Indeed, stating that these objects result from some classification of all number-like objects seems misleading at best: Most mathematicians would, just as one e

Indeed, stating that these objects result from some classification of all number-like objects seems misleading at best: Most mathematicians would, just as one example, certainly consider p-adic numbers as number like objects. It would be better to point out what really matters in the hierarchy N, Z, Q, R, C: Everybody will accept that N is an interesting object for counting. Z is the quotient group of N under addition, more elementary: In Z you can compute arbitrary differences of elements of N, and it is the smallest "reasonable" such object. Similarly, Q is the quotient field of Z, i.e. "the smallest reasonable object containing Z that allows division of non-zero elements". R is the completion of Q under a natural metric (i.e., "fills the gap on the number line of Q"), and therefore allows one to have a reasonable notion of limits and calculus. C finally is the algebraic closure of R, so that every non-constant polynomial has a root in C. Thus, these objects are all constructed from the natural numbers to satisfy certain "niceness" properties. Of course, it would take quite some space to explain this in a very elementary fashion. So perhaps it is instead better to simply refrain from such a statement in a textbook aiming for shortness above all, or at least make sure that any statement in this direction is substantiated and supported by a suitable reference to a place where your readers can learn more. A textbook that gains conciseness from vague statements and half-truths would seem to be much worse than a textbook of twice the length that explains its content well.