Class 10 -Prove √5 is a Irrational number.

 

Prove √5 is a Irrational number.

Proof:

Let √5 be a rational number.

So, $\sqrt{5}=\frac{p}{q}$

On squaring both sides

$5=\frac{p^2}{q^2}$
$q^2=\frac{p^2}{5}$

$\Rightarrow 5$ is a factor of $p^2$

$\Rightarrow 5$ is a factor of p.

Now, again let p = 5c.

So, $\sqrt{5}=\frac{5c}{q}$

On squaring both sides
$5=\frac{25c^2}{q^2}$
$q^2=5c^2$
$c^2=\frac{q^2}{5}$
$\Rightarrow 5$ is factor of $q^2$
$\Rightarrow 5$ is a factor of q.

Here 5 is a common factor of p, q  

 So p, q are not co-prime.

Therefore our assumption is wrong. 

√$\sqrt{5}$ is an irrational number.