# Real NumbersExercise – 1.3

1. Prove that 5 is irrational.

Solution:

Let5 is a rational number.

So 5 will divide b2, Therefore 5 will divide b .…………(iii)

Now, Equations (i) and (ii) show that a and b have a common factor of 5 which is contrary to our assumption (since a and b are co-prime numbers i.e. a and b have no common factor other than 1.)

So this contradiction proves that contrary to our imagination, 5 is an irrational number.

2. Prove that 3+25 is irrational.

Solution:

Let 3+25 is rational

However, we know that 5 is an irrational number, So this contradiction proves that contrary to our imagination, 3+25 is an irrational number.

3. Prove that the following are irrationals:

(ii) 75

Solution:

Let 75 is rational

However, we know that 5 is an irrational number, So this contradiction proves that contrary to our imagination, 75 is an irrational number.

(iii) 6+2

Solution:

Let 6+2 is rational

However, we know that 2 is an irrational number, So this contradiction proves that contrary to our imagination, 6+2 is an irrational number.