\,b"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="a\;b"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mspace" style="width: 0.278em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="a\quad b"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mspace" style="width: 1em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="a\qquad b"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mspace" style="width: 2em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\{x\in s\mid x\text{ is extra large}\}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">{</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">∈</span></span><span class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">∣</span></span><span class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.519em;"> is extra large</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">}</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\hat x"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.213em; padding-bottom: 0.06em; padding-left: 0.064em;"><span class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">^</span></span></span><span class="mjx-op"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\widehat{xy}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.257em; padding-bottom: 0.06em; padding-left: 0.034em;"><span class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.519em; padding-bottom: 0.225em;">ˆ</span></span></span><span class="mjx-op"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\bar x"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.096em; padding-bottom: 0.06em; padding-left: 0.064em;"><span class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.298em;">¯</span></span></span><span class="mjx-op"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\overline{xyz}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="font-size: 70.7%; height: 0.096em; padding-bottom: 0.283em; padding-top: 0.141em;"><span class="mjx-mo" style="vertical-align: top;"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin: 0px -0.07em 0px -0.069em;">¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; letter-spacing: -0.242em; margin: 0px 0.031em 0px -0.17em;">¯¯¯¯¯¯</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.301em; padding-bottom: 0.833em; margin-right: -0.07em; margin-bottom: 0px; margin-top: 0px;">¯</span></span></span></span><span class="mjx-op"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\vec x"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.248em; padding-bottom: 0.06em; padding-left: 0.064em;"><span class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-vec-R" style="padding-top: 0.519em; padding-bottom: 0.225em;">→</span></span></span><span class="mjx-op"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\overrightarrow{xy}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="padding-bottom: 0.06em;"><span class="mjx-mo"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.222em; padding-bottom: 0.35em; margin: 0px -0.299em 0px -0.084em;">−</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.222em; padding-bottom: 0.35em; margin: 0px -0.056em 0px -0.271em;">→</span></span></span></span><span class="mjx-op"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\overleftrightarrow{xy}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="padding-bottom: 0.06em;"><span class="mjx-mo"><span class="mjx-delim-h"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.222em; padding-bottom: 0.35em; margin: 0px -0.41em 0px -0.055em;">←</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.222em; padding-bottom: 0.35em; margin: 0px -0.056em 0px -0.41em;">→</span></span></span></span><span class="mjx-op"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\frac d{dx}x\dot x = \dot x^2 + x\ddot x"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.916em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.295em; top: -1.41em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.295em; bottom: -0.704em;"><span class="mjx-mrow" style=""><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 0.916em;" class="mjx-line"></span></span><span style="height: 1.495em; vertical-align: -0.498em;" class="mjx-vsize"></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.17em; padding-bottom: 0.06em; padding-left: 0.064em;"><span class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">˙</span></span></span><span class="mjx-op"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.17em; padding-bottom: 0.06em; padding-left: 0.064em;"><span class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">˙</span></span></span><span class="mjx-op"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.6em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-munderover"><span class="mjx-stack"><span class="mjx-over" style="height: 0.165em; padding-bottom: 0.06em; padding-left: 0.064em;"><span class="mjx-mo" style="vertical-align: top;"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">¨</span></span></span><span class="mjx-op"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\$"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">$</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\{"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">{</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\_"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.291em; padding-bottom: 0.372em;">_</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\backslash"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">∖</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\LaTeX"><span class="mjx-mrow" style="width: 2.72em;" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">L</span></span><span class="mjx-mspace" style="margin-right: -0.325em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.327em; padding: 0px 0.469em 0px 0px;"><span class="mjx-box" style="width: 0px; margin: -0.463em 0px -0.016em; position: relative; top: -0.21em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mstyle" style="font-size: 80%;"><span class="mjx-mrow" style="font-size: 88.4%;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.298em;">A</span></span></span></span></span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.17em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.12em;">T</span></span><span class="mjx-mspace" style="margin-right: -0.14em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.516em; padding: 0px 0.764em 0.246em 0px;"><span class="mjx-box" style="width: 0px; margin: -0.705em 0px -0.025em; position: relative; top: 0.221em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.115em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.024em;">X</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\mathrm{B}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">B</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\sin"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">sin</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\operatorname{Spec} A"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.519em;">Spec</span></span><span class="mjx-mo"><span class="mjx-char"></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.298em;">A</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="_5C_3"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char"></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">5</span></span></span></span><span class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.045em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.045em;">C</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\TeX"><span class="mjx-mrow" style="width: 2.065em;" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.12em;">T</span></span><span class="mjx-mspace" style="margin-right: -0.14em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.516em; padding: 0px 0.764em 0.246em 0px;"><span class="mjx-box" style="width: 0px; margin: -0.705em 0px -0.025em; position: relative; top: 0.221em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.115em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.024em;">X</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\TeX"><span class="mjx-mrow" style="width: 2.065em;" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.12em;">T</span></span><span class="mjx-mspace" style="margin-right: -0.14em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.516em; padding: 0px 0.764em 0.246em 0px;"><span class="mjx-box" style="width: 0px; margin: -0.705em 0px -0.025em; position: relative; top: 0.221em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.115em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.024em;">X</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\LaTeX"><span class="mjx-mrow" style="width: 2.72em;" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">L</span></span><span class="mjx-mspace" style="margin-right: -0.325em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.327em; padding: 0px 0.469em 0px 0px;"><span class="mjx-box" style="width: 0px; margin: -0.463em 0px -0.016em; position: relative; top: -0.21em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mstyle" style="font-size: 80%;"><span class="mjx-mrow" style="font-size: 88.4%;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.298em;">A</span></span></span></span></span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.17em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.12em;">T</span></span><span class="mjx-mspace" style="margin-right: -0.14em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.516em; padding: 0px 0.764em 0.246em 0px;"><span class="mjx-box" style="width: 0px; margin: -0.705em 0px -0.025em; position: relative; top: 0.221em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.115em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.024em;">X</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\mathrm{d}x"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">d</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label=" | |
| \begin{matrix} | |
| 1 & x & x^2 \\ | |
| 1 & y & y^2 \\ | |
| 1 & z & z^2 \\ | |
| \end{matrix} | |
| "><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mtable" style="vertical-align: -1.742em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.252em;"><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.148em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0.5em; width: 0.572em;"><span class="mjx-mrow" style="margin-top: -0.148em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; width: 0.961em;"><span class="mjx-mrow" style="margin-top: -0.148em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.482em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span class="mjx-mrow" style="margin-top: -0.148em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.148em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.148em;"><span class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.006em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.082em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.252em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span class="mjx-mrow" style="margin-top: -0.148em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.148em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.148em;"><span class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.003em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.076em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span></span></span> | |
| </div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\begin{pmatrix}1&2\\3&4\\ \end{pmatrix}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">(</span></span><span class="mjx-mtable" style="vertical-align: -0.95em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.2em;"><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.2em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">4</span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">)</span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\begin{bmatrix}1&2\\3&4\\ \end{bmatrix}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">[</span></span><span class="mjx-mtable" style="vertical-align: -0.95em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.2em;"><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.2em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">4</span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">]</span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\begin{Bmatrix}1&2\\3&4\\ \end{Bmatrix}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">{</span></span><span class="mjx-mtable" style="vertical-align: -0.95em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.2em;"><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.2em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">4</span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">}</span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\begin{vmatrix}1&2\\3&4\\ \end{vmatrix}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo" style="vertical-align: -0.676em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">∣</span><span class="mjx-char MJXc-TeX-main-R" style="line-height: 0.402em; margin-bottom: 0.014em; margin-top: -0.114em;">∣</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">∣</span></span></span><span class="mjx-mtable" style="vertical-align: -0.95em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.2em;"><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.2em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">4</span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-mo" style="vertical-align: -0.676em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">∣</span><span class="mjx-char MJXc-TeX-main-R" style="line-height: 0.402em; margin-bottom: 0.014em; margin-top: -0.114em;">∣</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">∣</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\begin{Vmatrix}1&2\\3&4\\ \end{Vmatrix}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo" style="vertical-align: -0.675em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">∥</span><span class="mjx-char MJXc-TeX-main-R" style="line-height: 0.4em; margin-bottom: 0.014em; margin-top: -0.114em;">∥</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">∥</span></span></span><span class="mjx-mtable" style="vertical-align: -0.95em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.2em;"><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.2em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">4</span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-mo" style="vertical-align: -0.675em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">∥</span><span class="mjx-char MJXc-TeX-main-R" style="line-height: 0.4em; margin-bottom: 0.014em; margin-top: -0.114em;">∥</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">∥</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\cdots"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.004em; padding-bottom: 0.298em;">⋯</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\ddots"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 1.256em; padding-bottom: 0.298em;">⋱</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\vdots"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 1.035em; padding-bottom: 0.372em;">⋮</span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="\begin{pmatrix} | |
| 1 & a_1 & a_1^2 & \cdots & a_1^n \\ | |
| 1 & a_2 & a_2^2 & \cdots & a_2^n \\ | |
| \vdots & \vdots& \vdots & \ddots & \vdots \\ | |
| 1 & a_m & a_m^2 & \cdots & a_m^n | |
| \end{pmatrix}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo" style="vertical-align: -2.276em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">⎛</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.554em; margin-bottom: -0.1em; margin-top: 0px;">⎜ | |
| ⎜ | |
| ⎜ | |
| ⎜ | |
| ⎜</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.961em; padding-bottom: 0.888em;">⎝</span></span></span><span class="mjx-mtable" style="vertical-align: -2.945em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.393em;"><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.14em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0.5em; width: 1.185em;"><span class="mjx-mrow" style="margin-top: -0.14em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0.5em; width: 1.185em;"><span class="mjx-mrow" style="margin-top: -0.14em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-stack" style="vertical-align: -0.315em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0.5em; width: 1.282em;"><span class="mjx-mrow" style="margin-top: -0.14em;"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.004em; padding-bottom: 0.298em;">⋯</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; width: 1.185em;"><span class="mjx-mrow" style="margin-top: -0.14em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-stack" style="vertical-align: -0.315em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.593em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span class="mjx-mrow" style="margin-top: -0.14em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.14em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.14em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-stack" style="vertical-align: -0.315em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.14em;"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.004em; padding-bottom: 0.298em;">⋯</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.14em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-stack" style="vertical-align: -0.315em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 2.145em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span class="mjx-mrow" style="margin-top: 0.22em;"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 1.035em; padding-bottom: 0.372em;">⋮</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: 0.22em;"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 1.035em; padding-bottom: 0.372em;">⋮</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: 0.22em;"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 1.035em; padding-bottom: 0.372em;">⋮</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 1.256em; padding-bottom: 0.298em;">⋱</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span class="mjx-mrow" style="margin-top: 0.22em;"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 1.035em; padding-bottom: 0.372em;">⋮</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.26em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px;"><span class="mjx-mrow" style="margin-top: -0.14em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.14em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.14em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-stack" style="vertical-align: -0.157em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.14em;"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.004em; padding-bottom: 0.298em;">⋯</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.14em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-stack" style="vertical-align: -0.157em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-mo" style="vertical-align: -2.276em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">⎞</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.554em; margin-bottom: -0.1em; margin-top: 0px;">⎟ | |
| ⎟ | |
| ⎟ | |
| ⎟ | |
| ⎟</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.961em; padding-bottom: 0.888em;">⎠</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label=" | |
| \left[\begin{array}{cc|c} | |
| 1&2&3\\ | |
| 4&5&6 | |
| \end{array}\right] | |
| "><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.551em; padding-bottom: 1.551em;">[</span></span><span class="mjx-mtable" style="vertical-align: -1.171em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.4em; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.463em 0px 0.5em; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.4em 0px 0.463em; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">4</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">5</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.463em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">6</span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.551em; padding-bottom: 1.551em;">]</span></span></span></span></span></span></span> | |
| </div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label=" | |
| \begin{pmatrix} | |
| a & b \\ | |
| c & d\\ | |
| \hline | |
| 1 & 0\\ | |
| 0 & 1 | |
| \end{pmatrix} | |
| "><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo" style="vertical-align: -1.902em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">⎛</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.506em; margin-bottom: -0.1em; margin-top: 0px;">⎜ | |
| ⎜ | |
| ⎜ | |
| ⎜</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.961em; padding-bottom: 0.888em;">⎝</span></span></span><span class="mjx-mtable" style="vertical-align: -2.571em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.4em; width: 0.529em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.4em 0px 0.5em; width: 0.523em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em; border-bottom: 1.3px solid;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em; border-bottom: 1.3px solid;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-mo" style="vertical-align: -1.902em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">⎞</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.506em; margin-bottom: -0.1em; margin-top: 0px;">⎟ | |
| ⎟ | |
| ⎟ | |
| ⎟</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.961em; padding-bottom: 0.888em;">⎠</span></span></span></span></span></span></span></span> | |
| </div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\bigl( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \bigr)"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">(</span></span></span></span><span class="mjx-mstyle" style="font-size: 80%;"><span class="mjx-mrow" style="font-size: 88.4%;"><span class="mjx-mtable" style="vertical-align: -0.605em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 0.855em;"><span class="mjx-mtd" style="padding: 0px 0.167em 0px 0px; width: 0.529em;"><span class="mjx-mrow" style="margin-top: -0.281em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0.167em; width: 0.523em;"><span class="mjx-mrow" style="margin-top: -0.281em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 0.855em;"><span class="mjx-mtd" style="padding: 0.1em 0.167em 0px 0px;"><span class="mjx-mrow" style="margin-top: -0.281em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.1em 0px 0px 0.167em;"><span class="mjx-mrow" style="margin-top: -0.281em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">)</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="\begin{align} | |
| \sqrt{37} & = \sqrt{\frac{73^2-1}{12^2}} \\ | |
| & = \sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\ | |
| & = \sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\ | |
| & = \frac{73}{12}\sqrt{1 - \frac{1}{73^2}} \\ | |
| & \approx \frac{73}{12}\left(1 - \frac{1}{2\cdot73^2}\right) | |
| \end{align}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mtable" style="vertical-align: -7.817em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 3.26em;"><span class="mjx-mtd" style="padding: 0px 0px 0px 0px; text-align: right; width: 1.833em;"><span class="mjx-mrow" style="margin-top: 0.87em;"><span class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">√</span></span><span class="mjx-box" style="padding-top: 0.066em; border-top: 1px solid;"><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">37</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0px; text-align: left; width: 9.019em;"><span class="mjx-mrow"><span class="mjx-mi"></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-msqrt MJXc-space3"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.551em; padding-bottom: 1.551em;">√</span></span><span class="mjx-box" style="padding-top: 0.157em; border-top: 1px solid;"><span class="mjx-mrow"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.311em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.311em; top: -1.593em;"><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">73</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.605em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-denominator" style="width: 3.311em; bottom: -0.972em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">12</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 3.311em;" class="mjx-line"></span></span><span style="height: 2.565em; vertical-align: -0.972em;" class="mjx-vsize"></span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 3.41em;"><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: 0.854em;"><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: left;"><span class="mjx-mrow"><span class="mjx-mi"></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-msqrt MJXc-space3"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.551em; padding-bottom: 1.551em;">√</span></span><span class="mjx-box" style="padding-top: 0.141em; border-top: 1px solid;"><span class="mjx-mrow"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.589em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.589em; top: -1.593em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">73</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.605em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span><span class="mjx-denominator" style="width: 1.589em; bottom: -0.972em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">12</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.589em;" class="mjx-line"></span></span><span style="height: 2.565em; vertical-align: -0.972em;" class="mjx-vsize"></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.004em; padding-bottom: 0.298em;">⋅</span></span><span class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 3.311em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.311em; top: -1.593em;"><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">73</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.605em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-denominator" style="width: 3.311em; bottom: -1.004em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">73</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.605em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 3.311em;" class="mjx-line"></span></span><span style="height: 2.597em; vertical-align: -1.004em;" class="mjx-vsize"></span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 3.426em;"><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: 0.87em;"><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: left;"><span class="mjx-mrow"><span class="mjx-mi"></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-msqrt MJXc-space3"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.551em; padding-bottom: 1.551em;">√</span></span><span class="mjx-box" style="padding-top: 0.157em; border-top: 1px solid;"><span class="mjx-mrow"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.589em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.589em; top: -1.593em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">73</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.605em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span><span class="mjx-denominator" style="width: 1.589em; bottom: -0.972em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">12</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.589em;" class="mjx-line"></span></span><span style="height: 2.565em; vertical-align: -0.972em;" class="mjx-vsize"></span></span></span></span></span></span><span class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.551em; padding-bottom: 1.551em;">√</span></span><span class="mjx-box" style="padding-top: 0.141em; border-top: 1px solid;"><span class="mjx-mrow"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.311em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.311em; top: -1.593em;"><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">73</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.605em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-denominator" style="width: 3.311em; bottom: -1.004em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">73</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.605em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 3.311em;" class="mjx-line"></span></span><span style="height: 2.597em; vertical-align: -1.004em;" class="mjx-vsize"></span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 3.41em;"><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: 0.742em;"><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: left;"><span class="mjx-mrow"><span class="mjx-mi"></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.378em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">73</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.756em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">12</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.2em;" class="mjx-line"></span></span><span style="height: 2.134em; vertical-align: -0.756em;" class="mjx-vsize"></span></span><span class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 1.551em; padding-bottom: 1.551em;">√</span></span><span class="mjx-box" style="padding-top: 0.254em; border-top: 1px solid;"><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.589em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.589em; top: -1.368em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span><span class="mjx-denominator" style="width: 1.589em; bottom: -1.004em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">73</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.605em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.589em;" class="mjx-line"></span></span><span style="height: 2.371em; vertical-align: -1.004em;" class="mjx-vsize"></span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 2.629em;"><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: 0.475em;"><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: left;"><span class="mjx-mrow"><span class="mjx-mi"></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">≈</span></span><span class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.2em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.2em; top: -1.378em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">73</span></span></span><span class="mjx-denominator" style="width: 1.2em; bottom: -0.756em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">12</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.2em;" class="mjx-line"></span></span><span style="height: 2.134em; vertical-align: -0.756em;" class="mjx-vsize"></span></span><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">(</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 2.811em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.811em; top: -1.368em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span><span class="mjx-denominator" style="width: 2.811em; bottom: -1.004em;"><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.004em; padding-bottom: 0.298em;">⋅</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">73</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.605em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 2.811em;" class="mjx-line"></span></span><span style="height: 2.371em; vertical-align: -1.004em;" class="mjx-vsize"></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">)</span></span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label=" | |
| \begin{align} f(x)&=\left(x^3\right)+\left(x^3+x^2+x^1\right)+\left(x^3+x^2\right)\\ f'(x)&=\left(\left(3x^2+2x+1\right)+\left(3x^2+2x\right)\right)\\ f''(x)&=\left(6x+2\right)\\ \end{align} | |
| "><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mtable" style="vertical-align: -1.876em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.426em;"><span class="mjx-mtd" style="padding: 0px 0px 0px 0px; text-align: right; width: 2.392em;"><span class="mjx-mrow" style="margin-top: -0.098em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0px; text-align: left; width: 15.959em;"><span class="mjx-mrow" style="margin-top: -0.098em;"><span class="mjx-mi"></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mrow MJXc-space3"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">(</span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">)</span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mrow MJXc-space2"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">(</span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">)</span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mrow MJXc-space2"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">(</span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">)</span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.576em;"><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: -0.098em;"><span class="mjx-msup"><span class="mjx-base" style="margin-right: -0.06em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0.181em; padding-right: 0.071em;"><span class="mjx-mo" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.298em;">′</span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.098em;"><span class="mjx-mi"></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mrow MJXc-space3"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">)</span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mrow MJXc-space2"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">)</span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">)</span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.252em;"><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: -0.173em;"><span class="mjx-msup"><span class="mjx-base" style="margin-right: -0.06em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0.181em; padding-right: 0.071em;"><span class="mjx-mo" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.298em;">′′</span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.173em;"><span class="mjx-mi"></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mrow MJXc-space3"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">6</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="{}{}{}{}{}{}{}{}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"></span></span><span class="mjx-texatom"><span class="mjx-mrow"></span></span><span class="mjx-texatom"><span class="mjx-mrow"></span></span><span class="mjx-texatom"><span class="mjx-mrow"></span></span><span class="mjx-texatom"><span class="mjx-mrow"></span></span><span class="mjx-texatom"><span class="mjx-mrow"></span></span><span class="mjx-texatom"><span class="mjx-mrow"></span></span><span class="mjx-texatom"><span class="mjx-mrow"></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\Psi"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">Ψ</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\delta"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.007em;">δ</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\zeta"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.033em;">ζ</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\ge"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.446em;">≥</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\subseteq"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.446em;">⊆</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\LaTeX"><span class="mjx-mrow" style="width: 2.72em;" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">L</span></span><span class="mjx-mspace" style="margin-right: -0.325em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.327em; padding: 0px 0.469em 0px 0px;"><span class="mjx-box" style="width: 0px; margin: -0.463em 0px -0.016em; position: relative; top: -0.21em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mstyle" style="font-size: 80%;"><span class="mjx-mrow" style="font-size: 88.4%;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.298em;">A</span></span></span></span></span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.17em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.12em;">T</span></span><span class="mjx-mspace" style="margin-right: -0.14em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.516em; padding: 0px 0.764em 0.246em 0px;"><span class="mjx-box" style="width: 0px; margin: -0.705em 0px -0.025em; position: relative; top: 0.221em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.115em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.024em;">X</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\LaTeX"><span class="mjx-mrow" style="width: 2.72em;" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">L</span></span><span class="mjx-mspace" style="margin-right: -0.325em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.327em; padding: 0px 0.469em 0px 0px;"><span class="mjx-box" style="width: 0px; margin: -0.463em 0px -0.016em; position: relative; top: -0.21em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mstyle" style="font-size: 80%;"><span class="mjx-mrow" style="font-size: 88.4%;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.298em;">A</span></span></span></span></span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.17em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.12em;">T</span></span><span class="mjx-mspace" style="margin-right: -0.14em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.516em; padding: 0px 0.764em 0.246em 0px;"><span class="mjx-box" style="width: 0px; margin: -0.705em 0px -0.025em; position: relative; top: 0.221em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.115em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.024em;">X</span></span></span></span></span></span></div> | |
| <div>f(n) = | |
| <span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="\begin{cases} | |
| n/2, & \text{if $n$ is even} \\ | |
| 3n+1, & \text{if $n$ is odd} | |
| \end{cases}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">{</span></span><span class="mjx-mtable" style="vertical-align: -0.897em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.175em;"><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: left; width: 3.1em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 4.578em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mrow"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">if </span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span></span><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;"> is even</span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.119em;"><span class="mjx-mtd" style="padding: 0.1em 0.5em 0px 0px; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.1em 0px 0px 0.5em; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mrow"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">if </span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span></span><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;"> is odd</span></span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-mo" style="width: 0.12em;"></span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label=" | |
| \left. | |
| \begin{array}{l} | |
| \text{if $n$ is even:}&n/2\\ | |
| \text{if $n$ is odd:}&3n+1 | |
| \end{array} | |
| \right\} | |
| =f(n) | |
| "><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo" style="width: 0.12em;"></span><span class="mjx-mtable" style="vertical-align: -0.988em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.275em;"><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: left; width: 4.856em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mrow"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">if </span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span></span><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;"> is even:</span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 2.822em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.2em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mrow"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">if </span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span></span><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;"> is odd:</span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.256em; padding-bottom: 1.256em;">}</span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span></span></span> | |
| </div> | |
| <div>f(n) = | |
| <span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="\begin{cases} | |
| \frac{n}{2}, & \text{if $n$ is even} \\[2ex] | |
| 3n+1, & \text{if $n$ is odd} | |
| \end{cases}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo" style="vertical-align: -0.511em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.667em; padding-bottom: 0.298em; margin-bottom: -0.209em;">⎧</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">⎨</span><span class="mjx-char MJXc-TeX-size4-R" style="margin-top: -0.426em; padding-bottom: 1.182em;">⎩</span></span></span><span class="mjx-mtable" style="vertical-align: -1.435em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.82em;"><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: left; width: 3.1em;"><span class="mjx-mrow" style="margin-top: -0.18em;"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.566em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.8em; top: -1.159em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.8em; bottom: -0.665em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 0.566em;" class="mjx-line"></span></span><span style="height: 1.29em; vertical-align: -0.47em;" class="mjx-vsize"></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 4.578em;"><span class="mjx-mrow" style="margin-top: -0.18em;"><span class="mjx-mrow"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">if </span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span></span><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;"> is even</span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.549em;"><span class="mjx-mtd" style="padding: 0.53em 0.5em 0px 0px; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.53em 0px 0px 0.5em; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mrow"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">if </span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span></span><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;"> is odd</span></span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-mo" style="width: 0.12em;"></span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="\begin{array}{c|lcr} | |
| n & \text{Left} & \text{Center} & \text{Right} \\ | |
| \hline | |
| 1 & 0.24 & 1 & 125 \\ | |
| 2 & -1 & 189 & -8 \\ | |
| 3 & -20 & 2000 & 1+10i | |
| \end{array}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mtable" style="vertical-align: -2.587em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.452em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.463em 0px 0.4em; border-bottom: 1.3px solid; width: 0.6em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.463em; border-bottom: 1.3px solid; text-align: left; width: 1.778em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">Left</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.5em; border-bottom: 1.3px solid; width: 2.947em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">Center</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.4em 0px 0.5em; border-bottom: 1.3px solid; text-align: right; width: 3.067em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.519em;">Right</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.463em; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0.24</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em; text-align: right;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">125</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.463em; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">189</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em; text-align: right;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">8</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.463em; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">20</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2000</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em; text-align: right;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">10</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span></span></span> | |
| </div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label=" | |
| % outer vertical array of arrays | |
| \begin{array}{c} | |
| % inner horizontal array of arrays | |
| \begin{array}{cc} | |
| % inner array of minimum values | |
| \begin{array}{c|cccc} | |
| \text{min} & 0 & 1 & 2 & 3\\ | |
| \hline | |
| 0 & 0 & 0 & 0 & 0\\ | |
| 1 & 0 & 1 & 1 & 1\\ | |
| 2 & 0 & 1 & 2 & 2\\ | |
| 3 & 0 & 1 & 2 & 3 | |
| \end{array} | |
| & | |
| % inner array of maximum values | |
| \begin{array}{c|cccc} | |
| \text{max}&0&1&2&3\\ | |
| \hline | |
| 0 & 0 & 1 & 2 & 3\\ | |
| 1 & 1 & 1 & 2 & 3\\ | |
| 2 & 2 & 2 & 2 & 3\\ | |
| 3 & 3 & 3 & 3 & 3 | |
| \end{array} | |
| \end{array} | |
| \\ | |
| % inner array of delta values | |
| \begin{array}{c|cccc} | |
| \Delta&0&1&2&3\\ | |
| \hline | |
| 0 & 0 & 1 & 2 & 3\\ | |
| 1 & 1 & 0 & 1 & 2\\ | |
| 2 & 2 & 1 & 0 & 1\\ | |
| 3 & 3 & 2 & 1 & 0 | |
| \end{array} | |
| \end{array} | |
| "><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mtable" style="vertical-align: -6.992em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 7.242em;"><span class="mjx-mtd" style="padding: 0px 0px 0px 0px; width: 19.128em;"><span class="mjx-mrow"><span class="mjx-mtable" style="vertical-align: -3.271em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 7.042em;"><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; width: 8.8em;"><span class="mjx-mrow"><span class="mjx-mtable" style="vertical-align: -3.271em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.463em 0px 0.4em; border-bottom: 1.3px solid; width: 1.667em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">min</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.463em; border-bottom: 1.3px solid; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.5em; border-bottom: 1.3px solid; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.5em; border-bottom: 1.3px solid; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.4em 0px 0.5em; border-bottom: 1.3px solid; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.463em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.463em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.463em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.463em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; width: 8.994em;"><span class="mjx-mrow"><span class="mjx-mtable" style="vertical-align: -3.271em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.463em 0px 0.4em; border-bottom: 1.3px solid; width: 1.861em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">max</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.463em; border-bottom: 1.3px solid; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.5em; border-bottom: 1.3px solid; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.5em; border-bottom: 1.3px solid; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.4em 0px 0.5em; border-bottom: 1.3px solid; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.463em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.463em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.463em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.463em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 7.242em;"><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0px;"><span class="mjx-mrow"><span class="mjx-mtable" style="vertical-align: -3.271em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.221em 0.463em 0px 0.4em; border-bottom: 1.3px solid; width: 0.833em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.372em;">Δ</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.463em; border-bottom: 1.3px solid; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.5em; border-bottom: 1.3px solid; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.5em; border-bottom: 1.3px solid; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.4em 0px 0.5em; border-bottom: 1.3px solid; width: 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.463em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.463em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.463em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="border-right: 1.3px solid; padding: 0.2em 0.463em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.463em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span></span></span> | |
| </div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\mathsf{Show\ Math\ As\ }\blacktriangleright\mathsf{\ TeX\ Commands}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.519em; padding-bottom: 0.298em;">S</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">h</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">o</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">w</span></span><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-sans-R" style="margin-top: -0.217em; padding-bottom: 0.298em;"> </span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">M</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">h</span></span><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-sans-R" style="margin-top: -0.217em; padding-bottom: 0.298em;"> </span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">A</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-sans-R" style="margin-top: -0.217em; padding-bottom: 0.298em;"> </span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">▶</span></span><span class="mjx-texatom MJXc-space2"><span class="mjx-mrow"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-sans-R" style="margin-top: -0.217em; padding-bottom: 0.298em;"> </span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">T</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">X</span></span><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-sans-R" style="margin-top: -0.217em; padding-bottom: 0.298em;"> </span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.519em; padding-bottom: 0.298em;">C</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">o</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-sans-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="all italics, weird-looking spacing, an' odd apostrophes"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">l</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">l</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">l</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">w</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">r</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">l</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">o</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">o</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">k</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.003em;">g</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.003em;">g</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.144em; padding-bottom: 0.519em;">,</span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-msup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">n</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mo" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.298em;">′</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">o</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">o</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">r</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">o</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">h</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\LaTeX"><span class="mjx-mrow" style="width: 2.72em;" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">L</span></span><span class="mjx-mspace" style="margin-right: -0.325em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.327em; padding: 0px 0.469em 0px 0px;"><span class="mjx-box" style="width: 0px; margin: -0.463em 0px -0.016em; position: relative; top: -0.21em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mstyle" style="font-size: 80%;"><span class="mjx-mrow" style="font-size: 88.4%;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.298em;">A</span></span></span></span></span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.17em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.12em;">T</span></span><span class="mjx-mspace" style="margin-right: -0.14em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.516em; padding: 0px 0.764em 0.246em 0px;"><span class="mjx-box" style="width: 0px; margin: -0.705em 0px -0.025em; position: relative; top: 0.221em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.115em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.024em;">X</span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label=" | |
| \begin{array}{ll} \hfill\mathrm{Bad}\hfill & \hfill\mathrm{Better}\hfill \\ \hline \\ e^{i\frac{\pi}2} \quad e^{\frac{i\pi}2}& e^{i\pi/2} \\ \int_{-\frac\pi2}^\frac\pi2 \sin x\,dx & \int_{-\pi/2}^{\pi/2}\sin x\,dx \\ \end{array} | |
| "><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mtable" style="vertical-align: -3.433em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.4em; border-bottom: 1.3px solid; width: 5.078em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">d</span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.4em 0px 0.5em; border-bottom: 1.3px solid; width: 5.565em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">r</span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em; text-align: left;"><span style="margin-top: -0.2em;"></span></span></span><span class="mjx-mtr" style="height: 1.878em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em; text-align: left;"><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.73em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.644em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 83.3%; width: 0.773em; top: -1.064em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">π</span></span></span><span class="mjx-denominator" style="font-size: 83.3%; width: 0.773em; bottom: -0.688em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span style="border-bottom: 1px solid; top: -0.296em; width: 0.644em;" class="mjx-line"></span></span><span style="height: 1.46em; vertical-align: -0.573em;" class="mjx-vsize"></span></span></span></span></span></span><span class="mjx-mspace" style="width: 1em; height: 0px;"></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.73em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.932em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 83.3%; width: 1.118em; top: -1.294em;"><span class="mjx-mrow" style=""><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">π</span></span></span></span><span class="mjx-denominator" style="font-size: 83.3%; width: 1.118em; bottom: -0.688em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span style="border-bottom: 1px solid; top: -0.296em; width: 0.932em;" class="mjx-line"></span></span><span style="height: 1.652em; vertical-align: -0.573em;" class="mjx-vsize"></span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em; text-align: left;"><span class="mjx-mrow" style="margin-top: 0.278em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">π</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 2.666em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em; text-align: left;"><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.453em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.324em; padding-right: 0.071em;"><span class="mjx-mfrac" style=""><span class="mjx-box MJXc-stacked" style="width: 0.644em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 83.3%; width: 0.773em; top: -1.064em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">π</span></span></span><span class="mjx-denominator" style="font-size: 83.3%; width: 0.773em; bottom: -0.688em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span style="border-bottom: 1px solid; top: -0.296em; width: 0.644em;" class="mjx-line"></span></span><span style="height: 1.46em; vertical-align: -0.573em;" class="mjx-vsize"></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.644em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 83.3%; width: 0.773em; top: -1.064em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">π</span></span></span><span class="mjx-denominator" style="font-size: 83.3%; width: 0.773em; bottom: -0.688em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span style="border-bottom: 1px solid; top: -0.296em; width: 0.644em;" class="mjx-line"></span></span><span style="height: 1.46em; vertical-align: -0.573em;" class="mjx-vsize"></span></span></span></span></span></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">sin</span></span><span class="mjx-mo"><span class="mjx-char"></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em; text-align: left;"><span class="mjx-mrow" style="margin-top: 0.28em;"><span class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.256em; padding-left: 0.324em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">π</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">π</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">sin</span></span><span class="mjx-mo"><span class="mjx-char"></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="\begin{array}{cc} | |
| \mathrm{Bad} & \mathrm{Better} \\ | |
| \hline \\ | |
| e^{i\frac{\pi}2} \quad e^{\frac{i\pi}2}& e^{i\pi/2} \\ | |
| \int_{-\frac\pi2}^\frac\pi2 \sin x\,dx & \int_{-\pi/2}^{\pi/2}\sin x\,dx \\ | |
| \end{array}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mtable" style="vertical-align: -3.433em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.4em; border-bottom: 1.3px solid; width: 5.078em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">d</span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.4em 0px 0.5em; border-bottom: 1.3px solid; width: 5.565em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">r</span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span style="margin-top: -0.2em;"></span></span></span><span class="mjx-mtr" style="height: 1.878em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em;"><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.73em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.644em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 83.3%; width: 0.773em; top: -1.064em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">π</span></span></span><span class="mjx-denominator" style="font-size: 83.3%; width: 0.773em; bottom: -0.688em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span style="border-bottom: 1px solid; top: -0.296em; width: 0.644em;" class="mjx-line"></span></span><span style="height: 1.46em; vertical-align: -0.573em;" class="mjx-vsize"></span></span></span></span></span></span><span class="mjx-mspace" style="width: 1em; height: 0px;"></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.73em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.932em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 83.3%; width: 1.118em; top: -1.294em;"><span class="mjx-mrow" style=""><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">π</span></span></span></span><span class="mjx-denominator" style="font-size: 83.3%; width: 1.118em; bottom: -0.688em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span style="border-bottom: 1px solid; top: -0.296em; width: 0.932em;" class="mjx-line"></span></span><span style="height: 1.652em; vertical-align: -0.573em;" class="mjx-vsize"></span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: 0.278em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">π</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 2.666em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em;"><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.453em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.324em; padding-right: 0.071em;"><span class="mjx-mfrac" style=""><span class="mjx-box MJXc-stacked" style="width: 0.644em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 83.3%; width: 0.773em; top: -1.064em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">π</span></span></span><span class="mjx-denominator" style="font-size: 83.3%; width: 0.773em; bottom: -0.688em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span style="border-bottom: 1px solid; top: -0.296em; width: 0.644em;" class="mjx-line"></span></span><span style="height: 1.46em; vertical-align: -0.573em;" class="mjx-vsize"></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.644em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 83.3%; width: 0.773em; top: -1.064em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">π</span></span></span><span class="mjx-denominator" style="font-size: 83.3%; width: 0.773em; bottom: -0.688em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span style="border-bottom: 1px solid; top: -0.296em; width: 0.644em;" class="mjx-line"></span></span><span style="height: 1.46em; vertical-align: -0.573em;" class="mjx-vsize"></span></span></span></span></span></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">sin</span></span><span class="mjx-mo"><span class="mjx-char"></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: 0.28em;"><span class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.256em; padding-left: 0.324em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">π</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">π</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">sin</span></span><span class="mjx-mo"><span class="mjx-char"></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="\begin{array}{cc} | |
| \mathrm{Bad} & \mathrm{Better} \\ | |
| \hline \\ | |
| \{x|x^2\in\Bbb Z\} & \{x\mid x^2\in\Bbb Z\} \\ | |
| \end{array}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mtable" style="vertical-align: -1.934em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.4em; border-bottom: 1.3px solid; width: 4.7em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">d</span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.4em 0px 0.5em; border-bottom: 1.3px solid; width: 5.256em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">r</span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span style="margin-top: -0.2em;"></span></span></span><span class="mjx-mtr" style="height: 1.548em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.148em;"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">{</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">∈</span></span><span class="mjx-texatom MJXc-space3"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">Z</span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">}</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.148em;"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">{</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">∣</span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">∈</span></span><span class="mjx-texatom MJXc-space3"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.446em; padding-bottom: 0.298em;">Z</span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">}</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="\begin{array}{cc} | |
| \mathrm{Bad} & \mathrm{Better} \\ | |
| \hline \\ | |
| \int\int_S f(x)\,dy\,dx & \iint_S f(x)\,dy\,dx \\ | |
| \int\int\int_V f(x)\,dz\,dy\,dx & \iiint_V f(x)\,dz\,dy\,dx | |
| \end{array}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mtable" style="vertical-align: -2.801em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.4em; border-bottom: 1.3px solid; width: 8.276em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">d</span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.4em 0px 0.5em; border-bottom: 1.3px solid; width: 7.417em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">r</span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span style="margin-top: -0.2em;"></span></span></span><span class="mjx-mtr" style="height: 1.63em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.169em;"><span class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∫</span></span><span class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.138em;"><span class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∫</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.517em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.032em;">S</span></span></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.169em;"><span class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∬</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.517em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.298em; padding-right: 0.032em;">S</span></span></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.651em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.169em;"><span class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∫</span></span><span class="mjx-mo MJXc-space1" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∫</span></span><span class="mjx-msubsup MJXc-space1"><span class="mjx-base" style="margin-right: -0.138em;"><span class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∫</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.517em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.186em;">V</span></span></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.169em;"><span class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∭</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.517em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.186em;">V</span></span></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\TeX"><span class="mjx-mrow" style="width: 2.065em;" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.12em;">T</span></span><span class="mjx-mspace" style="margin-right: -0.14em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.516em; padding: 0px 0.764em 0.246em 0px;"><span class="mjx-box" style="width: 0px; margin: -0.705em 0px -0.025em; position: relative; top: 0.221em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.115em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.024em;">X</span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="\begin{array}{cc} | |
| \mathrm{Bad} & \mathrm{Better} \\ | |
| \hline \\ | |
| \iiint_V f(x)dz dy dx & \iiint_V f(x)\,dz\,dy\,dx | |
| \end{array}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mtable" style="vertical-align: -1.986em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.421em;"><span class="mjx-mtd" style="padding: 0.221em 0.5em 0px 0.4em; border-bottom: 1.3px solid; width: 6.917em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">d</span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.221em 0.4em 0px 0.5em; border-bottom: 1.3px solid; width: 7.417em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">B</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.372em;">t</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">r</span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.4em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span style="margin-top: -0.2em;"></span></span></span><span class="mjx-mtr" style="height: 1.651em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0.4em;"><span class="mjx-mrow" style="margin-top: -0.169em;"><span class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∭</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.517em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.186em;">V</span></span></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0.4em 0px 0.5em;"><span class="mjx-mrow" style="margin-top: -0.169em;"><span class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∭</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.517em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.186em;">V</span></span></span></span><span class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mspace" style="width: 0.167em; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\left\{x\middle | \frac{x^2}{2} \in \mathbb{z}\right\}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">{</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-texatom"><span class="mjx-mrow"></span></span><span class="mjx-mo" style="vertical-align: -0.379em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em; margin-bottom: -0.291em;">∣</span><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em; margin-top: -0.291em;">∣</span></span></span><span class="mjx-texatom"><span class="mjx-mrow"></span></span><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.87em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.23em; top: -1.631em;"><span class="mjx-msubsup" style=""><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="font-size: 83.3%; vertical-align: 0.435em; padding-left: 0px; padding-right: 0.06em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.23em; bottom: -0.665em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 0.87em;" class="mjx-line"></span></span><span style="height: 1.624em; vertical-align: -0.47em;" class="mjx-vsize"></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">∈</span></span><span class="mjx-texatom MJXc-space3"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.151em; padding-bottom: 0.372em;">z</span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">}</span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label=" | |
| \require{cancel}\begin{array}{rl} | |
| \verb|y+\cancel{x}| & y+\cancel{x}\\ | |
| \verb|\cancel{y+x}| & \cancel{y+x}\\ | |
| \verb|y+\bcancel{x}| & y+\bcancel{x}\\ | |
| \verb|y+\xcancel{x}| & y+\xcancel{x}\\ | |
| \verb|y+\cancelto{0}{x}| & y+\cancelto{0}{x}\\ | |
| \verb+\frac{1\cancel9}{\cancel95} = \frac15+& \frac{1\cancel9}{\cancel95} = \frac15 \\ | |
| \end{array} | |
| "><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mtable" style="vertical-align: -4.571em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.253em;"><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: right; width: 19.425em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">y+\cancel{x}</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 3.44em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-menclose MJXc-space2"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span><svg style="width: 0.972em; height: 0.903em; vertical-align: -0.236em; margin-left: -0.972em;"><g stroke-width="1px"><line x1="0.5px" y1="11.3px" x2="12.2px" y2="0.5px"></line></g></svg></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.638em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: -0.192em;"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">\cancel{y+x}</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.192em;"><span class="mjx-menclose"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span><svg style="width: 2.69em; height: 1.238em; vertical-align: -0.43em; margin-left: -2.69em;"><g stroke-width="1px"><line x1="0.5px" y1="15.8px" x2="35.5px" y2="0.5px"></line></g></svg></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.453em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">y+\bcancel{x}</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-menclose MJXc-space2"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span><svg style="width: 0.972em; height: 0.903em; vertical-align: -0.236em; margin-left: -0.972em;"><g stroke-width="1px"><line x1="0.5px" y1="0.5px" x2="12.2px" y2="11.3px"></line></g></svg></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.453em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">y+\xcancel{x}</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-menclose MJXc-space2"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span><svg style="width: 0.972em; height: 0.903em; vertical-align: -0.236em; margin-left: -0.972em;"><g stroke-width="1px"><line x1="0.5px" y1="11.3px" x2="12.2px" y2="0.5px"></line></g></svg><svg style="width: 0.972em; height: 0.903em; vertical-align: -0.236em; margin-left: -0.972em;"><g stroke-width="1px"><line x1="0.5px" y1="0.5px" x2="12.2px" y2="11.3px"></line></g></svg></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.606em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: -0.047em;"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">y+\cancelto{0}{x}</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.047em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msup MJXc-space2"><span class="mjx-base"><span class="mjx-menclose"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span><svg style="width: 0.972em; height: 0.903em; vertical-align: -0.236em; margin-left: -0.972em;"><g stroke-width="1px"><g fill="currentColor" transform="translate(0.5 11.8) rotate(-41.56771594726492)"><line x1="0.5px" y1="0" x2="13px" y2="0"></line><path d="M 13,0 L 12,2L 17,0 L 12,-2" stroke="none"></path></g></g></svg></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.557em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mpadded" style=""><span class="mjx-block" style="width: 0px; margin-top: -0.209em; padding: 0px 0.5em 0px 0px;"><span class="mjx-box" style="width: 0px; margin: -0.691em 0px -0.047em; position: relative; top: -0.1em;"><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span><span class="mjx-strut"></span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 2.238em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: 0.269em;"><span class="mjx-mtext"><span class="mjx-char" style="padding-top: 0.519em; padding-bottom: 0.077em;"><span class="mjx-charbox MJXc-TeX-type-R" style="padding-bottom: 0.233em; margin-right: 0.275em;">\frac{1\cancel9}{\cancel95} </span><span class="mjx-charbox MJXc-TeX-type-R" style="padding-bottom: 0.233em; margin-right: 0.275em;">= </span><span class="mjx-charbox MJXc-TeX-type-R" style="padding-bottom: 0.233em;">\frac15</span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span class="mjx-mrow"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.131em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.6em; top: -1.794em;"><span class="mjx-mrow" style=""><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-menclose"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">9</span></span></span></span><svg style="width: 0.9em; height: 1.138em; vertical-align: -0.247em; margin-left: -0.9em;"><g stroke-width="1px"><line x1="0.5px" y1="14.4px" x2="11.2px" y2="0.5px"></line></g></svg></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.6em; bottom: -1.087em;"><span class="mjx-mrow" style=""><span class="mjx-menclose"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">9</span></span></span></span><svg style="width: 0.9em; height: 1.138em; vertical-align: -0.247em; margin-left: -0.9em;"><g stroke-width="1px"><line x1="0.5px" y1="14.4px" x2="11.2px" y2="0.5px"></line></g></svg></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">5</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.131em;" class="mjx-line"></span></span><span style="height: 2.038em; vertical-align: -0.769em;" class="mjx-vsize"></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.372em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.687em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">5</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 0.495em;" class="mjx-line"></span></span><span style="height: 1.456em; vertical-align: -0.486em;" class="mjx-vsize"></span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span></span></span> | |
| </div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label=" | |
| \require{enclose}\begin{array}{rl} | |
| \verb|\enclose{horizontalstrike}{x+y}| & \enclose{horizontalstrike}{x+y}\\ | |
| \verb|\enclose{verticalstrike}{\frac xy}| & \enclose{verticalstrike}{\frac xy}\\ | |
| \verb|\enclose{updiagonalstrike}{x+y}| & \enclose{updiagonalstrike}{x+y}\\ | |
| \verb|\enclose{downdiagonalstrike}{x+y}| & \enclose{downdiagonalstrike}{x+y}\\ | |
| \verb|\enclose{horizontalstrike,updiagonalstrike}{x+y}| & \enclose{horizontalstrike,updiagonalstrike}{x+y}\\ | |
| \end{array} | |
| "><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mtable" style="vertical-align: -3.89em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.438em;"><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: right; width: 25.2em;"><span class="mjx-mrow" style="margin-top: -0.192em;"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">\enclose{horizontalstrike}{x+y}</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 2.69em;"><span class="mjx-mrow" style="margin-top: -0.192em;"><span class="mjx-menclose"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span></span><span style="border-top: 1px solid; height: 0.582em; vertical-align: -0.43em; width: 2.69em; margin-left: -2.69em;" class="mjx-box"></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 2.127em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: 0.02em;"><span class="mjx-mtext"><span class="mjx-char" style="padding-top: 0.519em; padding-bottom: 0.225em;"><span class="mjx-charbox MJXc-TeX-type-R" style="padding-bottom: 0.233em; margin-right: 0.275em;">\enclose{verticalstrike}{\frac </span><span class="mjx-charbox MJXc-TeX-type-R" style="padding-bottom: 0.233em;">xy}</span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span class="mjx-mrow"><span class="mjx-menclose"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.546em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.772em; top: -1.159em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.772em; bottom: -0.718em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 0.546em;" class="mjx-line"></span></span><span style="height: 1.327em; vertical-align: -0.507em;" class="mjx-vsize"></span></span></span></span><span style="border-left: 1px solid; height: 1.727em; vertical-align: -0.707em; width: 0.556em; margin-left: -0.63em;" class="mjx-box"></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.638em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: -0.192em;"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">\enclose{updiagonalstrike}{x+y}</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.192em;"><span class="mjx-menclose"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span></span><svg style="width: 2.69em; height: 1.238em; vertical-align: -0.43em; margin-left: -2.69em;"><g stroke-width="1px"><line x1="0.5px" y1="15.8px" x2="35.5px" y2="0.5px"></line></g></svg></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.638em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: -0.192em;"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">\enclose{downdiagonalstrike}{x+y}</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.192em;"><span class="mjx-menclose"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span></span><svg style="width: 2.69em; height: 1.238em; vertical-align: -0.43em; margin-left: -2.69em;"><g stroke-width="1px"><line x1="0.5px" y1="0.5px" x2="35.5px" y2="15.8px"></line></g></svg></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.438em;"><span class="mjx-mtd" style="padding: 0.2em 0.5em 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: -0.192em;"><span class="mjx-mtext"><span class="mjx-char MJXc-TeX-type-R" style="padding-top: 0.519em; padding-bottom: 0.519em;">\enclose{horizontalstrike,updiagonalstrike}{x+y}</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0.5em; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.192em;"><span class="mjx-menclose"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span></span></span><span style="border-top: 1px solid; height: 0.582em; vertical-align: -0.43em; width: 2.69em; margin-left: -2.69em;" class="mjx-box"></span><svg style="width: 2.69em; height: 1.238em; vertical-align: -0.43em; margin-left: -2.69em;"><g stroke-width="1px"><line x1="0.5px" y1="15.8px" x2="35.5px" y2="0.5px"></line></g></svg></span><span class="mjx-strut"></span></span></span></span></span></span></span></span></span></span> | |
| </div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="19/95 = 1/5"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">19</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">95</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">5</span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\cancelto{\cancelto{\cancelto{x^{2+x}}{\cancelto{x^2}{x}+4}}4}0"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msup"><span class="mjx-base"><span class="mjx-menclose"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span><svg style="width: 0.9em; height: 1.138em; vertical-align: -0.247em; margin-left: -0.9em;"><g stroke-width="1px"><g fill="currentColor" transform="translate(0.5 14.9) rotate(-50.9583926190483)"><line x1="0.5px" y1="0" x2="14.6px" y2="0"></line><path d="M 14.6,0 L 13.6,2L 18.6,0 L 13.6,-2" stroke="none"></path></g></g></svg></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.874em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mpadded" style=""><span class="mjx-block" style="width: 0px; margin-top: 1.834em; padding: 0px 6.229em 0.125em 0px;"><span class="mjx-box" style="width: 0px; margin: -2.734em 0px -0.225em; position: relative; top: -0.1em;"><span class="mjx-mrow"><span class="mjx-msup"><span class="mjx-base"><span class="mjx-menclose"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">4</span></span></span></span><svg style="width: 0.9em; height: 1.127em; vertical-align: -0.225em; margin-left: -0.9em;"><g stroke-width="1px"><g fill="currentColor" transform="translate(0.5 14.8) rotate(-50.66687801800788)"><line x1="0.5px" y1="0" x2="14.5px" y2="0"></line><path d="M 14.5,0 L 13.5,2L 18.5,0 L 13.5,-2" stroke="none"></path></g></g></svg></span></span><span class="mjx-sup" style="font-size: 83.3%; vertical-align: 0.696em; padding-left: 0px; padding-right: 0.06em;"><span class="mjx-mpadded" style=""><span class="mjx-block" style="width: 0px; margin-top: 1.585em; padding: 0px 6.344em 0.336em 0px;"><span class="mjx-box" style="width: 0px; margin: -2.485em 0px -0.436em; position: relative; top: -0.1em;"><span class="mjx-mrow"><span class="mjx-msup"><span class="mjx-base"><span class="mjx-menclose"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-msup"><span class="mjx-base"><span class="mjx-menclose"><span class="mjx-box" style="padding: 0.2em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span><svg style="width: 0.972em; height: 0.903em; vertical-align: -0.236em; margin-left: -0.972em;"><g stroke-width="1px"><g fill="currentColor" transform="translate(0.5 11.8) rotate(-41.56771594726492)"><line x1="0.5px" y1="0" x2="13px" y2="0"></line><path d="M 13,0 L 12,2L 17,0 L 12,-2" stroke="none"></path></g></g></svg></span></span><span class="mjx-sup" style="vertical-align: 0.363em; padding-left: 0px; padding-right: 0.05em;"><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: 0.154em; padding: 0px 1.122em 0px 0px;"><span class="mjx-box" style="width: 0px; margin: -1.054em 0px -0.036em; position: relative; top: -0.1em;"><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="vertical-align: 0.363em; padding-left: 0px; padding-right: 0.05em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">4</span></span></span></span><svg style="width: 3.822em; height: 2.153em; vertical-align: -0.436em; margin-left: -3.822em;"><g stroke-width="1px"><g fill="currentColor" transform="translate(0.5 28.7) rotate(-28.779234196747712)"><line x1="0.5px" y1="0" x2="54.6px" y2="0"></line><path d="M 54.6,0 L 53.6,2L 58.6,0 L 53.6,-2" stroke="none"></path></g></g></svg></span></span><span class="mjx-sup" style="vertical-align: 1.331em; padding-left: 0px; padding-right: 0.05em;"><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: 0.154em; padding: 0px 2.472em 0px 0px;"><span class="mjx-box" style="width: 0px; margin: -1.054em 0px -0.036em; position: relative; top: -0.1em;"><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span><span class="mjx-sup" style="vertical-align: 0.363em; padding-left: 0px; padding-right: 0.05em;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span></span></div> | |
| <div><span class="mjpage"><span class="mjx-chtml"><span class="mjx-math" aria-label="\LaTeX"><span class="mjx-mrow" style="width: 2.72em;" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">L</span></span><span class="mjx-mspace" style="margin-right: -0.325em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.327em; padding: 0px 0.469em 0px 0px;"><span class="mjx-box" style="width: 0px; margin: -0.463em 0px -0.016em; position: relative; top: -0.21em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mstyle" style="font-size: 80%;"><span class="mjx-mrow" style="font-size: 88.4%;"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.298em;">A</span></span></span></span></span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.17em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.12em;">T</span></span><span class="mjx-mspace" style="margin-right: -0.14em; width: 0px; height: 0px;"></span><span class="mjx-mpadded"><span class="mjx-block" style="width: 0px; margin-top: -0.516em; padding: 0px 0.764em 0.246em 0px;"><span class="mjx-box" style="width: 0px; margin: -0.705em 0px -0.025em; position: relative; top: 0.221em;"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span></span></span></span></span><span class="mjx-strut"></span></span></span><span class="mjx-mspace" style="margin-right: -0.115em; width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.024em;">X</span></span></span></span></span></span></div> | |
| <div> | |
| <span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label=" | |
| \left\{ | |
| \begin{array}{c} | |
| a_1x+b_1y+c_1z=d_1 \\ | |
| a_2x+b_2y+c_2z=d_2 \\ | |
| a_3x+b_3y+c_3z=d_3 | |
| \end{array} | |
| \right. | |
| "><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo" style="vertical-align: -0.771em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.667em; padding-bottom: 0.298em;">⎧</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.101em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">⎨</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.101em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="margin-top: -0.217em; padding-bottom: 1.182em;">⎩</span></span></span><span class="mjx-mtable" style="vertical-align: -1.695em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.23em;"><span class="mjx-mtd" style="padding: 0px 0px 0px 0px; width: 8.781em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.003em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.43em;"><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0px;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.003em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.23em;"><span class="mjx-mtd" style="padding: 0.2em 0px 0px 0px;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.003em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-mo" style="width: 0.12em;"></span></span></span></span></span></span> | |
| </div> | |
| <div> | |
| <span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label=" | |
| \begin{cases} | |
| a_1x+b_1y+c_1z=d_1 \\ | |
| a_2x+b_2y+c_2z=d_2 \\ | |
| a_3x+b_3y+c_3z=d_3 | |
| \end{cases} | |
| "><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo" style="vertical-align: -0.571em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.667em; padding-bottom: 0.298em; margin-bottom: -0.149em;">⎧</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">⎨</span><span class="mjx-char MJXc-TeX-size4-R" style="margin-top: -0.366em; padding-bottom: 1.182em;">⎩</span></span></span><span class="mjx-mtable" style="vertical-align: -1.495em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.13em;"><span class="mjx-mtd" style="padding: 0px 0px 0px 0px; text-align: left; width: 8.781em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.003em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.23em;"><span class="mjx-mtd" style="padding: 0.1em 0px 0px 0px; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.003em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.13em;"><span class="mjx-mtd" style="padding: 0.1em 0px 0px 0px; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.003em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-mo" style="width: 0.12em;"></span></span></span></span></span></span> | |
| </div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="\left\{\begin{aligned} a_1x+b_1y+c_1z&=d_1+e_1 \\ | |
| a_2x+b_2y&=d_2 \\ | |
| a_3x+b_3y+c_3z&=d_3 \end{aligned} \right. "><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo" style="vertical-align: -0.671em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.667em; padding-bottom: 0.298em;">⎧</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.001em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">⎨</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.001em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="margin-top: -0.217em; padding-bottom: 1.182em;">⎩</span></span></span><span class="mjx-mtable" style="vertical-align: -1.595em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.18em;"><span class="mjx-mtd" style="padding: 0px 0px 0px 0px; text-align: right; width: 6.539em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0px; text-align: left; width: 4.32em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mi"></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.003em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.33em;"><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mi"></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.003em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-strut"></span></span></span></span><span class="mjx-mtr" style="height: 1.18em;"><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: right;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0.15em 0px 0px 0px; text-align: left;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mi"></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.003em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span><span class="mjx-mo" style="width: 0.12em;"></span></span></span></span></span></span></div> | |
| <div><span class="mjpage mjpage__block"><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label=" | |
| \left\{\begin{array}{ll}a_1x+b_1y+c_1z &=d_1+e_1 \\ a_2x+b_2y &=d_2 \\ a_3x+b_3y+c_3z &=d_3 \end{array} \right. | |
| "><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mrow"><span class="mjx-mo" style="vertical-align: -0.771em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.667em; padding-bottom: 0.298em;">⎧</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.101em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.961em; padding-bottom: 0.961em;">⎨</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.101em; margin-bottom: 0.087em; margin-top: -0.137em;">⎪</span><span class="mjx-char MJXc-TeX-size4-R" style="margin-top: -0.217em; padding-bottom: 1.182em;">⎩</span></span></span><span class="mjx-mtable" style="vertical-align: -1.695em; padding: 0px 0.167em;"><span class="mjx-table"><span class="mjx-mtr" style="height: 1.23em;"><span class="mjx-mtd" style="padding: 0px 0.5em 0px 0px; text-align: left; width: 6.539em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em; padding-right: 0.006em;">y</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.003em;">z</span></span><span class="mjx-strut"></span></span></span><span class="mjx-mtd" style="padding: 0px 0px 0px 0.5em; text-align: left; width: 4.042em;"><span class="mjx-mrow" style="margin-top: -0.2em;"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.003em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372 |