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 is important be cause it is a fundamental attribute of nature and very useful. Without PI it is impossible to comprehend or to calculate many simple things, like how a circuit works or electricity is generated or how elementary particles interact with each other, or even how to accurately design a wheel.
Pi is important be cause it is a fundamental attribute of nature and very useful. Without PI it is impossible to comprehend or to calculate many simple things, like how a circuit works or electricity is generated or how elementary particles interact with each other, or even how to accurately design a wheel.
So, Pi is, by definition, related to circles and therefore to spheres as well.
So anything circular, like planetary orbits, is also related to Pi.
And things that are cyclic, like [simple harmonic motion](https://en.wikipedia.org/wiki/Simple_harmonic_motion).
And a sphere is the collection of all points that are a certain distance from a central point. So anything that propagates equally in all directions (such as gravitational and electric fields) also deals with spheres, and therefore relate to Pi.
There are many more such examples.
So, Pi is, by definition, related to circles and therefore to spheres as well.
So anything circular, like planetary orbits, is also related to Pi.
And things that are cyclic, like [simple harmonic motion](https://en.wikipedia.org/wiki/Simple_harmonic_motion).
And a sphere is the collection of all points that are a certain distance from a central point. So anything that propagates equally in all directions (such as gravitational and electric fields) also deals with spheres, and therefore relate to Pi.
There are many more such examples.
So, Pi is, by definition, related to circles and therefore to spheres as well.
So anything circular, like planetary orbits, is also related to Pi.
And things that are cyclic, like [simple harmonic motion](https://en.wikipedia.org/wiki/Simple_harmonic_motion).
And a sphere is the collection of all points that are a certain distance from a central point. So anything that propagates equally in all directions (such as gravitational and electric fields) also deals with spheres, and therefore relate to Pi.
There are many more such examples.
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> I understand Pi is the ratio of circumference to diameter. But why is that fact make it important and useful.
Because the properties of circles are hidden everywhere in maths. Why ? Let’s look at the equation of a circle: x^2 + y^2 = 1
Said otherwise, if you draw the function y = sqrt(1-x^2 ) you will have an half-circle, and if you draw the function y = -sqrt(1-x^2 ) you will have the second half of the circle.
So as soon as you have squares somewhere, and that there is a notion of area or length, then a Pi might appear.
So let’s take the normal distribution, it’s formula is exp(-x^2 /2)/sqrt(2Pi). The reason why there is a Pi is that the normal distribution is defined as “I want a constant C multiplied by exp(-x^2 /2) so that the area under the curve is exactly equal to 1”, and since you’re talking about an area of something that has a square in it, it’s no surprise that a Pi might appear somewhere when trying to determine the perfect value for C.
And additionally, the notion of “area under the curve” appears everywhere in maths, because it’s linked to calculus and it in some sense linked to the opposite of “differentiating functions”.
> I understand Pi is the ratio of circumference to diameter. But why is that fact make it important and useful.
Because the properties of circles are hidden everywhere in maths. Why ? Let’s look at the equation of a circle: x^2 + y^2 = 1
Said otherwise, if you draw the function y = sqrt(1-x^2 ) you will have an half-circle, and if you draw the function y = -sqrt(1-x^2 ) you will have the second half of the circle.
So as soon as you have squares somewhere, and that there is a notion of area or length, then a Pi might appear.
So let’s take the normal distribution, it’s formula is exp(-x^2 /2)/sqrt(2Pi). The reason why there is a Pi is that the normal distribution is defined as “I want a constant C multiplied by exp(-x^2 /2) so that the area under the curve is exactly equal to 1”, and since you’re talking about an area of something that has a square in it, it’s no surprise that a Pi might appear somewhere when trying to determine the perfect value for C.
And additionally, the notion of “area under the curve” appears everywhere in maths, because it’s linked to calculus and it in some sense linked to the opposite of “differentiating functions”.
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