Prove $a + b\sqrt{2}$ is irrational

Question

Suppose that a and b are non-zero rational numbers. How can I show that $a+b√2$ is not a rational number. You may assume that $√2$ is not a rational number.

I thought that finding contradictions in $a+b√2=m/n$ might work, but I can't figure out how.

Answer 1

HINT: Your idea of using proof by contradiction is fine. To try that, assume that $a+b\sqrt2$ is rational. This means that there are integers $m$ and $n$ such that $$a+b\sqrt2=\frac{m}n\;.$$ Solve this equation for $\sqrt2$ in terms of $a,b,m$, and $n$. Do you see what the contradiction is going to be? If not, the conclusion of the argument is spoiler-protected below; mouse-over to see it.

We have $\sqrt2=\frac1b\left(\frac{m}n-a\right)$. The difference of two rationals is rational, and $\frac1b$ is rational since $b$ is rational and not zero, so $\sqrt2$ is the product of two rational numbers and is therefore rational. This of course contradicts the known irrationality of $\sqrt2$.