Question

Prove that the following is irrational:

`1/sqrt2`

Answer 1

`1/sqrt2`

`1/sqrt2 xx sqrt2/sqrt2 = sqrt2/2`

Let a = `(1/2)sqrt2` be a rational number.

∴ `1/2 (sqrt2)` is rational.

Let `1/2 (sqrt2) = a/b`, such that a and b are co-prime integer and b ≠ 0.

∴ `sqrt2 = (2a)/b`      ...(1)

Since the division of two integers is rational.

∴ `(2a)/b` is rational.

From (1), `sqrt2` is rational, which contradicts the fact that `sqrt2` is irrational.

∴ Our assumption is wrong.

Thus, `1/sqrt2` is irrational.