Ex 1.3, 2 - Prove that 3 + 2 root 5 is irrational - Ex 1.3

  1. Chapter 1 Class 10 Real Numbers
  2. Serial order wise
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Ex 1.3 , 2 Prove that 3 + 2 root 5 √5 is irrational. We have to prove 3 + 2 root 5√5 is irrational Let us assume the opposite, i.e., 3 + 2√5 is rational Hence, 3 + 2√5 can be written in the form π‘Ž/𝑏 where a and b (bβ‰  0) are co-prime (no common factor other than 1) Hence, 3 + 2√5 = π‘Ž/𝑏 2√5 = π‘Ž/𝑏 - 3 2√5 = (π‘Ž βˆ’ 3𝑏)/𝑏 √5 = 1/2 Γ— (π‘Ž βˆ’ 3𝑏)/𝑏 √5 = (π‘Ž βˆ’ 3𝑏)/2𝑏 Here, (π‘Ž βˆ’ 3𝑏)/2𝑏 is a rational number But √5 is irrational Since, Rational β‰  Irrational This is a contradiction ∴ Our assumption is incorrect Hence 3 + 2√5 is irrational Hence proved

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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.