Prove that is irrational.

Question

KnowledgeBoat · Accepted Answer

Let us assume, to the contrary, that is a rational number. If is rational, that means it can be written in the form of , where a and b are integers that have no common factor other than 1 and b 0. i.e., a and b are co-prime numbers. Squaring both sides, ⇒ 5b² = a² ...........(1) ⇒ This means 5 divides a². From this, 5 also divides a. Then a = 5c, for some integer 'c'. On squaring, we get ⇒ a² = 25c² Substituting above value of a² in equation (1) ⇒ 5b² = 25c² ⇒ b² = ⇒ b² = 5c² ⇒ c² = This means b² is divisible by 5 and so b is also divisible by 5. Therefore, a and b have 5 as common factor but this contradicts the fact that a and b are co-prime. This contradiction has arisen because of our incorrect assumption that is a rational number. Hence, proved that is irrational.

Mathematics

Prove that is irrational.

Irrational Numbers

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