Question

Show that the following numbers are irrational.
(i) $\frac{1}{\sqrt{2}}$
(ii) $7\sqrt{5}$
(iii) $6+\sqrt{2}$
(iv) $3-\sqrt{5}$

Answer 1

(i) Let us assume that is rational .Then , there exist positive co primes a and b such that

(ii) Let us assume that is rational .Then , there exist positive co primes a and b such that

We know that is an irrational number

Here we see that is a rational number which is a contradiction

(iii) Let us assume that is rational. Then , there exist positive co primes a and b such that

Here we see that is a rational number which is a contradiction as we know that is an irrational number

(iv) Let us assume that is rational .Then, there exist positive co primes a and b such that

Here we see that is a rational number which is a contradiction as we know that is an irrational number