Prove that 5 is irrational.

Let us assume, on the contrary, that 5 is a rational number. Therefore, we can find two co-prime integers a , b ( b ≠ 0 ) such that 5 = a b . 5 = a b ⇒ a = 5 b ⇒ a 2 = 5 b 2 Therefore, a 2 is divisible by 5 , then a is also divisible by 5 . So, a 2 = 5 k 2 = 5 5 k 2 = 5 b 2 ⇒ b 2 = 5 k 2 This means that b 2 is divisible by 5 and hence, b is divisible by 5 . This implies that a and b have 5 as a common factor. And this is a contradiction to the fact that a and b are co-prime. So, our assumption that 5 is rational is wrong. Hence, 5 cannot be a rational number. Therefore, 5 is irrational.

Prove that 3 + 2 5 is irrational.

Let us assume, on the contrary, that 3 + 2 5 is rational. Therefore, we can find two integers a , b ( b ≠ 0 ) such that 3 + 2 5 = a b 2 5 = a b - 3 5 = 1 2 a b - 3 Since a and b are integers, 1 2 a b - 3 will also be rational and therefore, 5 is rational. This contradicts the fact that 5 is irrational. Hence, our assumption that 3 + 2 5 is rational is false. Therefore, 3 + 2 5 is irrational.

Prove that the following is irrational. 1 2

Given : 1 2 Let us assume that 1 2 is rational. Therefore, we can consider co-primes as a , b ( b ≠ 0 ) such that 1 2 = a b 2 = b a b a is rational as a and b are integers. Therefore, 2 is rational. This contradicts the fact that 2 is irrational. Hence, our assumption is false and 1 2 is irrational.

Prove that the following are irrational. 7 5

Given, 7 5 Let us assume that 7 5 is rational. Therefore, we can find two integers a , b ( b ≠ 0 ) such that 7 5 = a b ∴ 5 = a 7 b a 7 b is rational as a and b are integers. Therefore, 5 should be rational. This contradicts the fact that 5 is irrational. Therefore, our assumption that 7 5 is rational is false. Hence, 7 5 is irrational.

Prove that the following is irrational. 6 + 2

Let 6 + 2 be rational. Therefore, we can find two integers a , b b ≠ 0 such that, 6 + 2 = a b 2 = a b - 6 Since a and b are integers, a b - 6 is a rational number and hence, 2 should be rational. This contradicts the fact that 2 is irrational. Therefore, our assumption is false and hence, 6 + 2 is irrational.

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MATHEMATICS - Class X>Chapter 1 - Real Numbers>Exercise 1.3

Exercise 1.3

Author:Jharkhand Board

Jharkhand Board Mathematics Solutions for Exercise 1.3

Simple step-by-step solutions to Exercise 1.3 questions of Real Numbers from MATHEMATICS - Class X. Also get 3D topic explainers, cheat sheets, and unlimited doubts solving on EMBIBE.