prove root 6 is irrational

Introduction

We have learned that  is irrational and thousands of sites on the internet will show you the proof that  is irrational. In this post, we show that  is irrational. This is a prerequisite for the next post which will show that  is irrational.

You will probably argue that since  is irrational and  is irrational, then their product should be irrational. This is not always the case. The product of two irrational numbers is not always irrational.  For instance,  is irrational but  is rational. And now, we proceed the the theorem and its proof.

Theorem

 is irrational.

Proof

To prove the theorem above, we use proof by contradiction.

Assuming that  is rational. Then,  where   and  are integers and  is in lowest terms. This means that  and cannot be both even (Why?) (*)

Squaring both sides, we have .

Multiplying both sides by , we have . It follows that  is even and   is even. (**)

If  is even, then it can be expressed as  where  is an integer. Substituting to the equation above, we have which simplifies to

.

Dividing both sides by  gives . This implies that  is even which means that  is even (Why?). (***)

From *, we assumed that  and  cannot be both even. But from ** and ***  and  are even. A contradiction!

Therefore,  is irrational.