Prove that 1 2 + 3 is an irrational number.

Let us assume that 1 2 + 3 is a rational number. Now, let 1 2 + 3 = a b , where a , b are co-primes and b ≠ 0 . ⇒ 2 + 3 = b a ⇒ 3 = b a - 2 Since a , b are rational numbers, then b a - 2 is also a rational number. ⇒ 3 is a rational number But, we know that, 3 is an irrational number. This shows our initial assumption was incorrect. Therefore, 1 2 + 3 is an irrational number.

Prove that 3 + 5 is an irrational number.

Assume that 3 + 5 is a rational number equal to ' a ' . ⇒ a = 3 + 5 . Squaring both sides, ⇒ a 2 = 3 + 5 2 ⇒ a 2 = 3 + 5 + 2 3 5 ⇒ a 2 = 8 + 2 15 ⇒ 2 15 = a 2 - 8 ⇒ 15 = a 2 - 8 2 Now, a 2 - 8 2 is a rational number, since a is a rational number. ⇒ 15 is a rational number. But, we know that, 15 is an irrational number. Hence, our assumption is wrong. Therefore, 3 + 5 is an irrational number.

Prove that 2 3 - 7 is an irrational.

Let us assume that 2 3 - 7 is a rational number. Now, let 2 3 - 7 = a b , where a , b are co-primes and b ≠ 0 . ⇒ 2 3 = a b + 7 ⇒ 3 = 1 2 a b + 7 Since a , b are integers, then a b + 7 is also a rational number. ⇒ 1 2 a b + 7 is a rational number. ⇒ 3 is a rational number. But, we know that, 3 is an irrational number. This shows our initial assumption was incorrect. Therefore, 2 3 - 7 is an irrational number.

Prove that 5 - 3 is an irrational number.

Let us assume that 5 - 3 is a rational number. Now, let 5 - 3 = a b , where a , b are co-primes and b ≠ 0 . ⇒ 3 = 5 - a b . Since a , b are integers, then 5 - a b is also a rational number. ⇒ 3 is a rational number. But, we know that, 3 is an irrational number. This shows our initial assumption was incorrect. Therefore, 5 - 3 is an irrational number.

E M B I B E

New Mathematics for Class X>Chapter 1 - Number System>REVISION EXERCISE

H K Dass, Rama Verma and, Bhagwat Swarup Sharma Solutions for Chapter: Number System, Exercise 5: REVISION EXERCISE

Author:H K Dass, Rama Verma & Bhagwat Swarup Sharma

H K Dass Mathematics Solutions for Exercise - H K Dass, Rama Verma and, Bhagwat Swarup Sharma Solutions for Chapter: Number System, Exercise 5: REVISION EXERCISE

Attempt the free practice questions on Chapter 1: Number System, Exercise 5: REVISION EXERCISE with hints and solutions to strengthen your understanding. New Mathematics for Class X solutions are prepared by Experienced Embibe Experts.