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 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.
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.
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.
What do we have to change about our universe so that Pi is no longer everywhere?
The universal laws of our universe is uniform, meaning that physics is the same everywhere in our universe. At least, that’s our assumption, and as far as we can observe, that is the case. Part if this observation is that, there is no special “direction” — information propagate at the same speed in all directions.
With this in mind, let’s look at what happens when anything happens in our universe, what rule governs how information about that event propagates outwards? That’s right, since information travels at the same speed in all directions, it forms a sphere ( in 3D universe), with the origin point at the center, which means Pi is involved.
To get rid of Pi, we can try to imagine what would it take so information propagates outwards in a cube? How about making it so that the speed of light is faster along a magical universal set of 6 cardinal directions (up, down, front, back, left, right), and gets slower linearly…
What do we have to change about our universe so that Pi is no longer everywhere?
The universal laws of our universe is uniform, meaning that physics is the same everywhere in our universe. At least, that’s our assumption, and as far as we can observe, that is the case. Part if this observation is that, there is no special “direction” — information propagate at the same speed in all directions.
With this in mind, let’s look at what happens when anything happens in our universe, what rule governs how information about that event propagates outwards? That’s right, since information travels at the same speed in all directions, it forms a sphere ( in 3D universe), with the origin point at the center, which means Pi is involved.
To get rid of Pi, we can try to imagine what would it take so information propagates outwards in a cube? How about making it so that the speed of light is faster along a magical universal set of 6 cardinal directions (up, down, front, back, left, right), and gets slower linearly…
What do we have to change about our universe so that Pi is no longer everywhere?
The universal laws of our universe is uniform, meaning that physics is the same everywhere in our universe. At least, that’s our assumption, and as far as we can observe, that is the case. Part if this observation is that, there is no special “direction” — information propagate at the same speed in all directions.
With this in mind, let’s look at what happens when anything happens in our universe, what rule governs how information about that event propagates outwards? That’s right, since information travels at the same speed in all directions, it forms a sphere ( in 3D universe), with the origin point at the center, which means Pi is involved.
To get rid of Pi, we can try to imagine what would it take so information propagates outwards in a cube? How about making it so that the speed of light is faster along a magical universal set of 6 cardinal directions (up, down, front, back, left, right), and gets slower linearly…
It is, essentially, the conversion factor between polar and rectilinear (Cartesian) systems, the constant that is needed to convert an arcuate (circles and friends) shape into a linear output. An awful lot of math revolves around a choice of coordinates where each axis is perpendicular to the others. Circles don’t really fit perpendicular coordinates, so to deal with the effects of a circle in a rectilinear system, we have to do a conversion. Pi is the constant required to make that conversion.
Describing a circle in x-y-z space is hard, but it is also hard to describe a rectangle in polar (magnitude, direction) coordinates. Whichever system you choose, there will be shapes that are hard to describe without a means to convert, and pi is part of that means, the constant that is used to make the conversion.
It is, essentially, the conversion factor between polar and rectilinear (Cartesian) systems, the constant that is needed to convert an arcuate (circles and friends) shape into a linear output. An awful lot of math revolves around a choice of coordinates where each axis is perpendicular to the others. Circles don’t really fit perpendicular coordinates, so to deal with the effects of a circle in a rectilinear system, we have to do a conversion. Pi is the constant required to make that conversion.
Describing a circle in x-y-z space is hard, but it is also hard to describe a rectangle in polar (magnitude, direction) coordinates. Whichever system you choose, there will be shapes that are hard to describe without a means to convert, and pi is part of that means, the constant that is used to make the conversion.
It is, essentially, the conversion factor between polar and rectilinear (Cartesian) systems, the constant that is needed to convert an arcuate (circles and friends) shape into a linear output. An awful lot of math revolves around a choice of coordinates where each axis is perpendicular to the others. Circles don’t really fit perpendicular coordinates, so to deal with the effects of a circle in a rectilinear system, we have to do a conversion. Pi is the constant required to make that conversion.
Describing a circle in x-y-z space is hard, but it is also hard to describe a rectangle in polar (magnitude, direction) coordinates. Whichever system you choose, there will be shapes that are hard to describe without a means to convert, and pi is part of that means, the constant that is used to make the conversion.
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.
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