# How to Prove That the Square Root of 2 Is Irrational

*I am a former maths teacher and owner of DoingMaths. I love writing about maths, its applications, and fun mathematical facts.*

## What Is an Irrational Number?

In mathematics, any real number can be described as either rational **or** irrational.

A rational number is any number that can be written as a fraction of two whole numbers. For example 2.5 can be written as 5/2, while 0.333333... can be written as 1/3, hence both of these numbers are rational.

Any real number that is not rational, hence can't be written as a fraction of whole numbers, is irrational. Famous examples of these include π, e and √2.

Something useful to note is that when written in decimal form, rational numbers either terminate or recur, while irrational numbers go on forever without repetition or pattern.

## Proof by Contradiction

We have mentioned that √2 is an irrational number, but how can we prove this?

In this article, we are going to use a proof by contradiction based on work by the Greek mathematician Euclid (mid-4th century BC). Proof by contradiction works by assuming the opposite of what you want to prove is true and then working through the mathematical steps until you come to a contradiction of your original assumption.

## Proving That Root 2 Is Irrational

Let's assume that √2 is rational and therefore can be written as a fraction in lowest terms p/q, where p and q are integers and q ≠ 0.

√2 = p/q

Square both sides

2 = p^{2}/q^{2}

Multiply both sides by q^{2}

2q^{2} = p^{2}

As p^{2} is equal to two times a whole number, it must be even. This further implies that p is even, hence can be written as 2n for some whole number n.

Therefore 2q^{2} = (2n)^{2}

2q^{2} = 4n^{2}

q^{2} = 2n^{2}

q^{2} is also two times a whole number, hence also even. Again, this implies that q is also even.

We have now shown that p and q are both even, but if this is true then p/q cannot be in its simplest form as we would be able to divide numerator and denominator by 2. We therefore have a contradiction with our original statement, hence this original statement is wrong and so √2 must be irrational.

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

**© 2021 David**

## Comments

**David (author)** from West Midlands, England on March 28, 2021:

Thank you

**Fozia Naz** from Pakistan on March 28, 2021:

Great work