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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


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

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.