Prove that $(√5 - 1)/2$ is irrational.

Question

Please help me prove that $(√5 - 1)/2$ is irrational. I know how to prove √5 is irrational:

Assume that √5 is rational meaning

√5 = $p/q$ $p,q$ $are$ $Z$ $and$ $q≠0$

$p^2/q^2 = 5$

$q^2 = p^2/5$

Therefore, 5 must be a factor of p.

$Let$ $p = 5c$

$q^2 = (5c)^2/5$

$q^2 = 25c^2/5$

$c^2 = q^2/5$

Therefore, 5 must also be a factor of q. CONTRADICTION!

Thanks.

Answer 1

Let $\phi=\frac{\sqrt5-1}2$. Then rearranging gives$$\sqrt5=2\phi+1$$ If $\phi$ is rational, what does the above equality say about $\sqrt5$?

Answer 2

Try This : (If you know how to prove $\sqrt{5}$ is irrational)

Suppose that $\frac{\sqrt{5}-1}{2}$ is rational.

Then there are $m ,n\neq0 \in \mathbb Z $ such that

$$\frac{\sqrt{5}-1}{2}=\frac{m-n}{2n}$$

Thus we get that $\sqrt{5}=\frac{m}{n}$.

Now proceed the way that you used to prove that $\sqrt{5} $ is irrational.