Question

√5 is an irrational number​

Answer 1

Rational number :

● The number which can be represented in the form of $\dfrac{p}{q}$ is basically termed as Rational number.

For example- $\dfrac{3}{2} , \dfrac{7}{9}$

Irrational number :

● The number which cannot be represented in the form of $\dfrac{p}{q}$ is basically termed as irrational number.

☆ $\sqrt{5}$ is not a rational number, so it is a irrational number.

Answer 2

Let √5 be a rational number in the form of p/q which is at its simplest form.

So, √5 = p/q

5 = p²/q²

5q² = p²

Hence, 5 is factor of p²

and 5 is factor of p.

Now let m be any natural number.

5m = p

Squaring  both the sides we get,

5²m² = p²

25m² = 5q²

5m² = q²

Thus, 5 is factor of q²

and 5 is factor of q.

This concludes that 5 is factor of both p and q which proved our statement wrong that p/q is in simplest form. Due to this contradiction √5 is an irrational number.