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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> 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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