How do we know the digits of pi?

Next Thursday is going to be Tau day. If you remember, there are many mathematicians that want the number tau, which is equal to 2π, to replace pi because of it being more natural and simple.

In honor of this special occasion, I decided to write a post about pi. I had always wondered how we know the digits of pi go on forever, and more importantly, how we figure them out.

Turns out, there is a formula. One of the common formulas is:

π = 4(1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 +...)

Let me show you how we can derive this. First off, I am going to ask a completely irrelevant question, but you will see its relevance in a moment. What is the arch-tangent of 1?

To figure this out, you would punch into your scientific calculator 1, and then the tanh button. You would get:

tanh(1) ≈ .76159...

But how did the calculator get the answer? It had to use a formula. It used the formula:

x^1/1 - x^3/3 + x^5/5 - x^7/7 + x^9/9 -...

Now, multiply this number by 4. The calculator approximated the arch-tangent, so you won't clearly see the pattern. However, the true answer is:

4tanh(1) = π

So, we plug one into the tangent formula, and we get:

1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 +...

Multiply that mess by four and we have pi.

4(1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 +...)

I found it really cool that pi has a formula this simple that mathematicians can use to calculate digits. This also proves that it doesn't terminate since the third, seventh, ninth, eleventh, and many more of them go on forever.