Question

is the number $-1 + \sqrt{5}$ rational or irrational? \
rational \
irrational

Explanation:

Step1: Recall rational/irrational definitions

A rational number can be expressed as $\frac{p}{q}$ ($p,q\in\mathbb{Z}, q
eq0$), irrational otherwise. $\sqrt{5}$ is irrational (since 5 is not a perfect square, so $\sqrt{5}$ cannot be written as a fraction of integers).

Step2: Analyze $-1 + \sqrt{5}$

The sum of a rational number ($-1$) and an irrational number ($\sqrt{5}$) is irrational. Because if $-1 + \sqrt{5}$ were rational, then $\sqrt{5}=(-1 + \sqrt{5})+1$ would be the sum of two rationals, hence rational, which contradicts $\sqrt{5}$ being irrational.

Answer:

irrational