May I suggest one reference? Check out The Number Systems: Foundations of Algebra and Analysis by Solomon Feferman.
If you click on the link, you can preview this book, to see, e.g., if the Table of Contents, etc., seem promising in answering your questions. Perhaps you can obtain it at a library, or through inter-library loan.
For "quicker" access to some discussion of your interest, see:
$(1)$ Number (Wikipedia)
It begins by describing Natural Numbers, referring both to set theory, and to the Peano axioms.
Then it moves to integers, where you'll find a link to the integer entry, which describes how one can construct integers from natural numbers.
Then it moves to the rational numbers, where you'll find a link to the rational number entry, which describes how one can construct rational numbers from the integers...,
Also see the entry on numbers that follows:
$(2)$ Number (Encyclopedia of Mathematics)
In particular, scroll down the entry, until you find the following passages:
"Throughout the 19th century, and into the early 20th century, deep changes were taking place in mathematics. Conceptions about the objects and the aims of mathematics were changing. The axiomatic method of constructing mathematics on set-theoretic foundations was gradually taking shape. In this context, every mathematical theory is the study of some algebraic system. In other words, it is the study of a set with distinguished relations, in particular algebraic operations, satisfying some predetermined conditions, or axioms.
From this point of view every number system is an algebraic system. For the definition of concrete number systems it is convenient to use the notion of an "extension of an algebraic system" . This notion makes precise in a natural way the principle of permanence of formal computing laws, which was formulated above..."
You can read then how the numbers, along with appropriate operations and elements like an additive identity (and/or multiplicative identity), etc, comprise algebraic systems and extensions of algebraic systems, with "axioms" related to each system/extended from predecessor systems.
The details, and additional links and references, are left for the reader to explore!