Tagged Questions
6
votes
2answers
98 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
96 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
360 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 ...
7
votes
6answers
264 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 ...
2
votes
4answers
295 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 ...
4
votes
1answer
131 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 ...
21
votes
5answers
1k 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? ...
8
votes
1answer
229 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?
10
votes
2answers
720 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
217 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
151 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
626 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 ...
20
votes
3answers
2k 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 ...
52
votes
3answers
2k 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 ...
3
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 ...
9
votes
6answers
2k 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 ...
