A Proof that Square Root of 2 Plus Square Root of 3 is Irrational

Introduction

In the previous post, we have learned that $latex \sqrt{6}$ is irrational.  In this post, we will use that theorem to show that $latex \sqrt{2} + \sqrt{3}$ is also irrational. This proof was explained by Ivan Niven in his book Numbers: Rational and Irrational.

A common argument that might arise is that since $latex \sqrt{2}$ is irrational and $latex \sqrt{3}$ is irrational, then their sum is irrational. Like multiplication, the set of irrational numbers is not closed under addition. The sum of two irrational numbers is not necessarily irrational. For instance, $latex \sqrt{2}$ is irrational, $latex -\sqrt{2}$ is irrational, but $latex \sqrt{2} + (-\sqrt{2}) = 0$ is rational.

Theorem

$latex \sqrt{2} + \sqrt{3}$ is irrational.  Continue reading