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 is what’s called a “mathematical constant”.
This means, it’s a number that appears in a surprisingly varied number of places. It’s a number that the *universe* has chosen as important, and not humanity. And given how interesting the number is, that only makes it all the more incredible that a precise a number as pi is can appear everywhere, even when you’re not looking for it.
It most commonly is used to represent the ratio of the circumference of a circle to its diameter. It is a non-repeating, non-terminating decimal (basically, it goes on seemingly infinitely) that has been calculated to over 31 trillion digits, making it one of the most well-known and fascinating mathematical constants.
As to why it’s important
1. It is a fundamental constant in mathematics: Pi appears in many mathematical formulas and equations in various fields of mathematics, such as geometry, trigonometry, calculus, and number theory. It is used to calculate the area, volume, and circumference of circles, spheres, and cylinders, among other shapes. Knowing Pi is key to understanding these things.
2. It has practical applications: Pi is used in many real-world applications, including engineering, physics, and astronomy. For example, it is used in calculations for designing bridges, buildings, and other structures, as well as in satellite and space navigation systems.
3. It has cultural significance: Pi has been studied and celebrated by many cultures throughout history. It has been represented in art, literature, music, and even in the design of buildings and monuments. Pi Day, which is celebrated on March 14th (3/14) every year, has become a popular celebration of math and science.
4. It is a challenging and fascinating mathematical concept: Calculating the digits of Pi has been a challenge for mathematicians throughout history, and it continues to be a topic of research and interest today. The quest to find the exact value of Pi has led to the development of new mathematical tools and techniques, and has inspired many people to pursue careers in mathematics and science.
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))
Pi is what’s called a “mathematical constant”.
This means, it’s a number that appears in a surprisingly varied number of places. It’s a number that the *universe* has chosen as important, and not humanity. And given how interesting the number is, that only makes it all the more incredible that a precise a number as pi is can appear everywhere, even when you’re not looking for it.
It most commonly is used to represent the ratio of the circumference of a circle to its diameter. It is a non-repeating, non-terminating decimal (basically, it goes on seemingly infinitely) that has been calculated to over 31 trillion digits, making it one of the most well-known and fascinating mathematical constants.
As to why it’s important
1. It is a fundamental constant in mathematics: Pi appears in many mathematical formulas and equations in various fields of mathematics, such as geometry, trigonometry, calculus, and number theory. It is used to calculate the area, volume, and circumference of circles, spheres, and cylinders, among other shapes. Knowing Pi is key to understanding these things.
2. It has practical applications: Pi is used in many real-world applications, including engineering, physics, and astronomy. For example, it is used in calculations for designing bridges, buildings, and other structures, as well as in satellite and space navigation systems.
3. It has cultural significance: Pi has been studied and celebrated by many cultures throughout history. It has been represented in art, literature, music, and even in the design of buildings and monuments. Pi Day, which is celebrated on March 14th (3/14) every year, has become a popular celebration of math and science.
4. It is a challenging and fascinating mathematical concept: Calculating the digits of Pi has been a challenge for mathematicians throughout history, and it continues to be a topic of research and interest today. The quest to find the exact value of Pi has led to the development of new mathematical tools and techniques, and has inspired many people to pursue careers in mathematics and science.
The [Wikipedia article](https://en.wikipedia.org/wiki/Pi) is very thorough on the history and range of mathematical abstractions to which pi has been applied. Specific to the normal distribution, there is a section in the article that doesn’t fully simply explain it. Apparently [Gauss reasoned his way into discovering the normal distribution function](https://en.wikipedia.org/wiki/Normal_distribution#History). But for it to be a probability distribution, the sum of all probabilities — the area under the curve — has to be 1. To make sure of that, we take our bare function, calculate the area under the curve (the integral), and so then the final Gaussian probability distribution is Gauss’s reasoned function shape divided by its area that we calculated. This process, by the way, is called “normalization”. (Do you think there are there enough uses of the word “normal” in math and stats? I suppose a psychologist would agree that nothing’s ever normal enough.)
So why the pi? The shape of the bell curve that Gauss reasoned-out was e^(-x^2). Finding the integral, as the Wikipedia article notes, requires a [change of variables — into *polar coordinates*](https://en.wikipedia.org/wiki/Gaussian_integral#By_polar_coordinates), which gives us our circle from which the pi drops out in the end. (A more detailed explanation requires knowing the basics of calculus — that can be provided/linked on request.)
In physics you see pi a lot — like almost everywhere. Often that’s because in physics we begin by turning everything into a circle or sphere (which of course has plenty of pi) — the reason is because then, unlike with a box, we can work in 2, 3, or more dimensions while only ever using a single variable: the radius.
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.
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))
The [Wikipedia article](https://en.wikipedia.org/wiki/Pi) is very thorough on the history and range of mathematical abstractions to which pi has been applied. Specific to the normal distribution, there is a section in the article that doesn’t fully simply explain it. Apparently [Gauss reasoned his way into discovering the normal distribution function](https://en.wikipedia.org/wiki/Normal_distribution#History). But for it to be a probability distribution, the sum of all probabilities — the area under the curve — has to be 1. To make sure of that, we take our bare function, calculate the area under the curve (the integral), and so then the final Gaussian probability distribution is Gauss’s reasoned function shape divided by its area that we calculated. This process, by the way, is called “normalization”. (Do you think there are there enough uses of the word “normal” in math and stats? I suppose a psychologist would agree that nothing’s ever normal enough.)
So why the pi? The shape of the bell curve that Gauss reasoned-out was e^(-x^2). Finding the integral, as the Wikipedia article notes, requires a [change of variables — into *polar coordinates*](https://en.wikipedia.org/wiki/Gaussian_integral#By_polar_coordinates), which gives us our circle from which the pi drops out in the end. (A more detailed explanation requires knowing the basics of calculus — that can be provided/linked on request.)
In physics you see pi a lot — like almost everywhere. Often that’s because in physics we begin by turning everything into a circle or sphere (which of course has plenty of pi) — the reason is because then, unlike with a box, we can work in 2, 3, or more dimensions while only ever using a single variable: the radius.
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.
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.
The [Wikipedia article](https://en.wikipedia.org/wiki/Pi) is very thorough on the history and range of mathematical abstractions to which pi has been applied. Specific to the normal distribution, there is a section in the article that doesn’t fully simply explain it. Apparently [Gauss reasoned his way into discovering the normal distribution function](https://en.wikipedia.org/wiki/Normal_distribution#History). But for it to be a probability distribution, the sum of all probabilities — the area under the curve — has to be 1. To make sure of that, we take our bare function, calculate the area under the curve (the integral), and so then the final Gaussian probability distribution is Gauss’s reasoned function shape divided by its area that we calculated. This process, by the way, is called “normalization”. (Do you think there are there enough uses of the word “normal” in math and stats? I suppose a psychologist would agree that nothing’s ever normal enough.)
So why the pi? The shape of the bell curve that Gauss reasoned-out was e^(-x^2). Finding the integral, as the Wikipedia article notes, requires a [change of variables — into *polar coordinates*](https://en.wikipedia.org/wiki/Gaussian_integral#By_polar_coordinates), which gives us our circle from which the pi drops out in the end. (A more detailed explanation requires knowing the basics of calculus — that can be provided/linked on request.)
In physics you see pi a lot — like almost everywhere. Often that’s because in physics we begin by turning everything into a circle or sphere (which of course has plenty of pi) — the reason is because then, unlike with a box, we can work in 2, 3, or more dimensions while only ever using a single variable: the radius.
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