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

Transcript

Ex 1.3 , 2 Prove that 3 + 2 root 5 โˆš5 is irrational. We have to prove 3 + 2 root 5โˆš5 is irrational Let us assume the opposite, i.e., 3 + 2โˆš5 is rational Hence, 3 + 2โˆš5 can be written in the form ๐‘Ž/๐‘ where a and b (bโ‰  0) are co-prime (no common factor other than 1) Hence, 3 + 2โˆš5 = ๐‘Ž/๐‘ 2โˆš5 = ๐‘Ž/๐‘ - 3 2โˆš5 = (๐‘Ž โˆ’ 3๐‘)/๐‘ โˆš5 = 1/2 ร— (๐‘Ž โˆ’ 3๐‘)/๐‘ โˆš5 = (๐‘Ž โˆ’ 3๐‘)/2๐‘ Here, (๐‘Ž โˆ’ 3๐‘)/2๐‘ is a rational number But โˆš5 is irrational Since, Rational โ‰  Irrational This is a contradiction โˆด Our assumption is incorrect Hence 3 + 2โˆš5 is irrational Hence proved

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