In honor of Pi day (3/14) and my increased fascination with mathematics as a whole, I decided to look under the hood and understand how the value of pi is calculated to be 3.14159… and write a program to do it.

Introducing: Pi-Py. Keep reading to learn more about the underlying process!

As it turns out, there are a number of approaches mathematicians have taken. For now, I chose to work with two methods: the Gregory-Leibniz series and the Nilakantha series. Here’s how they work.

Gregory-Leibniz

This method can accurately calculate $\pi$ to 5 decimal places after approximately 450,000 iterations. That’s a lot!

Using sigma notation:

$\sum\limits_{i=0}^\infty \frac{(-1)^n}{2n+1} = \frac{\pi}{4}$

This is basically fancy notation for a loop – a summation.

In this case, $i=0$ is the initial value, $\infty$ is the number of iterations, and $\frac{(-1)^n}{2n+1}$ is the body of the loop.

Another way of representing this, which is perhaps easier to understand more directly, is as follows:

$4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - ... = \pi$

(Note: I multiplied each number by 4 to get the value of $\pi$ rather than $\frac{\pi}{4}$ )

This can be easily computed in a loop by alternating addition and subtraction of odd numbers. View the code on GitHub to see how this was implemented!

Nilakantha

This method is considerably faster – by about 10,000 times, as a matter of fact. After only 45 iterations, $\pi&s=0&bg=00090a&fg=ffffff$ can be calculated to the same 5 decimal places as the Gregory-Leibniz method.

In the 15th Century, Indian astronomer/mathematician Nilakantha Somayaji published the following infinite series:

$3 + \frac{4}{2\times3\times4} - \frac{4}{4\times5\times6} + \frac{4}{6\times7\times8} - \frac{4}{8\times9\times10} + ... = \pi$

As the denominator value changes incrementally, this too can be easily computed using a simple loop.

View the code on GitHub to see how I implemented the Nilakantha infinite series!

The Nilakantha method converges at a much faster rate than the Gregory-Leibniz method, but there are even faster series which I have not yet studied or attempted to implement as life has kept me busy! Nevertheless, it has been an interesting journey, and I’ll be posting about other projects soon.

As a side-note, I had the chance to learn some LaTeX syntax as a result of writing this article, which really helped to make the material look presentable.