What is pi?

What is $\pi$?

The number $\pi$, which is approximately 3.14159, is the length of the circumference of a circle whose diameter is exactly 1. More generally, a circle of diameter d has a circumference $\pi d$. A simple approximation to $\pi$ is 22/7, but this is not exact. 22/7 is approximately 3.14285, which is wrong by the third decimal place. A better approximation is 355/113, which is 3.1415929 to seven places, whereas $\pi$ is 3.1415926.

How do we know that $\pi$ is not an exact fraction? However much you continue to improve the approximation x/y, by using ever larger numbers, you can never get to $\pi$ itself, only better and better approximations. A number that cannot be written exactly as a fraction is said to be irrational. The simplest proof that $\pi$ is irrational uses calculus, and it was found by Johann Lambert in 1770. Although we can’t write down an exact numerical representation of $\pi$, we can write down various formulas that define it precisely, and Lambert’s proof uses one of those formulas.

More strongly, $\pi$ is transcendental — it does not satisfy any algebraic equation that relates it to rational number. This was proved by Ferdinand Lindemann in 1882, also using calculus.

This fact that $\pi$ is transcendental implies that the classical geometric problem of ‘squaring the circle’ is impossible. This problem asks for a Euclidean construction of a square whose area is equal to that of a given circle (which turns out to be equivalent to constructing a line whose length is the circumference of the circle). A construction is called Euclidean if it can be performed using an unmarked ruler and a compass. Well, to be pedantic, a ‘pair of compasses’, which means a single instrument, much as a ‘pair of scissors’ comes as one gadget.

More later,

Nalin Pithwa

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