1. Real Numbers

Real Numbers

Class X

Chapter 1

NCERT Text Book Solution

Exercise 1.3

Q.No. 1. Prove that √5 is irrational.

Solution:

Let √5 is a rational number.

Therefore,

√5 = a/b

Suppose a and b have a common factor other than 1, then we can divide by the common factor, and assume that a and b are coprime.

So,

b√5 = a

Now squaring both side

5b² = a²

a² divisible by 5, therefore, a is divisible by 5.

Let a = 5c

Therefore,

5b² = 25c²

b² = 5c²

Hence, b² is divisible by 5, therefore, b is divisible by 5.

Therefore, a and b have at least 5 as a common factor.

But this contradicts the fact that a and b are coprime.

This contradiction has arisen because of our incorrect assumption that √5 is a rational.

So, we conclude that √5 is irrational. Proved.

Q.No. 2. Prove that 3 + 2√5 is irrational.

Solution:

From the solution of question no. 1. It is proved that √5 is irrational.

We know that product of rational and irrational number is irrational.

So, 2√5 is irrational.

We also know that sum of rational and irrational number is irrational.

So, 3 + 2√5 is irrational. Proved.

Q.No. 3. Prove that the following are irrationals:

(i) 1/√2

(ii) 7√5

(iii) 6 + √2

Solution:

(i) Let √2 be rational.

Therefore,

√2 = a/b 

Suppose a and b have a common factor other than 1, then we can divide by the common factor, and assume that a and b are coprime.

So, b√2 = a

Now squaring both side

2b² = a²

a² is divisible by 2, therefore a is divisible by 2.

Let a = 2c

Therefore,

2b² = 4c²

b² = 2c²

b² is divisible by 2, therefore, b is divisible by 2.

Therefore, a and b have at least 2 as a common factor.
But this contradicts the fact that a and b are coprime.
This contradiction has arisen because of our incorrect assumption that √2 is a rational.
So, we conclude that √2 is irrational.
We know that quotient of rational and irrational is irrational.
So, 1/√2 is irrational. Proved.

Solution:
(ii) 7√5
From the solution of question no. 1 it is proved that √5 is irrational.
We know that product of rational and irrational is irrational.
So, 7√5 is irrational. Proved.

Solution:
(iii) 6 + √2

From the solution of question no. 3 (i) it is proved that √2 is irrational.
We know that sum of rational and irrational is irrational.
So, 6 + √2 is irrational. Proved.