# JOST A MON

Mar 26, 2008

## Irrationality

I have often wondered why it was that the same Pythagorean proof for the irrationality of the square root of 2 was inflicted on students. Surely there were other proofs? The Wikipedia article on this shows three in all: the usual one, a slight variant, and a geometric construction.

Recently, though, I came across another one 1. It is also based on reduction ad absurdum, i.e., a proof by contradiction. It is due to T. Estermann (1902 - 1991), who claimed it to be the first new proof since Pythagoras.

First, establish the fact that $\inline \sqrt{2}$ exceeds $\inline 1$.

This is not difficult, as you can show that $\inline {1.4}^{2}=1.96$ and $\inline {1.5}^{2}=2.25$, and so $\inline \sqrt{2}$ lies between $\inline 1.4$ and $\inline 1.5$.

Clearly, then, the square root of 2 is not a whole number.

Consider the set S of all positive integers n such that $\inline \sqrt{2}n$ is also an integer.

Now, if S is not empty, then it has a least member. Why is that? Because any non-empty set of integers can be sorted in ascending order, and will therefore have a least element. Let this element be k. Then it follows that $\inline \sqrt{2}k$ is an integer.

Construct the number $\inline u=\left(\sqrt{2}-1 \right)k$.

What can we say about u?

1. u is smaller than k

2. u is positive, and it is an integer because $\inline \sqrt{2}k$ is an integer, subtracting k from which results in yet another integer.

3. Because $\inline \sqrt{2}u=2k-\sqrt{2}k$, we have that $\inline \sqrt{2}u$ is also a positive integer.

Therefore:

4. u is a member of S.

But this is a contradiction! We had that k was the smallest element of S, and now we have shown that u, which is smaller than k is also a member of S.

So our assumption that S is non-empty is false. In other words, S is empty.

In which case there is no positive integer n such that $\inline \sqrt{2}n$ is also an integer.

Therefore $\inline \sqrt{2}$ is not a fraction.

In other words, it is irrational.

Oooo, sir, as one might say.

Reference

1. J. Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics