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”.
>What does Pi have to do with a normal distribution. That’s just one example.
Unfortunately this is relatively high level calculus and doesn’t explain easily at 5-year-old level. But suffice to say it involves trigonometry and the polar coordinate system. Polar coordinates have some very nice mathematical properties that make them useful for dealing with repeating functions, etc.
>What does Pi have to do with a normal distribution. That’s just one example.
Unfortunately this is relatively high level calculus and doesn’t explain easily at 5-year-old level. But suffice to say it involves trigonometry and the polar coordinate system. Polar coordinates have some very nice mathematical properties that make them useful for dealing with repeating functions, etc.
>What does Pi have to do with a normal distribution. That’s just one example.
Unfortunately this is relatively high level calculus and doesn’t explain easily at 5-year-old level. But suffice to say it involves trigonometry and the polar coordinate system. Polar coordinates have some very nice mathematical properties that make them useful for dealing with repeating functions, etc.
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.”
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.”
Pi isn’t important because of its definition (the ratio of a circle’s circumference to its diameter).
Pi is important because how much it pops up and how often it’s useful.
It’s irrational, infinite, statistically random, apparently normal, transcendent and more.
As it is related to a circle, it is found in MANY geometry, trigonometry formulae and higher-dimensional analysis.
Basically, it’s so important because of so many equations and theories cannot be solved without it.
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.”
many natural phenomena moves in circles. From spinning galaxies, planetary orbits all the way down to electrons in a atom. Many other phenomena also they move in a sinusoidal way/harmonic movement (Electrical signals, soundwaves, radiowaves/light, mechanical transmission gears, springs for example) and if you remove time from that motion you are left with circles.
Pi isn’t important because of its definition (the ratio of a circle’s circumference to its diameter).
Pi is important because how much it pops up and how often it’s useful.
It’s irrational, infinite, statistically random, apparently normal, transcendent and more.
As it is related to a circle, it is found in MANY geometry, trigonometry formulae and higher-dimensional analysis.
Basically, it’s so important because of so many equations and theories cannot be solved without it.
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