The Story of Pi (π)

Transcript

The Story of Pi (π) Pi is the constant RATIO of a circle's CIRCUMFERENCE to its DIAMETER. π=c/d CIRCUMFERENCE/DIAMETER (1) PIIS A CONSTANT NUMBER: Its ratio is the same for all circles. (2) COMMON APPROXIMATIONS: (3) PI IS IRRATIONAL: The decimals do not repeat or end. 3.1415926535 ...How did we find the value of 𝝅? 𝜋 is the ratio of a circle's circumference to its diameter. For centuries, mathematicians tried to find a perfect fraction for it. Ancient Approximations The great Indian mathematician Äryabhața (499 CE) got incredibly close with the fraction 𝟑𝟗𝟐𝟕/𝟏𝟐𝟓𝟎, which equals 3.1416 . However, he was a genius who recognized this was only an āsanna (an approximation), hinting that a perfect fraction might be impossible to find. The Proof Much later, in 1761, a mathematician named Lambert finally proved formally what Āryabhața suspected: 𝝅 is irrational. No single fraction will ever perfectly equal 𝝅. Mādhava's Breakthrough So, how do you write an exact formula for an irrational number if you can't use a fraction? In the 14th century, Mādhava of Sangamagrama unlocked the secret: you use an infinite series. He discovered that you can calculate 𝜋 by adding and subtracting an endless sequence of fractions that follow a specific pattern (using odd numbers in the denominator): 𝝅=𝟒 ×(𝟏−𝟏/𝟑+𝟏/𝟓−𝟏/𝟕+…)