Prove root 5 is rational number

1.	Prove root 5 is rational number
Answer» To prove that\xa0√5 is irrational number\xa0Let us assume that\xa0√5 is rational\xa0Then\xa0√5 = (a and b are co primes, with only 1 common factor and b≠0)\xa0⇒\xa0√5 = (cross multiply)\xa0⇒ a =\xa0√5b\xa0⇒ a² = 5b² ------->\xa0α⇒ 5/a²\xa0(by theorem if p divides q then p can also divide q²)\xa0⇒ 5/a ----> 1\xa0⇒ a = 5c\xa0(squaring on both sides)\xa0⇒ a² = 25c² ---->\xa0β\xa0From equations\xa0α and\xa0β\xa0⇒ 5b² = 25c²⇒ b² = 5c²\xa0⇒ 5/b²\xa0(again by theorem)\xa0⇒ 5/b-------> 2\xa0we know that a and b are co-primes having only 1 common factor but from 1 and 2 we can that it is wrong.\xa0This contradiction arises because we assumed that\xa0√5 is a rational number\xa0∴ our assumption is wrong\xa0∴\xa0√5 is irrational number

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