Prove that 2 – √3 is irrational, given that √3 is irrational.

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  1. Class 10
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Transcript

Question 27 Prove that 2 – √3 is irrational, given that √3 is irrational.We have to prove 2 – √3 is irrational Let us assume the opposite, i.e., 2 – √3 is rational Hence, 2 – √3 can be written in the form 𝑎/𝑏 where a and b (b≠ 0) are co-prime (no common factor other than 1) Hence, 2 – √3 = 𝑎/𝑏 −√3 = 𝑎/𝑏 − 2 √3 = (−𝑎)/𝑏 + 2 √3 = (−𝑎 + 2𝑏)/𝑏 Here, (−𝑎 + 2𝑏)/𝑏 is a rational number But √3 is irrational Since, Rational ≠ Irrational This is a contradiction ∴ Our assumption is incorrect Hence, 2 – √𝟑 is irrational Hence proved.

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.