Where can I find a derivation of high school mathematics based on the simplest axioms, principles and definitions?

Question

Basically something that starts with the substitution principles, definition of an equation, of subtraction and division, axiom of opposites, your usual commutative, associative, distributive axioms, and covers all the way up to the end of high school trigonometry.

Dolciani's two volume "Algebra: Structure and Method" was good but I need another similar work.

Serge Lang's "Basic Mathematics" wasn't good enough.

Reading the first few chapter of abstract algebra books wasn't that helpful as they were too broad in scope.

Any other suggestions?

The book Calculus by Spivak takes the axioms for the real numbers as a starting point. It does also give a development of the trig functions. — littleO, Jul 21 '23 at 10:34
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. — Community, Jul 21 '23 at 10:40
Related: Books for middle-school-level algebra that covers some set theory and "axioms of algebra"? — Dave L. Renfro, Jul 21 '23 at 10:57
@Dr. Momo: Analysis I and II by Tao. --- Since the OP wrote "Dolciani's two volume "Algebra: Structure and Method" was good but I need another similar work", I seriously doubt Tao's books would be suitable. contents for Dolciani Volume 1 AND contents for Dolciani Volume 2 — Dave L. Renfro, Jul 21 '23 at 15:53
@DaveL.Renfro I don't know... Tao develops the usual number systems (naturals, integers, rationals, and reals) from scratch and gives a definite definition of trigonometric functions, etc. There is an appendix about the equality relation and logic. That seems to be what the OP is asking for? — Dr. Momo, Jul 21 '23 at 16:16
@Dr. Momo: I suppose we need to know whether the OP wants a book for someone still learning (or recently learned) school algebra and trigonometry vs. a book for someone at the advanced undergraduate / beginning graduate university level for a mathematics major. If the latter, then Tao will work, but then so would several hundred other books for a similar mathematical maturity level. — Dave L. Renfro, Jul 21 '23 at 16:53
One possibility, but it stops well before trigonometry, is The Anatomy of Mathematics by Kershner/Wilcox (1950 1st edition -- also here; 1974 2nd edition). Another possibility, this one being primarily exponential and trigonometric functions, is The Non-Algebraic Elementary Functions. A Rigorous Approach by André Louis Yandl (1964). — Dave L. Renfro, Jul 21 '23 at 16:55

score 2 · Answer 1 · answered Jul 21 '23 at 10:38

You might like some of Hung-Hsi Wu's books such as 'Pre-Algebra', 'Algebra and Geometry', and 'Pre-calculus, Calculus and Beyond', available from the AMS. The books are aimed at teachers of mathematics, but everything is done rigorously from scratch. Wu goes into more detail than I have seen in any other books covering the content of school mathematics.

To get a flavour of his work, there is an old draft of his Pre-Algebra book available on his website here.

Where can I find a derivation of high school mathematics based on the simplest axioms, principles and definitions?

1 Answers1