Happy 𝜋 Day everyone! Since it's 𝜋 day, I thought it would be nice to go over the number 𝜋 and why it is so important.
So what is 𝜋? 𝜋 is the ratio of the circumference of a circle to it's diameter; in any circle the ratio is the same. 𝜋 is equal to 3.1415923.1415926535897932384626433832795028841971693993751058209... yeah, it's a long number. 𝜋 is a special type of number: it's an irrational number. This means it cannot be expressed as either a decimal or a fraction. That's why we use the weird Greek symbol.
The earliest records of the number 𝜋 that we saw were from the Babylonians. While they never explicitly mentioned 𝜋 or 3.14, they had a formula for the area of circle. Since A=𝜋r², we can use their formula to find their value of 𝜋. They put it at 3; obviously not the best.
The next civilization to attempt finding a value of 𝜋 was Egypt. They got a lot closer, with a value of 3.1605 for 𝜋. Unlike the Babylonians, they overshot the value of 𝜋, but since the margin of error was a lot less, this was a marked improvement.
The next big step in finding the value of 𝜋 goes to the ancient Greeks, and it was a massive step. Archimedes, one of the greatest inventors of his time, came up with the first recorded method for calculating 𝜋 by inscribing a regular polygon inside of a circle and finding its area. He then circumscribed another regular polygon about the circle and found its area. By doing this, you could find the upper and lower bounds for the value of 𝜋. With some dedicated calculation, he found that the value of 𝜋 was in between 3 and 1/7(3.145287) and 3 and 10/71(3.14084507042), which is much closer to the original value than the Egyptian and Babylonian methods. The more sides the polygons had, the closer the values got to 𝜋. In fact, the digits which the upper and lower bounds shared were correct values of 𝜋, so Archimedes was the first person to calculate that 𝜋 started with 3.14.
This method was used for hundreds of years; in fact it was there until Newton came along. Newton, and independently Gottfried Leibniz, invented Calculus, which gave us infinite series. Some of these series led to 𝜋, such as the famous 1-⅓+⅕-⅐+...=𝜋/4. The more you added, the more accurate of an answer you got. This is much faster than attempting to find the area of a regular polygon with 9000 sides. The only problem with this formula is that it took more than 300 additions to get to an accuracy of 3.14! Faster and more accurate formulas were eventually found, and soon, nobody was attempting to find the area of a regular polygon with 1 million sides.
So why is 𝜋 everywhere? That is because circles are everywhere. circular and cyclical motion is everywhere, so 𝜋 is found everywhere. However, some formulas which have no relation whatsoever to circles contain 𝜋 in them. This is because 𝜋 is also present in trigonometry. When dealing with sine, cosine, and tangent, it is often much simpler to deal with radians than with degrees. 1 radian is a unit of angle, equal to an angle at the center of a circle whose arc is equal in length to the radius. Since circumference is 2𝜋r, the circumference of a circle is equal to 2𝜋r radians. This is why 𝜋 appears in almost every field of math and science.
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