Yes. If you're not using two's complement arithmetic, you end up with the bizarre situation that there are two different representation for zero. OTOH, two's complement takes care of that. I think most systems nowadays use two's complement, so this is seldom an issue. Even the IEEE floating point uses two's complement for both the mantissa and exponent. I suspect, but am not certain, that this is a big issue with legacy systems and no issue at all with current systems so long as they follow the standard.

I was thinking about discussing rational numbers next, because that issue actually came up in a discussion a few months back - and it also helps to explain the term to my tutees (or whatever you might call the students I tutor).

Yes, they are called rational numbers because they can be expressed as the ratio (division) of two integers, not because they are more sensible or useful than irrational numbers.

Typically, authors use the symbol Q (in blackboard bold) or just a bold R to represent the rational numbers. According to wiki, the Q stands for "quotient" which makes sense, though I never
considered it before. The real numbers are typically represented by a cursive R.

You're right, of course, that the cardinality (size) of the Rationals is the same as the Integers. Also interesting is the fact that although there are an infinite number of integers (and rationals); the set of irrationals is infinitely larger.