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How to Read Mathematical Expressions Out Loud

## Basic

Expression How to Read This
$$a + b$$ a plus b
$$a - b$$ a minus b
$$a \times b$$ a times b
$$a / b$$ a over b
$$a &gt; b$$ a is greater than b
$$a &lt; b$$ a is less than b
$$a = b$$ a is equal to b
$$a \neq b$$ a is not equal to b
$$a \geq b$$ a is greater than or equal to b
$$a \leq b$$ a is less than or equal to b
$$x^2$$ x squared
$$x^3$$ x cubed
$$x^n$$ x to the power of n
$$\sqrt{x}$$ the square root of x
$$\sqrt[n]{x}$$ the n-th root of x
$$\log x$$ the logarithm of x (base 10)
$$\ln x$$ the natural logarithm of x (base e)
$$e^x$$ e to the power of x
$$|x|$$ the modulus of x
$$s(t)$$ s of t(s as a function of t)
$$f(x)$$ f of x

## Angle Operations

Expression How to Read This
$$\sin \theta$$ sine of theta
$$\cos \theta$$ cosine of theta
$$\tan \theta$$ tangent of theta
$$\csc \theta$$ cosecant of theta
$$\sec \theta$$ secant of theta
$$\cot \theta$$ cotangent of theta
$$\sin^{-1} x$$ arcsine of x or inverse sine of x
$$\cos^{-1} x$$ arccosine of x or inverse cosine of x
$$\tan^{-1} x$$ arctangent of x or inverse tangent of x
$$\sin(\alpha \pm \beta)$$ sine of (alpha plus or minus beta)
$$\cos(\alpha \pm \beta)$$ cosine of (alpha plus or minus beta)
$$\tan(\alpha \pm \beta)$$ tangent of (alpha plus or minus beta)
$$\theta \pm \phi$$ theta plus or minus phi
$$\theta \cdot \phi$$ theta times phi
$$\theta / \phi$$ theta over phi

## Logic

Expression How to Read This
$$a := b$$ - a is defined by(given as) b (colon equal)
$$\triangleq$$ delta equal, interchangeable with $:=$
(more of personal preference, tradition, or convention)
$$\doteq$$ is approximately equal to
$$\equiv$$ - is equivalent to
- is identically equal to
(identical under all circumstances)
$$\exists$$ there exists
$$\nexists$$ there does not exist
$$\forall$$ for all

## Sets

Expression How to Read This
$$S_1 \Rightarrow S_2$$ S sub one implies S sub two
$$S_1 \Leftrightarrow S_2$$ S sub one is equivalent to S sub two
$\{a, b, c\}$ the set of a, b, and c
$$a \in A$$ a is an element of set A
$\emptyset$ or $\{\}$ empty set
$$A \subset$$ B A is a proper subset of
$$A \subseteq B$$ A is a subset of B
$$A \cup B$$ A union B
$$A \cap B$$ A intersection B
$$A \setminus B$$ A complement of B
$$\exists p \in \mathbb{P}, p = 2$$ there exists a prime number p such that p equals two.
$$\forall x \in \mathbb{R}, x + 0 = x$$ for all x in the set of real numbers, x plus zero equals x.
$$\mathbb{N}$$ the set of all natural numbers
$$\mathbb{Z}$$ the set of all integers
$$\mathbb{Q}$$ the set of all rational numbers
$$\mathbb{R}$$ the set of all real numbers
$$\mathbb{C}$$ the set of all complex numbers
$$\mathbb{P}$$ the set of all prime numbers

## Intervals

Expression How to Read This
$$[a, b]$$ closed interval from a to b
$$(a, b)$$ open interval from a to b
$$[a, b)$$ the interval from a to b, inclusive of a and exclusive of b.
$$(a, b]$$ the interval from a to b, exclusive of a and inclusive of b.
$$(a, \infty)$$ the half line starting at a and extending to infinity, exclusive of a.
$$(-\infty, a)$$ the half line starting at negative infinity and extending to a, exclusive of a.

## Factorials and Binomial Coefficients

Expression How to Read This
$$n!$$ n factorial (the product of all positive integers up to n)
$\binom{n}{k}$ or $nC_k$ n choose k (the number of ways to choose k elements from a set of n elements)

## Statistics

Expression How to Read This
$$\overline{x}$$ x bar (the average of x)
$$\sigma$$ sigma (standard deviation)
$$\mu$$ mu (mean, sound ends with an "ew" as in "new")
$$r$$ r (correlation coefficient)
$$p$$ p-value (probability)
$$\chi^2$$ chi-squared (a statistical test)
$$\hat{p}$$ p-hat (sample proportion)
$$H_0$$ H-naught(null hypothesis)
$$H_a$$ H-a(alternative hypothesis)

## Sum and Product

Expression How to Read This
$$\sum_{n=1}^{\infty} a_n$$ the sum of a sub n from n equals 1 to infinity
$$\sum_{k=0}^n \binom{n}{k}$$ the sum from k equals zero to n of n choose k
$$\left(\sum_{k=1}^n a_k b_k\right)^2$$ the square of the sum of the products of a sub k and b sub k from k equals 1 to n
$$\prod_{k=1}^n k$$ the product from k equals one to n of k (n factorial)

## Functions

Expression How to Read This
$$\varphi$$ phi (variant)
$$\varphi(0) = 0$$ phi of zero equals zero
$$f : A \to B$$ - f is a function that maps from the set A to the set B
- f is a function from A to B(more succinct and standard way)
- the function f maps each element of set A to an element in set B(more explanatory)
$$x \mapsto f(x)$$ x is mapped to f(x)
$$g: \mathbb{R} \to \mathbb{C}, \quad \theta \mapsto g(\theta) := e^{i\theta}$$ g from R(real numbers) to C(complex numbers), theta maps to g of theta, defined as(given by) e to the i theta
$$f: \mathbb{R} \to \mathbb{R}, \quad x \mapsto f(x) := 1 + x^2$$ f from R(real numbers) to R, x maps to f of x, defined as(given by) one plus x squared
$$h: \mathbb{C} \to [0, \infty), \quad z \mapsto h(z) := |z|$$ function h maps complex numbers C to the set of non-negative real number from zero to infinity, z maps to function h of z equals the modulus of z
$$\text{im}(f)$$ image or range of f
$$\text{im}(f) = B$$ the image(range) of f equals B
(the image (or range) of the function f is equal to the set B)
$$f: A \to \text{im}(f)$$ f from A to the image of f
(the image of the function, includes all the outputs the function can produce as its input varies over the set A.)
$$f^{-1}$$ inverse function of f
$$f^{-1}(b) := a$$ f inverse of b is defined as(given by) a
$$f_{i,j}$$ f sub i comma j

## Calculus Operation Notations

Expression How to Read This
$$\frac{d}{dx}$$ d by dx (the derivation with respect to x)
$$\frac{d}{dx} f(x)$$ the derivative of f with respect to x
$$f'(x)$$ f prime of x (the derivative of f with respect to x)
$$\int f(x) dx$$ the integral of f with respect to x
$$\int_a^b f(x) dx$$ the definite integral of f from a to b with respect to x
$$\lim_{x \to a} f(x)$$ the limit of f as x approaches a

## Vector calculus and physics

Expression How to Read This
$$\nabla$$ nabla or del
$$\nabla f$$ gradient of(nabla) f
$$\nabla \cdot \mathbf{F}$$ divergence of(nabla dot) F
$$\nabla \times \mathbf{F}$$ curl of(nabla cross) F
$$\partial$$ partial
$$\frac{\partial}{\partial x} f(x)$$ the partial derivative of f with respect to x
$$\frac{\partial f}{\partial x}$$ partial of f with respect to x
$$\frac{\partial^2 f}{\partial x^2}$$ - the second partial derivative of f with respect to x squared
- the second partial of f with respect to x
$$\frac{\partial}{\partial x_j} \left( \frac{\partial f}{\partial x_i} \right)$$ The partial derivative with respect to x sub j of the partial derivative of f with respect to x sub i
$$\frac{\partial^2 f}{\partial x_j \partial x_i}$$ The second partial derivative of f with respect to x sub j and x sub i
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