Let's say you're a university lecturer who regularly teaches remedial and/or college algebra courses. A standard textbook for such a course usually starts out with a series of facts about real number properties, but doesn't distinguish between facts that are definitions, versus facts that can be proven from more basic assumptions. A small number of examples:
- What is the definition of subtraction and division in use?
- Is the fact that $(-1)x = -x$ a definition, or a theorem?
- Is the fact that $a^0 = 1$ a definition or a theorem?
- What about the fact that $a/b = c/d \Leftrightarrow ad = bc$?
- Is the fact that we can add a value to both sides of an equation and maintain equivalence a definition, or provable from simpler assumptions?
Over the years I've spent a good amount of time disentangling these issues, but I'm wondering if there is an existing book that I could recommend to others. Say: We start by taking the real numbers as a roughly undefined term, and define the equality relation clearly. Explicate the definitions of the various operations in use, and the order of the operations. Establish the properties of commutativity, association, and distribution as axioms. From that point forward, clarify what facts are definitions, versus what are theorems, and present explicit proofs for the latter. Even if we don't present the information to our students exactly this way, I think it would be a relief for a mathematically-trained instructor to know what really must come first in the logical sequence, and what can really be proven to an inquisitive student, versus the few things that are effectively "just because".
Is there such a resource that formally develops topics from the start of a basic algebra course?