# 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 can be uniquely represented as a product of prime numbers. For example:

The theorem tells us two things. The first is that *any* positive number greater than 1 can be expressed as a product of one or more prime numbers. The second thing it tells us is that, for any particular number, there is *only one way* to do it. For instance, 45 can be expressed as 3 squared times 5, and there is no other way to express 45 as a product of primes.

The reason this only applies to numbers greater than 1 is that 1 itself is not a prime number.

For many numbers, the same prime number might appear as a factor more than once. For example, 1960 has three factors of 2, and two factors of 7. These can either be written as repeated multiplications or as powers (both…