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Answer» To prove that√5 is irrational numberLet us assume that√5 is rationalThen√5 =(a and b are co primes, with only 1 common factor and b≠0)⇒√5 = (cross multiply)⇒ a =√5b⇒ a² = 5b² ------->α⇒ 5/a²(by theorem if p divides q then p can also divide q²)⇒ 5/a ----> 1⇒ a = 5c(squaring on both sides)⇒ a² = 25c² ---->βFrom equationsα andβ⇒ 5b² = 25c²⇒ b² = 5c² ⇒ 5/b²(again by theorem)⇒ 5/b-------> 2 we know that a and b are co-primes having only 1 common factor but from 1 and 2 we can that it is wrong.This contradiction arises because we assumed that√5 is a rational number∴ our assumption is wrong∴√5 is irrational number |
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