9
votes
6answers
174 views

When the approximation $\pi\simeq 3.14$ is NOT sufficent

It's common at schools to use $3.14$ as an appropriate approximation of $\pi$. However, here it's stated that for some purposes, $\pi$ should be approximated to $32$ decimal places. I need an example ...
13
votes
0answers
341 views

On Shanks' quartic approximation $\pi \approx \frac{6}{\sqrt{3502}}\ln(2u)$

In Mathworld's "Pi Approximations", (line 58), Weisstein mentions one by the mathematician Daniel Shanks that differs by a mere $10^{-82}$, $$\pi \approx ...
10
votes
1answer
378 views

A strange “pattern” in the continued fraction convergents of pi?

From the simple continued fraction of $\pi$, one gets the convergents, $$p_n = \frac{3}{1}, \frac{22}{7}, \frac{333}{106}, \frac{355}{113}, \frac{103993}{33102}, \frac{104348}{33215}, ...
30
votes
7answers
1k views

Pi Estimation using Integers

I ran across this problem in a high school math competition: "You must use the integers $1$ to $9$ and only addition, subtraction, multiplication, division, and exponentiation to approximate the ...
6
votes
2answers
121 views

Get minimum N in approximation of pi

The following expression is an approximation of PI, where N determines the precision. $$\pi (N) = \frac{4}{N} \sum_{i=1}^{N}\frac{1}{1 +\left ( \frac{i -\frac{1}{2}}{N} \right )^{2}}$$ If I want to ...
2
votes
2answers
122 views

Approximating $\pi$ in Binary

I am interested in creating a Java program that generates digits of $\pi$ (in Binary though). To be clear, the number I'm looking for begins: $11.00100100 \dots$ I am unsure of the most efficient way ...
2
votes
2answers
409 views

Where is the error on this approximation to $\pi$

I came across this website (see also) where the author (a supposedly alternative mathematician) claims to have a better approximation to $\pi$. $$\pi\approx 3.1547…$$ Can someone tell me where is ...
8
votes
6answers
285 views

Approximating $\pi$ with least digits

Do you a digit efficient way to approximate $\pi$? I mean representing many digits of $\pi$ using only a few numeric digits and some sort of equation. Maybe mathematical operations also count as ...
3
votes
4answers
319 views

What is the correct value of $\pi$

I have seen that: $\pi = 22/7$ $\pi = 3.14\ldots$ $\pi = 17 - \sqrt{192}$. $22/7 \gt \pi$ $22/7 \lt\pi$ My brain storming doubt was, is A = B? Is B = C? Is C= A = B? how D and E are correct or ...
5
votes
1answer
157 views

Why does $\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$

I read on Wikipedia that $$\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$$ to a high degree of accuracy. Why is this true? Is this pure coincidence or is there some mathematical ...
23
votes
5answers
3k views

How do you calculate the decimal expansion of an irrational number?

Just curious, how do you calculate an irrational number? Take $\pi$ for example. Computers have calculated $\pi$ to the millionth digit and beyond. What formula/method do they use to figure this out? ...
9
votes
1answer
262 views

Why is $10\frac{\exp(\pi)-\log 3}{\log 2}$ almost an integer?

I read that $$10\frac{\exp(\pi)-\log 3}{\log 2} =318.000000033252\dots \approx 318$$ Is this simply a coincidence or can this somehow be explained?
12
votes
2answers
2k views

Euler's Approximation of pi.

I recently stumbled across the formula: $$\pi=20\arctan\frac{1}{7}+8\arctan\frac{3}{79}$$ developed by Euler, for approximating pi. I evaluated it to several thousand decimal places and up to that ...
2
votes
2answers
251 views

$\pi$ polynomials whose real zeros approximate $\pi$

Let's have the following polynomials $x^4+105x^2-1134=0$,$x^6+126^x4+10395x^2-115830=0$, $3x^8+550x^6+45045x^4+3378375x^2-38288250=0$ The positive real zeros of these equations are good ...
1
vote
1answer
309 views

How to select a sensible tolerance when making approximations in MATLAB

I'm approximating $\pi$ using a series in MATLAB. I can approximate to within a relative error of $3\times 10^{-10}$ of MATLAB's built-in value. How would I choose a sensible tolerance for my ...
2
votes
4answers
915 views

Approximating $\pi$ using Monte Carlo integration

I need to estimate $\pi$ using the following integration: $$\int_{0}^{1} \!\sqrt{1-x^2} \ dx$$ using monte carlo Any help would be greatly appreciated, please note that I'm a student trying to ...
40
votes
4answers
3k views

Motivation for Ramanujan's mysterious $\pi$ formula

The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or ...
61
votes
3answers
3k views

Is there an integral that proves $\pi > 333/106$?

The following integral, $$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$ is clearly positive, which proves that $\pi < 22/7$. Is there a similar integral which proves ...
4
votes
4answers
2k views

Why is $22/7$ a better approximation for $\pi$ than $3.14$?

This seems counterintuitive, but $22/7$ is closer to $\pi$ than $3.14=314/100$ which has a significantly greater denominator. Why is $22/7$ a better approximation for $\pi$ than $3.14$? This has ...
14
votes
6answers
8k views

Simple numerical methods for calculating the digits of $\pi$

Are there any simple methods for calculating the digits of $\pi$? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that ...