Question

prove that it was irrational

Answer 1

If a number is rational then it can be expressed in the form of p/q where q is not equal to 0. But the denominator of 1/√2 is an irrational number. and any number divided by an irrational number is irrational. Here 1 is being divided by√2, and thus the result will be an irrational number.



:D

Answer 2

If possible, let 1/√2 be rational. Then, there exist positive co-primes a and b such that

1/√2 = a/b
= 2a² = b²
= 2 | b²
= 2 | b
= b = 2c for some positive integer c.

:- 2a² = b²
= 2a² = 4c²
= a² = 2c²
= 2 | a²
= 2 | a

This is contradiction to that fact that a,b are co-primes.
Hence, 1/√2 is irrational.