Prove that $\sqrt{5}+1$ is irrational

Question

I'm guessing this is pretty basic,

but I've been struggling with this for a while now.

Any help would be appreciated.

Thanks.

Answer 1

HINT: Use proof by contradiction: Assume that $\sqrt{5} +1$ is rational.

Then $\sqrt{5} +1=\cfrac{p}{q}$ where $p\in \mathbb{Z}$ and $q\in \mathbb{N^{+}}$.

Answer 2

Suppose $\sqrt{5}+1 = r$ for some rational $r$. The set of all rationals with the usual addition and multiplication is a field, implying that $r-1$ is rational; this shows that $\sqrt{5}$ is rational. Then by well-ordering principle there is a least integer $a$ such that $a\sqrt{5}$ is an integer; but $(\sqrt{5}-2)a < a$ is an integer such that $[(\sqrt{5}-2)a]\sqrt{5}$ is an integer, a contradiction; therefore, the number $\sqrt{5}$ is irrational.