1. Show that the following numbers are irrational.(i)

Solution:

Consider

   

is a rational number

Let us assume

   

= r where r is a rational number

On further calculation we get

   

Since r is a rational number

   

is also a rational number

But we know that

   

is an irrational number

So our supposition is wrong.

Hence

   

is an irrational number.

Solution:

Let’s assume on the contrary that

   

is a rational number. Then, there exist positive integers a and b such that

   

= a/b where, a and b, are co-primes

⇒

   

⇒

   

 is rational [∵

   

, a and b are integers ∴

   

is a rational number]

This contradicts the fact that

   

is irrational. So, our assumption is incorrect.

Hence,

   

is an irrational number.