Question

Use method of contradiction to show that $\sqrt{3}$ and $\sqrt{5}$ are irrational numbers.

Answer 1

Let us suppose that and are rational numbers

and (Where a, b 7 and b, y 0 x , y)

Squaring both sides

a² and x² are odd as 3b² and 5y² are odd .

a and x are odd....(1)

Let a = 3c, x = 5z

a² = 9c², x² = 25z²

3b² = 9c², 5y² = 25z²(From equation )

b² =3c², y² = 5z²

b² and y² are odd as 3c² and 5z² are odd .

b and y are odd...(2)

From equation (1) and (2) we get a, b, x, y are odd integers.

i.e., a, b, and x, y have common factors 3 and 5 this contradicts our assumption that are rational i.e, a, b and x, y do not have any common factors other than.

is not rational

and are irrational.