Question 7 (OR 2 nd   question)

Show that 7 − √5 is irrational, give that √5 is irrational.

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Question 7 (OR 2nd question) Show that 7 − √5 is irrational, give that √5 is irrational. We have to prove 7 − √5 is irrational Let us assume the opposite, i.e., 7 − √5 is rational Hence, 7 − √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, 7 − √5 = 𝑎/𝑏 7 – 𝑎/𝑏 = √5 √5 = 7 – 𝑎/𝑏 √5 = (7𝑏 − 𝑎)/𝑏 Here, (7𝑏 − 𝑎)/𝑏 is a rational number But √5 is irrational Since, Rational ≠ Irrational This is a contradiction ∴ Our assumption is incorrect Hence, 7 − √5 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 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.