I think by the definition given at the beginning of the thread irrational numbers are numbers that can’t be expressed as a ratio of two natural numbers. i is irrational by that definition, but not a real.

But I must admit I haven’t read the whole post.

Looks like the set of reals is simply taken as the number universe (i.e. all numbers are already assumed to be real).

Just to clarify: there are lots of numbers that aren't real numbers (for example, imaginary numbers). Intuitively the real numbers are all the points along the number line, including rationals and irrationals (such as root 2, pi, or e). There are lots of other comments in this thread that give a good explanation of how that works formally.

If the students haven't yet encountered complex numbers, infinitesimals, infinities etc. then it's perfectly reasonable to say that all numbers are assumed to be real (as follows strictly from the definition in the book).

There is no such example. All real numbers are either rational or irrational.

∞