Question

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

Answer 1

Answer:

$\frac{1}{ \sqrt{2} } \times \frac{ \sqrt{2} }{ \sqrt{2} } \\ \frac{ \sqrt{2} }{2} = \frac{a}{b} \\ \sqrt{2} = \frac{a}{b} \times 2 \\ \sqrt{2} = \frac{2a}{b}$

it means

$\sqrt{2}$

is rational but this is a construction because. We know that

$\sqrt{2}$

is irrational. So our supposition is wrong.

Hence

$\frac{ \sqrt{2} }{2} \: is \: irrational$

b)

b)

$7 + \sqrt{5}$

Let if possible

$7 \sqrt{5}$

is rational

$7 + \sqrt{5} = \frac{a}{b} \\ \sqrt{5} = \frac{a}{7b}$

it means

$\sqrt{5}$

is rational number. but this is a construction because we know that

$\sqrt{5}$

is irrational. So our supposition is wrong. Hence

$7 + \sqrt{5}$

is irrational number