Question

Show that the following numbers are irrational.

`1/sqrt(2)`

Answer 1

Let us assume that `1/sqrt2` is rational. Then, there exist positive co-primes a and b such that

`1/sqrt2 = a/b`

`1/sqrt2 = (a/b)^2`

`⇒ 1/sqrt2 = a^2/b^2`

`⇒ b^2 = 2a^2`

`⇒ 2 | b^2 (because 2|2a^2)`

`⇒ 2|b`

`⇒ b= 2c  \text{for some posibive integer c}`

`⇒ 2a^2=b^2`

`⇒ 2a^2=4c^2(because a= pc)`

`⇒ a^2 = 2c^2`

`⇒ 2|a ^2 (2|2c^2)`

`⇒ 2|a`

`⇒ 2|a and 2|b`

Hence `1/sqrt2` is irrational.