The definition of rational numbers

Question

Very often I find this definition of rational numbers in my textbooks:

A rational number is a number determined by the ratio of some integer p to some nonzero natural number q.

But numbers $$\frac{-1}{-2};\frac{-2}{-4};...$$ are surely rational. Why the denominator has to be a natural number?

Answer 1

The keyword here is determined. If you prefer, replace the definition by

A rational number is a number that can be written as $\frac{p}{q}$, with $p$ an integer and $q$ a positive integer

$\frac{-1}{-2}$ can be written as $\frac{1}{2}$, so it's a rational number

$\frac{3\pi}{2\pi}$ can be written as $\frac{3}{2}$ so it's also a rational number

$5$ can be written as $\frac{10}{2}$ so it's also a rational number

But

$\sqrt{2}$ can't be written as a number of the form $\frac{p}{q}$, so it's not a rational number .