Question

prove that the square of any fraction is not 2​

Answer 1

Answer:

We use indirect reasoning.

Suppose x is a rational number whose square is 2.

Then x can be written in lowest terms as

a

b

, where a is an integer and b is a

positive integer.

Since x2 = 2, ⇣a

b

⌘2

= 2, so

a2

b2 = 2. Then a2 = 2b2

, so a2 is even.

But then a is even, so a = 2n for some integer n.

Then (2n)

2 = 2b2

, so 4n2 = 2b2.

Then 2n2 = b2

, so b2 is even, and thus b is even.

Then a and b both have 2 as a common factor,

so

a

b cannot be in lowest terms, a contradiction.

Thus x cannot be rational.

Answer 2

Explanation:

the squre of any fraton is not