Question

Show that 2 - sqrt(2) is irrational

Answer 1

Answer:

This proof can be done by contradiction.

Let us assume that 2 - √2 is rational number.

So, by the definition of rational number, we can write it as

$2 -\sqrt{2} = \dfrac{a}{b}$

where a & b are any integer.

⇒ $\sqrt{2} = 2 - \dfrac{a}{b}$

Since, a and b are integers $2 - \dfrac{a}{b}$ is also rational.

and therefore √2 is rational number.

This contradicts the fact that √2 is irrational number.

Hence our assumption that 2 - √2 is rational number is false.

Therefore, 2 - √2 is irrational number.