## Wednesday, 6 October 2021

### Is it true that square root of any positive integer greater than one, which is not a perfect square, is an irrational number?

This question was raised during a discussion with first year degree students.  The motivation was the proof showing the  irrationality of  $\sqrt{2}$.  The discussion took us to the following 'one line' proof: If $n$ is such a number having the rational expression $\sqrt{n}=\frac{a}{b}, \gcd(a,b)=1$, then by the fundamental theorem of arithmetic (unique prime factorization)  $b=1$ and $n$ is a square.