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.
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.