Root 2 is irrational — proof by prime factors

There are several proofs that the square root of 2 is irrational. The earliest dates back to Euclid, around 300 BC. In this article, we will look at an alternative proof, based on prime factors.

This proof has several advantages:

• It is simpler.
• It can be easily applied to other numbers.
• It provides a good explanation of why perfect squares are a special case.
• It can be easily applied to higher powers.
• It can also be used to prove the stronger statement that any integer root of an integer value is either an integer or irrational.

Proof by contradiction

We will use a proof by contradiction. We will start by assuming that the square root of 2 is rational, and prove that this leads to a contradiction, so root 2 cannot be rational.

If root 2 was rational, then by definition, there must be some positive integers a and b for which:

Squaring both sides gives:

So for root 2 to be rational, there must be positive integer values a and b such that:

If we can prove that no such integer values can exist, then we have proved that the root of 2 cannot be rational and therefore must be irrational.

Prime factors

According to the fundamental theorem of arithmetic, every positive integer greater than 1…