Question

Prove that the following
are irrationals:
(i) 1 /√2
(ii) 7√5
(ii) 6 + √2​

Answer 1

Step-by-step explanation:

(i) let us assume to the contrary , that 1/√2 is rational.

So that we can find integer a and b (b does not equal to 0)

Such that 1/√2 = a/b, where a and b are coprime.

Rearrange the equation, we get

b = a√2

Squaring on both side and rearranging, we get

b²= 2a²

b² is divisible by 2 and b is also divisible by 2.

So we can write b = 2c for some integer c.

substituting for b, we get 2a²= 4c² l, that is,

a = 2c²

this means that a² us divisible by 2 and so a also divisible by 2

:. a and b have at least 3 common factor.

but this contradicts the fact that a and b are co prime.

this contradiction has arisen because of our incorrect assumption that 1/√2 is rational.

so we conclude that 1/√2 is irrational.

(ii) same as (i)

(iii) same as (i)