Root 2 is irrational — proof by prime factors

Martin McBride
8 min readMar 4, 2024

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…