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christopherphan/pi_day_2024.ipynb

Last active March 14, 2024 04:02
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Pi Day 2024
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pi_day_2024.ipynb
{
"cells": [
{
"cell_type": "markdown",
"id": "56e701a6-d86b-41ab-b1e3-6cad4e140ab9",
"metadata": {},
"source": [
"# Happy π Day, 2024\n",
"\n",
"By [Christopher Phan](https://chrisphan.com/), 2024-03-14 / 2024-W11-4\n",
"\n",
"Happy π Day! It's March 14, i.e. 3/14 using US date conventions, or 03-14 using [ISO 8601](https://en.wikipedia.org/wiki/ISO_8601) (or [RFC 3339](https://www.rfc-editor.org/rfc/rfc3339), if you prefer) and omitting the year. Back on π Day 2021, I made [a JavaScript demo of a Monte Carlo method](https://chrisphan.com/pi_day_2021/) to approximate π. To celebrate this wonderful geek holiday in 2024, I will present another silly way to calculate an approximation for this wonderful irrational number.\n",
"\n",
"## About this document\n",
"\n",
"This document is a [Jupyter notebook](https://jupyter.org), which means that it includes cells with computer code and the output. For example:"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "8c0b8a24-d2b7-4f56-965a-32eda40870ea",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The value of π is approximately 3.14159."
]
}
],
"source": [
"# This is computer code\n",
"\n",
"using Printf\n",
"@printf \"The value of π is approximately %0.5f.\" pi\n",
"\n",
"# Just below is the output"
]
},
{
"cell_type": "markdown",
"id": "81695580-49ee-401c-9d8b-2d44df4d14cd",
"metadata": {},
"source": [
"A copy of this notebook is available [on GitHub](https://gist.github.com/christopherphan/6ebf27191c79ea5ff96b6016e5ab019e).",
"\n",
"## Julia\n",
"\n",
"In this notebook, I am using the language [Julia](https://julialang.org), because it is relatively easy to work with, has just-in-time compilation, and has built-in arbitrary precision numbers. (Also fun fact, the \"Ju\" in \"Jupyter\" stands for Julia. The other parts stand for Python and R.) \n",
"\n",
"The arbitrary precision numbers will help. The default in Julia (and many other languages) is to store integers as having a fixed number of bits, which makes computation faster. For example, `42` is a signed 64-bit integer."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "2f4537e0-4a41-46d7-9ef1-b8fb48bf2813",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Int64"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"typeof(42)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "35d9de8c-b1fc-4474-84dc-f5e17713dee8",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1! = 1\n",
"2! = 2\n",
"3! = 6\n",
"4! = 24\n",
"5! = 120\n",
"6! = 720\n",
"7! = 5040\n",
"8! = 40320\n",
"9! = 362880\n",
"10! = 3628800\n",
"11! = 39916800\n",
"12! = 479001600\n",
"13! = 6227020800\n",
"14! = 87178291200\n",
"15! = 1307674368000\n",
"16! = 20922789888000\n",
"17! = 355687428096000\n",
"18! = 6402373705728000\n",
"19! = 121645100408832000\n",
"20! = 2432902008176640000\n",
"21! = -4249290049419214848\n"
]
}
],
"source": [
"# calculation of 21!\n",
"a = 1\n",
"for k in range(1, 21)\n",
" a = a * k\n",
" @printf \"%i! = %i\\n\" k a\n",
"end"
]
},
{
"cell_type": "markdown",
"id": "27b1a21d-0764-47c0-9678-513d0da59085",
"metadata": {},
"source": [
"Note that $21!$ is shown to be negative! This is because of how 64-bit signed integers are represented internally. Basically, we ran out of room in the 64 bits to calculate $20! \\times 21$. The solution is to use \"big\" integers, which can have as many bits as necessary. (The trade-off is that they are a bit slower.)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "a97d4146-6583-44ae-b361-33fce2fd8000",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"BigInt"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"typeof(big(21))"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "6d06401f-f38a-4127-bb53-d9b6e01158b5",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1! = 1\n",
"2! = 2\n",
"3! = 6\n",
"4! = 24\n",
"5! = 120\n",
"6! = 720\n",
"7! = 5040\n",
"8! = 40320\n",
"9! = 362880\n",
"10! = 3628800\n",
"11! = 39916800\n",
"12! = 479001600\n",
"13! = 6227020800\n",
"14! = 87178291200\n",
"15! = 1307674368000\n",
"16! = 20922789888000\n",
"17! = 355687428096000\n",
"18! = 6402373705728000\n",
"19! = 121645100408832000\n",
"20! = 2432902008176640000\n",
"21! = 51090942171709440000\n"
]
}
],
"source": [
"# correct calculation of 21!\n",
"a = big(1)\n",
"for k in range(1, 21)\n",
" a = a * k\n",
" @printf \"%i! = %i\\n\" k a\n",
"end"
]
},
{
"cell_type": "markdown",
"id": "ec341222-84fa-40af-92fc-e425f33534f9",
"metadata": {},
"source": [
"Julia has a built-in `factorial` function:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "a6f49e9a-74ce-4fbd-8675-fb5499d5cd04",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"51090942171709440000"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"factorial(big(21))"
]
},
{
"cell_type": "markdown",
"id": "d189f2e8-4897-4cb0-bfb6-bb36ebe313b6",
"metadata": {},
"source": [
"As it happens, I don't have much experience writing code in Julia (I have mainly used Python, Rust, and Javascript), so this a good excuse to practice my Julia skills.\n",
"\n",
"## Cosine (Taylor's version)\n",
"\n",
"For this exercise, I will use the Taylor series of cosine centered at zero. Note that\n",
"$$\\frac{d^n}{dx^n} \\cos x = \\begin{cases} (-1)^{(n + 1)/2} \\sin x &\\text{ if $n$ is odd,}\\cr (-1)^{n/2} \\cos x &\\text{ if $n$ is even.}\\end{cases}$$\n",
"At $x = 0$, we have\n",
"$$\\left. \\frac{d^n}{dx^n} \\cos x \\right|_{x = 0} = \\begin{cases} 0 &\\text{ if $n$ is odd,}\\cr (-1)^{n/2} &\\text{ if $n$ is even.}\\end{cases}$$\n",
"Therefore, \"Taylor's version\" of cosine is\n",
"$$\\cos x = \\sum_{n = 0}^\\infty \\left. \\frac{d^n}{dx^n} \\cos x \\right |_{x = 0} \\frac{x^n}{n!} = \\sum_{n = 0}^\\infty (-1)^n \\frac{x^{2n}}{(2n)!}.$$\n",
"\n",
"(I really wish I was was clever enough to have come up with this \"Taylor's version\" joke, but alas, it's just something I saw on social media.)\n",
"\n",
"We'll set $$T_n(x) = \\frac{x^{2n}}{(2n)!}$$\n",
"and\n",
"$$C_n(t) = \\sum_{k = 0}^{n-1} (-1)^k T_k(x),$$\n",
"which is a (Taylor polynomial) approximation of $\\cos x$.\n",
"\n",
"We can implement this in Julia:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "555ee033-a5fc-432c-b2d4-f1e8a0bff04c",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"partial_cos (generic function with 1 method)"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Note: functions automatically return last value\n",
"\n",
"function partial_cos_term(n, x) # T_n(x)\n",
" x^(2 * n)//factorial(2 * n)\n",
"end \n",
"\n",
"# NOTE: Julia's range(0, N) returns values 0 through N inclusive\n",
"# (unlike Python)\n",
"\n",
"function partial_cos(n, x) # C_n(x)\n",
" sum([(-1)^(k) * partial_cos_term(k, x) for k in range(0, n - 1)]) \n",
"end"
]
},
{
"cell_type": "markdown",
"id": "b68d53fc-3b36-4c86-b232-66883628f06e",
"metadata": {},
"source": [
"The `//` operator in Julia represents exact (rational) division, e.g."
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "2f17ad83-806a-4bf9-99a3-a6de8f91fd33",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"7//6"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"1//2 + 2//3"
]
},
{
"cell_type": "markdown",
"id": "f6cb8098-36c6-40c4-a5d9-fb5d7e85f315",
"metadata": {},
"source": [
"## Squeezing $\\cos x$\n",
"\n",
"(The material in this section was inspired by an argument that appears on p. 3 of _Real and Complex Analysis, 3rd Edition_ by Walter Rudin (McGraw Hill 1987), often affectionaltely known as \"Papa Rudin\". Fun fact: Rudin's house is architectually signifcant enough to have its own [article in Wikipedia](https://en.wikipedia.org/wiki/Walter_Rudin_House).) \n",
"\n",
"Note that if $0 < x < 2k + 1$, then $$x^{2k + 2} < (2k + 1)(2k + 2) x^{2k}.$$\n",
"\n",
"So, $$T_{k + 1}(x) = \\frac{x^{2k + 2}}{(2k + 2)!} < \\frac{(2k + 1)(2k + 2)x^{2k}}{(2k + 2)!} = T_k(x).$$\n",
"\n",
"This gives us some inequalities for $C_n(x)$: For any $n \\geq 1$ and $0 < x \\leq 2n + 1$, we have\n",
"$$C_{2n + 2}(x) = C_{2n}(x) + T_{2n}(x) - T_{2n + 1}(x) > C_{2n}(t)$$\n",
"and\n",
"$$C_{2n + 3}(x) = C_{2n + 1}(x) - T_{2n + 1}(x) + T_{2n + 2}(x) < C_{2n + 1}(x).$$\n",
"\n",
"Therefore, for every $0 < x \\leq 2$, we have\n",
"$$C_2(x) < C_4(x) < C_6(x) < \\dots < \\cos t < \\dots < C_7(x) < C_5(x) < C_3(x).$$\n",
"\n",
"We can illustrate this with some plots:"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "4c343efb-0ae5-4944-85dd-f0b7f4d7d79c",
"metadata": {},
"outputs": [
{
"data": {
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]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Plots\n",
"using LaTeXStrings\n",
"\n",
"x = [2 * big(x) // 100 for x in range(1, 100)]\n",
"cosx = cos.(float.(x)) # func. applies func elementwise\n",
"y = [\n",
" log10.(float.(partial_cos.(big(3), x)) - cosx), \n",
" log10.(float.(partial_cos.(big(5), x)) - cosx),\n",
" log10.(float.(partial_cos.(big(7), x)) - cosx),\n",
" log10.(float.(partial_cos.(big(9), x)) - cosx),\n",
"]\n",
"plot(x, y, \n",
" title=\"Plot of \\$\\\\log_{10}(C_n(x) - \\\\cos(x))\\$ for odd \\$n\\$\",\n",
" label=[\"n = 3\" \"n = 5\" \"n = 7\" \"n = 9\"]\n",
")"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "e8686940-fd90-4045-9c7a-a27c8aee84c9",
"metadata": {},
"outputs": [
{
"data": {
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]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"y = [\n",
" log10.(cosx - float.(partial_cos.(big(2), x))),\n",
" log10.(cosx - float.(partial_cos.(big(4), x))), \n",
" log10.(cosx - float.(partial_cos.(big(6), x))),\n",
" log10.(cosx - float.(partial_cos.(big(8), x))), \n",
"]\n",
"plot(x, y, \n",
" title=\"Plot of \\$\\\\log_{10}(\\\\cos(x) - C_n(x))\\$ for even \\$n\\$\",\n",
" label=[\"n = 2\" \"n = 4\" \"n = 6\" \"n = 8\"]\n",
")"
]
},
{
"cell_type": "markdown",
"id": "4397aa79-c328-47a4-a62b-ad116105b4a1",
"metadata": {},
"source": [
"Furthermore, remember that for any real $x$, $\\lim_{n \\rightarrow \\infty} C_n(x) = \\cos x$. Therefore, for any real $0 < x \\leq 2$:\n",
"\n",
"* If $\\cos x > 0$, then there exists an integer $N > 0$ such that $C_{2n}(x) > 0$ for $n \\geq N$.\n",
"* If $\\cos x < 0$, then there exists an integer $N > 0$ such that $C_{2n + 1}(x) < 0$ for $n \\geq N$.\n",
"\n",
"## Approximating $\\pi/2$ means finding approximate solutions to $\\cos x = 0$ where $1 \\leq x \\leq 2$\n",
"\n",
"What does this have to do with approximating $\\pi$? Well, one definition of $\\pi$ (e.g. in the aforementioned Papa Rudin) is that $\\pi$ is the smallest positive real number such that $\\cos(\\pi/2) = 0$. \n",
"\n",
"Note that $\\cos 0 = 1$, and that when $0 < x \\leq 1$, we have $$0 < 1 - \\frac{x^2}{2} = C_2(x) < \\cos x,$$\n",
"and thus, $\\cos x > 0$ when $0 \\leq x \\leq 1$.\n",
"\n",
"Therefore, $\\pi/2 > 1$.\n",
"\n",
"Now, $-\\sin x < 0$ when $0 < x < \\pi$, so that means $\\cos$ is strictly decreasing on the interval $[1, 2]$. As \n",
"$$\\cos 2 < C_3(2) = 1 - \\frac{4}{2} + \\frac{16}{24} = -\\frac{1}{3} < 0,$$ we know that $\\cos$ has a unique zero on $[1, 2]$, i.e. $1 \\leq \\pi/2 \\leq 2$.\n",
"\n",
"## A way to tighten bounds\n",
"\n",
"So far, we've narrowed down the value of $\\pi/2$ to the interval $[1, 2]$. How do we make a bound better.\n",
"\n",
"For $1 \\leq a < b \\leq 2$ and $n \\geq 1$, let $P(n, a, b)$ be the statement \n",
"\n",
"> $C_{2n}(a) > 0$ and $C_{2n + 1}(b) < 0$. \n",
"\n",
"Note that:\n",
"\n",
"* We showed above that $P(1, 1, 2)$ is true.\n",
"* $P(n, a, b)$ implies $\\cos a > 0$ and $\\cos b < 0$, i.e. that $a < \\pi/2 < b$.\n",
"\n",
"Now, assume that $P(n, a, b)$ is true for some *rational* $1 \\leq a < b \\leq 2$ and $n \\geq 1$. Let $c = (a + b)/2$. Exactly one of the following conditions is true:\n",
"\n",
"1. $C_{2n + 3}(c) < 0$. Then $P(n + 1, a, c)$ is true.\n",
"2. $C_{2n + 3}(c) > 0$ and $C_{2n + 2}(c) > 0$. Then $P(n + 1, c, b)$ is true.\n",
"3. $C_{2n + 3}(c) > 0$ and $C_{2n + 2}(c) < 0$. In this case, $P(n + 1, a, b)$ is true.\n",
"\n",
"In cases (1) and (2), we've sucessfully cut in half the possible values of $\\pi/2$. Case (3) seems like let-down, but remember that for $n$ sufficently large, either $C_{2n} > 0$ or $C_{2n + 1} < 0$. So by incrementing $n$, we will eventually land in (1) or (2).\n",
"\n",
"By repeatedly iterating this process, starting with $n = 1$, $a = 1$ and $b = 2$, we can find $\\pi/2$ to any desired acccuracy.\n",
"\n",
"Here is the Julia implementation of this procedure:"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "18d2c97c-9d33-4d73-85fb-5d73fbce729e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"narrow (generic function with 1 method)"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function narrow(n, a, b)\n",
" c = (a + b) // 2\n",
" if partial_cos(2 * big(n) + 3, big(c)) < 0\n",
" (a, c)\n",
" elseif partial_cos(2 * big(n) + 1, big(c)) > 0\n",
" (c, b)\n",
" else\n",
" (a, b)\n",
" end\n",
"end"
]
},
{
"cell_type": "markdown",
"id": "9d83d929-ec56-4528-9847-7a5ffd4ac839",
"metadata": {},
"source": [
"## Working through the first few steps\n",
"\n",
"Let's work through the first few steps. We start with $P(1, 1, 2)$."
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "614511d9-e2ac-41ea-965b-52a3c69f833b",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(3//2, 2)"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"narrow(1, 1, 2)"
]
},
{
"cell_type": "markdown",
"id": "a950cb02-e069-4fe6-9a35-67a6b67ac649",
"metadata": {},
"source": [
"So, we know that $P(2, 3/2, 2)$ is true. We've narrowed the value of $\\pi/2$ to $[3/2, 2]$. We repeat:"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "9e7d160d-f7ab-477d-8635-08a1fd5462e6",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(3//2, 7//4)"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"narrow(2, 3//2, 2)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "f1ec3edc-d56f-4411-b18d-d20a36be2825",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(3//2, 13//8)"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"narrow(3, 3//2, 7//4)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "10f117bd-bac5-4baf-b0d6-b092d6b9ea7f",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(25//16, 13//8)"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"narrow(4, 3//2, 13//8)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "9356b228-3aaf-4ee9-8f4f-bf1ce99f7af8",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(25//16, 51//32)"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"narrow(5, 25//16, 13//8)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "f5a9c9f9-4932-4cb5-8d32-3adca720fc07",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(25//16, 101//64)"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"narrow(6, 25//16, 51//32)"
]
},
{
"cell_type": "markdown",
"id": "d0fcd057-f499-4ea2-bcae-4125aa3ff9b2",
"metadata": {},
"source": [
"## Approximating $\\pi$\n",
"\n",
"Now it's time to automate this.\n",
"\n",
"One last thing before seeing the conclusion: when we determine that $P(n, a, b)$ is true, then we are approximating $\\pi/2 \\approx (a + b)/2$. But this is the same thing as approximating $\\pi \\approx a + b$. \n",
"\n",
"I will use some apply `narrow` repeatedly to show some $P(80, u, v)$. I will use some Julia code to make a table showing, for each $P(n, a, b)$ obtained:\n",
"\n",
"1. $n$\n",
"2. The numerator of the rational approximation $a + b$ of $\\pi$\n",
"3. The denominator of the rational approximation $a + b$ of $\\pi$\n",
"4. A decimal representation of $a + b$\n",
"5. $\\log_{10}(b - a)$, which gives you an idea how many decimal places of precision we have achieved."
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "410ca733-e759-4cca-bc0e-f1ea44e4f088",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"| n | numerator | denominator | decimal | log_10(b-a) |\n",
"| ---: | -----------------------: | -----------------------: | :------------------------ | :---------- |\n",
"| 2 | 3 | 1 | 3.00 | 0.00000 |\n",
"| 3 | 7 | 2 | 3.50 | -0.30103 |\n",
"| 4 | 13 | 4 | 3.25 | -0.60206 |\n",
"| 5 | 25 | 8 | 3.12 | -0.90309 |\n",
"| 6 | 51 | 16 | 3.19 | -1.20412 |\n",
"| 7 | 101 | 32 | 3.16 | -1.50515 |\n",
"| 8 | 201 | 64 | 3.14 | -1.80618 |\n",
"| 9 | 403 | 128 | 3.15 | -2.10721 |\n",
"| 10 | 805 | 256 | 3.14 | -2.40824 |\n",
"| 11 | 1609 | 512 | 3.14 | -2.70927 |\n",
"| 12 | 3217 | 1024 | 3.14 | -3.01030 |\n",
"| 13 | 6433 | 2048 | 3.141 | -3.31133 |\n",
"| 14 | 12867 | 4096 | 3.141 | -3.61236 |\n",
"| 15 | 25735 | 8192 | 3.141 | -3.91339 |\n",
"| 16 | 51471 | 16384 | 3.142 | -4.21442 |\n",
"| 17 | 102943 | 32768 | 3.1416 | -4.51545 |\n",
"| 18 | 205887 | 65536 | 3.1416 | -4.81648 |\n",
"| 19 | 411775 | 131072 | 3.1416 | -5.11751 |\n",
"| 20 | 823549 | 262144 | 3.14159 | -5.41854 |\n",
"| 21 | 1647099 | 524288 | 3.14159 | -5.71957 |\n",
"| 22 | 3294199 | 1048576 | 3.14159 | -6.02060 |\n",
"| 23 | 6588397 | 2097152 | 3.141593 | -6.32163 |\n",
"| 24 | 13176795 | 4194304 | 3.141593 | -6.62266 |\n",
"| 25 | 26353589 | 8388608 | 3.141593 | -6.92369 |\n",
"| 26 | 52707179 | 16777216 | 3.141593 | -7.22472 |\n",
"| 27 | 105414357 | 33554432 | 3.1415927 | -7.52575 |\n",
"| 28 | 210828715 | 67108864 | 3.1415927 | -7.82678 |\n",
"| 29 | 421657429 | 134217728 | 3.1415927 | -8.12781 |\n",
"| 30 | 843314857 | 268435456 | 3.14159266 | -8.42884 |\n",
"| 31 | 1686629713 | 536870912 | 3.14159265 | -8.72987 |\n",
"| 32 | 3373259427 | 1073741824 | 3.14159265 | -9.03090 |\n",
"| 33 | 6746518853 | 2147483648 | 3.141592654 | -9.33193 |\n",
"| 34 | 13493037705 | 4294967296 | 3.141592654 | -9.63296 |\n",
"| 35 | 26986075409 | 8589934592 | 3.141592654 | -9.93399 |\n",
"| 36 | 53972150819 | 17179869184 | 3.141592654 | -10.23502 |\n",
"| 37 | 107944301637 | 34359738368 | 3.1415926536 | -10.53605 |\n",
"| 38 | 215888603273 | 68719476736 | 3.1415926536 | -10.83708 |\n",
"| 39 | 431777206545 | 137438953472 | 3.1415926536 | -11.13811 |\n",
"| 40 | 863554413089 | 274877906944 | 3.14159265359 | -11.43914 |\n",
"| 41 | 1727108826179 | 549755813888 | 3.14159265359 | -11.74017 |\n",
"| 42 | 3454217652357 | 1099511627776 | 3.14159265359 | -12.04120 |\n",
"| 43 | 6908435304715 | 2199023255552 | 3.141592653590 | -12.34223 |\n",
"| 44 | 13816870609431 | 4398046511104 | 3.141592653590 | -12.64326 |\n",
"| 45 | 27633741218861 | 8796093022208 | 3.141592653590 | -12.94429 |\n",
"| 46 | 55267482437723 | 17592186044416 | 3.141592653590 | -13.24532 |\n",
"| 47 | 110534964875445 | 35184372088832 | 3.1415926535898 | -13.54635 |\n",
"| 48 | 221069929750889 | 70368744177664 | 3.1415926535898 | -13.84738 |\n",
"| 49 | 442139859501777 | 140737488355328 | 3.1415926535898 | -14.14841 |\n",
"| 50 | 884279719003555 | 281474976710656 | 3.14159265358979 | -14.44944 |\n",
"| 51 | 1768559438007111 | 562949953421312 | 3.14159265358979 | -14.75047 |\n",
"| 52 | 3537118876014221 | 1125899906842624 | 3.14159265358979 | -15.05150 |\n",
"| 53 | 7074237752028441 | 2251799813685248 | 3.141592653589794 | -15.35253 |\n",
"| 54 | 14148475504056881 | 4503599627370496 | 3.141592653589793 | -15.65356 |\n",
"| 55 | 28296951008113761 | 9007199254740992 | 3.141592653589793 | -15.95459 |\n",
"| 56 | 56593902016227523 | 18014398509481984 | 3.141592653589793 | -16.25562 |\n",
"| 57 | 113187804032455045 | 36028797018963968 | 3.1415926535897933 | -16.55665 |\n",
"| 58 | 226375608064910089 | 72057594037927936 | 3.1415926535897932 | -16.85768 |\n",
"| 59 | 452751216129820177 | 144115188075855872 | 3.1415926535897932 | -17.15871 |\n",
"| 60 | 905502432259640355 | 288230376151711744 | 3.14159265358979324 | -17.45974 |\n",
"| 61 | 1811004864519280711 | 576460752303423488 | 3.14159265358979324 | -17.76077 |\n",
"| 62 | 3622009729038561421 | 1152921504606846976 | 3.14159265358979324 | -18.06180 |\n",
"| 63 | 7244019458077122843 | 2305843009213693952 | 3.141592653589793239 | -18.36283 |\n",
"| 64 | 14488038916154245685 | 4611686018427387904 | 3.141592653589793239 | -18.66386 |\n",
"| 65 | 28976077832308491369 | 9223372036854775808 | 3.141592653589793238 | -18.96489 |\n",
"| 66 | 57952155664616982739 | 18446744073709551616 | 3.141592653589793238 | -19.26592 |\n",
"| 67 | 115904311329233965479 | 36893488147419103232 | 3.1415926535897932385 | -19.56695 |\n",
"| 68 | 231808622658467930957 | 73786976294838206464 | 3.1415926535897932385 | -19.86798 |\n",
"| 69 | 463617245316935861913 | 147573952589676412928 | 3.1415926535897932385 | -20.16901 |\n",
"| 70 | 927234490633871723825 | 295147905179352825856 | 3.14159265358979323846 | -20.47004 |\n",
"| 71 | 1854468981267743447651 | 590295810358705651712 | 3.14159265358979323846 | -20.77107 |\n",
"| 72 | 3708937962535486895301 | 1180591620717411303424 | 3.14159265358979323846 | -21.07210 |\n",
"| 73 | 7417875925070973790601 | 2361183241434822606848 | 3.141592653589793238462 | -21.37313 |\n",
"| 74 | 14835751850141947581203 | 4722366482869645213696 | 3.141592653589793238463 | -21.67416 |\n",
"| 75 | 29671503700283895162407 | 9444732965739290427392 | 3.141592653589793238463 | -21.97519 |\n",
"| 76 | 59343007400567790324813 | 18889465931478580854784 | 3.141592653589793238463 | -22.27622 |\n",
"| 77 | 118686014801135580649625 | 37778931862957161709568 | 3.1415926535897932384626 | -22.57725 |\n",
"| 78 | 237372029602271161299249 | 75557863725914323419136 | 3.1415926535897932384626 | -22.87828 |\n",
"| 79 | 474744059204542322598499 | 151115727451828646838272 | 3.1415926535897932384626 | -23.17931 |\n",
"| 80 | 949488118409084645196999 | 302231454903657293676544 | 3.14159265358979323846265 | -23.48034 |\n"
]
}
],
"source": [
"largest_k = big(80)\n",
"\n",
"# create an array of strings\n",
"table = Array{String, 2}(undef, largest_k + 1, 5)\n",
"\n",
"# create header row\n",
"# Everything is 1-indexed in Julia. \n",
"# It messes me up sometimes!\n",
"table[1, 1] = \"n\"\n",
"table[1, 2] = \"numerator\"\n",
"table[1, 3] = \"denominator\"\n",
"table[1, 4] = \"decimal\"\n",
"table[1, 5] = \"log_10(b-a)\"\n",
"\n",
"for k in range(1, 5)\n",
" table[2, k] = \"----\"\n",
"end\n",
"\n",
"# fill in the numbers in the table\n",
"lower = big(1)\n",
"upper = big(2)\n",
"k = big(2)\n",
"while k < largest_k + 1\n",
" pi_midpt = lower + upper # approximation for pi\n",
" pi_flt = float(pi_midpt) # as a float\n",
" lg_err = log10(float(upper - lower))\n",
" # OK, so because (lower, upper) is an interval for pi/2\n",
" # (2 * lower, 2 * upper) is an interval for pi\n",
" # Hence |lower + upper - pi| < upper - lower.\n",
" # Let err = upper - lower.\n",
" # Now, to guarantee |lower + upper - pi| to be precise\n",
" # to N digits, we need:\n",
" # err < 0.5 * 10^(-N)\n",
" # 2*err < 10^(-N)\n",
" # log10(2) + log10(err) < -N\n",
" # N < -log10(2) - log10(err) \n",
" # \n",
" # So, we'll display the larger of \n",
" # floor(-log10(2) - log10(err)) or 2 digits\n",
" # past the decimal point.\n",
" num_digits = max(Int(floor(-log10(2) - lg_err)), 2)\n",
"\n",
" # populate table row\n",
" table[k + 1, 1] = @sprintf(\"%2i\", k)\n",
" table[k + 1, 2] = @sprintf(\"%i\", numerator(pi_midpt))\n",
" table[k + 1, 3] = @sprintf(\"%i\", denominator(pi_midpt))\n",
" table[k + 1, 4] = @sprintf(\"%.*f\", num_digits, pi_flt)\n",
" table[k + 1, 5] = @sprintf(\"%.5f\", lg_err)\n",
" k += 1\n",
" if k <= largest_k # otherwise the loop is ending\n",
" (lower, upper) = narrow(k, lower, upper)\n",
" end\n",
"end\n",
"\n",
"# determine the maximum width of the text in each column\n",
"max_widths = [\n",
" maximum([length(table[k, j]) \n",
" for k in range(1, largest_k + 1)]) for j in range(1, 5)\n",
"]\n",
"\n",
"# the second row of the table will have entries like \n",
"# \"-------:\" or \":-------\" (markdown syntax\n",
"# for right and left justification, respectively)\n",
"for k in range(1, 3)\n",
" # String concatenation is represented by * in Julia, not +\n",
" # This makes WAY more sense; ask your friendly local algebraist why.\n",
" table[2, k] = \"-\"^(max_widths[k] - 1) * \":\"\n",
"end\n",
"for k in range(4, 5)\n",
" table[2, k] = \":\" * \"-\"^(max_widths[k] - 1)\n",
"end\n",
"for k in range(1, largest_k + 1)\n",
" # We use some printf spacing goodness to make everything\n",
" # look really nice\n",
" for j in range(1, 3)\n",
" @printf \"| %*s \" max_widths[j] table[k, j] \n",
" end\n",
" for j in range(4, 5)\n",
" @printf \"| %-*s \" max_widths[j] table[k, j]\n",
" end\n",
" print(\"|\\n\")\n",
"end"
]
},
{
"cell_type": "markdown",
"id": "654dcf4b-54e5-4681-86de-d8ca03347214",
"metadata": {},
"source": [
"The approximation of $\\pi$ we found is $\\pi \\approx 3.14159265358979323846265$.\n",
"\n",
"## Thank you\n",
"\n",
"I hope you have enjoyed this &pi; Day journey. I hope you are able to celebrate &pi; Day with a yummy treat as well."
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.10.2",
"language": "julia",
"name": "julia-1.10"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.10.2"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
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