Prove under root 2 is irrational

Let us assume, to the contrary, that √2 is rational. So integers r and s ≠ 0 such that √2 =r/s. Suppose r and s have a common factor other than 1. Then we divide by the common factor to get √2=a/b, where a and b are coprime. So b√2=a. Squaring on both the sides and rearranging,2b²=a². Therefore,2 divides a². Now, by the theorem 1.3, 2 divides a. So, a=2c for some integer c. Substituting for a, we get 2b²=4c,i.e.,b²=2c². This means that 2 divides b² and so 2 divides b. Therefore,a and b have at least 2 as a common factor. But this contradicts the fact that a and b have no common factors other than 1. This contradiction has arisen because of our incorrect assumption that √2 is rational. So, we conclude that √2 is irrational.

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