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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