Question

Provide a beautiful proof of the following claim: The square of a rational number is a rational number.

Answer 1

This fact about rational numbers is very simple. Remember
`x` is a rational number if it is a number of the form
`(p)/(q)` where
`p,q` are integers and
`q\\eq 0`. To prove this result about rational numbers you can consider a rational number
`x=(p)/(q)`. Then, the square of
`x` is given by

`x^(2)=x\cdot x=(p)/(q)\cdot (p)/(q)=(p^2)/(q^(2))`

Note that
`x^(2)` satisfies the definition of a rational numbers.

Explanation: