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

First, several trigonometric functions are simplified for multiples of pi. Trigonometric functions are widely used in most sciences.

Then, there’s Euler’s formula e^{ix} = cos(x) + i*sin(x). Note that e^{i*pi} = cos(pi) + i*sin(pi) = -1+ i*0 = -1. Add one to see e^{i*pi} + 1 = 0. This equation has (almost) every interesting number: pi, e, additive identity, and multiplicative identity. (The only one I would say is missing is the golden ratio.)

So, to simplify my point, it is the number that makes calculations of trigonometric functions “simpler.”