I am having great difficulty, however, accepting that it is correct that
[MATH]p,\ q \in \mathbb Z,\ \text { and } q < 0 \implies \dfrac{p}{q} \not \in \mathbb Q,[/MATH]
which leads to the rather odd result that
[MATH]\dfrac{-\ p}{-\ q} \in \mathbb Q \text { and } \dfrac{-\ p}{-\ q} \equiv \dfrac{p}{q} \not \in \mathbb Q.[/MATH]
This is why I suggested a distinction between contexts, and specifically between definitions and other uses.
A key word in any definition, I think, is "can": "A rational number is any number that
can be represented by p/q, where ...". There, we have no need to specify a
preferred representation, much less a
unique representation (which requires mentioning lowest terms). In particular, if a fraction
is written with a negative denominator like 2/-3, it still
can be written with a positive denominator, -2/3, so it is still a rational number even if you were to call for positive denominators. But even so, I don't think I've seen a
definition that says q>0. That would be said in other contexts.
A common defect in informal "definitions" is a failure to clearly distinguish a
number from its
representation. We should not make it sound like a rational number
is a particular representation p/q. But the following (I can't make myself link to less respectable sources!) are all good, mostly:
Wikipedia: In mathematics, a rational number is any number that
can be expressed as the quotient or fraction p/q of two integers, a numerator p and a
non-zero denominator q.
-- good, though "non-zero" could in my opinion be left unsaid
MathWorld: A rational number is a number that
can be expressed as a fraction p/q where p and q are integers and
q!=0.
A rational number p/q is said to have numerator p and denominator q.
-- accidentally equates the number with a representation, as if a number has one specific denominator.
Math Open Reference: A rational number is one that
can be written as the ratio of two integers.
-- good
Dictionary.com: a number that
can be expressed exactly by a ratio of two integers.
-- good
Math Is Fun: A number that
can be made by dividing two integers (an integer is a number with no fractional part).
-- good
None of these mentions being positive; most don't mention non-zero. All are fine as definitions.