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Prove that√2 is rational

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Prove that√2 is rational

Posted by Mohd Arman 1 year, 8 months ago

CBSE > Class 10 > Mathematics

2 answers

Ayush Singh 1 year, 8 months ago

√2 = p/q, where 'p' and 'q' are integers, q ≠ 0 and p, q have no common factors (except 1).

Prakhar Sharma 1 year, 8 months ago

Let us assume on the contrary that 2 is a rational number. Then, there exist positive integers a and b such that 2 = b a where, a and b, are co-prime i.e. their HCF is 1 ⇒( 2 ) 2 =( b a ) 2 ⇒2= b 2 a 2 ⇒2b 2 =a 2 ⇒2∣a 2 [∵2∣2b 2 and 2b 2 =a 2 ] ⇒2∣a...(i) ⇒a=2c for some integer c ⇒a 2 =4c 2 ⇒2b 2 =4c 2 [∵2b 2 =a 2 ] ⇒b 2 =2c 2 ⇒2∣b 2 [∵2∣2c 2 ] ⇒2∣b...(ii) From (i) and (ii), we obtain that 2 is a common factor of a and b. But, this contradicts the fact that a and b have no common factor other than 1. This means that our supposition is wrong. Hence, 2 is an irrational number.

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