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How to Prove That the Square Root of 2 Is Irrational

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

The Square Root of Two and Its Decimal Form

The Square Root of Two and Its Decimal Form

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.

Statue of Euclid, Oxford University Museum of Natural History, Oxford, UK

Statue of Euclid, Oxford University Museum of Natural History, Oxford, UK

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 = p2/q2

Multiply both sides by q2

2q2 = p2

As p2 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 2q2 = (2n)2

2q2 = 4n2

q2 = 2n2

q2 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


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

Thank you

Fozia Naz from Pakistan on March 28, 2021:

Great work