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.
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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))
It’s a number that shows up frequently in many different places.
It shows up frequently in many different formulae because it is related to circles, and circles and spheres are related to lots of other things, sometimes in not-so obvious ways.
Anything related to angles and rotations for example is obviously related to circles, but it turns out that powers in imaginary numbers are also related to rotations, which means that pi shows ups wherever complex numbers do. Spheres have something to do with how energy diffuses or radiates in space, so pi shows up in a lot of physics. Diffusion, in turn, has something to do with normal distributions, so pi shows up in statistics. Circles are also related to waves, so pi shows up in formulae about waves.
3blue1brown is a great YouTube channel for this. Basically, pi is like the definitive property of a circle. You can’t have a circle without pi being there.
But the reverse is pretty much always true: you can’t have pi without a circle hiding somewhere in the background. If an equation features pi or a problem has pi in the answer, there’s almost always a way to represent some part of the question with a circle. And since a circle is the most simple geometric shape (you can draw one with a pencil and a piece of string!), circles appear everywhere, therefore so does pi.
It’s a number that shows up frequently in many different places.
It shows up frequently in many different formulae because it is related to circles, and circles and spheres are related to lots of other things, sometimes in not-so obvious ways.
Anything related to angles and rotations for example is obviously related to circles, but it turns out that powers in imaginary numbers are also related to rotations, which means that pi shows ups wherever complex numbers do. Spheres have something to do with how energy diffuses or radiates in space, so pi shows up in a lot of physics. Diffusion, in turn, has something to do with normal distributions, so pi shows up in statistics. Circles are also related to waves, so pi shows up in formulae about waves.
It’s a number that shows up frequently in many different places.
It shows up frequently in many different formulae because it is related to circles, and circles and spheres are related to lots of other things, sometimes in not-so obvious ways.
Anything related to angles and rotations for example is obviously related to circles, but it turns out that powers in imaginary numbers are also related to rotations, which means that pi shows ups wherever complex numbers do. Spheres have something to do with how energy diffuses or radiates in space, so pi shows up in a lot of physics. Diffusion, in turn, has something to do with normal distributions, so pi shows up in statistics. Circles are also related to waves, so pi shows up in formulae about waves.
It’s not that pi in of itself is important. We didn’t just make it up. It’s that while solving a bunch of problems, you just happen to end up with pi. Especially in Calculus, you end up deriving pi all the time, anytime you’re working with circles or anything round, really.
It’s not that pi in of itself is important. We didn’t just make it up. It’s that while solving a bunch of problems, you just happen to end up with pi. Especially in Calculus, you end up deriving pi all the time, anytime you’re working with circles or anything round, really.
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