Question

prove that 1/√2is an irrational no ​

Answer 1

Answer:

Refer to the attachment

Answer 2

ur ans

Step-by-step explanation:

Let’s assume on the contrary that 1 √ 2 12 is a rational number. Then, there exist positive integers a and b such that 1 √ 2 12 = a b ab where, a and b, are co-primes ( 1 √ 2 12)2 = ( a b ab)2 1 2 12 = a 2 b 2 a2b2 2a2 = b2 2 | b2 [∵ 2|2b2 and 2a2 = b2] 2 | b ………… (ii) b = 2c for some integer c. b2 = 4c2 ⇒ 2a2 = 4c2 [∵ b2 = 2a2] a2 = 2c2 ⇒ 2 | a2 2 | a ……… (i) From (i) and (ii), we can infer that 2 is a common factor of a and b. But, this contradicts the fact that a and b are co-primes. So, our assumption is incorrect.