why Pi is important?

Question

I understand the mathematical definition of Pi, but why does it end up being used in so many formulas and applications in math, engineering, physics, etc? What does it unlock?

Edit: I understand Pi is the ratio of circumference to diameter. But why is that fact make it important and useful. For example it shows up in the equation for standard normal distribution. What does Pi have to do with a normal distribution. That’s just one example.

In: 3

Answer 1

The [Wikipedia article](https://en.wikipedia.org/wiki/Pi) is very thorough on the history and range of mathematical abstractions to which pi has been applied. Specific to the normal distribution, there is a section in the article that doesn’t fully simply explain it. Apparently [Gauss reasoned his way into discovering the normal distribution function](https://en.wikipedia.org/wiki/Normal_distribution#History). But for it to be a probability distribution, the sum of all probabilities — the area under the curve — has to be 1. To make sure of that, we take our bare function, calculate the area under the curve (the integral), and so then the final Gaussian probability distribution is Gauss’s reasoned function shape divided by its area that we calculated. This process, by the way, is called “normalization”. (Do you think there are there enough uses of the word “normal” in math and stats? I suppose a psychologist would agree that nothing’s ever normal enough.)

So why the pi? The shape of the bell curve that Gauss reasoned-out was e^(-x^2). Finding the integral, as the Wikipedia article notes, requires a [change of variables — into *polar coordinates*](https://en.wikipedia.org/wiki/Gaussian_integral#By_polar_coordinates), which gives us our circle from which the pi drops out in the end. (A more detailed explanation requires knowing the basics of calculus — that can be provided/linked on request.)

In physics you see pi a lot — like almost everywhere. Often that’s because in physics we begin by turning everything into a circle or sphere (which of course has plenty of pi) — the reason is because then, unlike with a box, we can work in 2, 3, or more dimensions while only ever using a single variable: the radius.