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
pi helps us understand circles, and things that oscillate/repeat.
As it turns out, nature is filled to the brim with things that are circles, or things that oscillate/repeat. So pi helps with setting up equations for those phenomena.
(Wrt the normal distribution, it’s just a normalization factor. This is veering out of ELI5 territory, but laymen don’t know the eq for a normal distribution anyways. You want the entire integral over a normal distribution to be equal to 1.
So you take the normal exp(-ax^(2)) function for a normal distribution, and multiply it by an unkown constant (K). integrate K*exp(-ax^(2)) from negative to positive infinity, you get K*sqrt(pi/a), which must be equal to 1. So that means that K is equal to sqrt(a/pi). (The pi comes rolling out of the integral because the exp() function is another one of those tied to osscilations. These things turn up A LOT))
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