Ex 1.2, 1 - Chapter 1 Class 10 Real Numbers

Transcript

Ex 1.2, 1 Prove that √5 is irrational. We have to prove √5 is irrational Let us assume the opposite, i.e., √5 is rational Hence, √5 can be written in the form 𝑎/𝑏 where a and b (b≠ 0) are co-prime (no common factor other than 1) Hence, √𝟓 = 𝒂/𝒃 √5 b = a Squaring both sides (√5b)2 = a2 5b2 = a2 𝒂^𝟐/𝟓 = b2 Hence, 5 divides a2 So, 5 shall divide a also Hence, we can say 𝑎/5 = c where c is some integer So, a = 5c Now we know that 5b2 = a2 Putting a = 3c 5b2 = (5c)2 5b2 = 25c2 b2 = 1/5 × 25c2 b2 = 5c2 𝒃^𝟐/𝟓 = c2 Hence, 5 divides b2 So, 5 divides b also By (1) and (2) 5 divides both a & b Hence 5 is a factor of a and b So, a & b have a factor 5 Therefore, a & b are not co-prime. Hence, our assumption is wrong ∴ By contradiction, √𝟓 is irrational