Question

Prove that $\sqrt{5}$ is irrational. [4 MARKS]

Answer 1

Concept : 1 Mark
Application :1 Mark
Method : 2 Mark

Let us assume, to the contrary, that $\sqrt{5}$ is rational.

So, we can find co-prime integers a and b $(\ne 0)$ such that $\sqrt{5}=\frac{a}{b}$

$\Rightarrow \sqrt{5}b=a$ Squaring on both sides , we get

$5b^2=a^2$ Therefore, 5 divides $a^2$

Therefore, 5 divides a

So, we can write a=5c for some integer c.

Substituting for a, we get $5b^2=25c^2$ $\Rightarrow b^2=5c^2$

This means that 5 divides $b^2$, and so 5 divides b.

Therefore, a and b have at least 5 as a common factor.

But this is contradiction to the assumption that $\sqrt{5}$ is rational.

So, we conclude that $\sqrt{5}$ is irrational.