Before posting I read questions that were similar to mine but I didn't find an answer that was quite what I was looking for.

I want to learn all high school math from the ground up, but I want books that focus extremely heavily on proving every little thing rigorously. I made another post which was closed for being too broad because I specifically requested books for these topics: Algebra, Geometry, Trigonometry, Probability and statistics, Pre-calculus and calculus

So I guess I need to create one question for each of these topics so it's less broad (I hope this isn't against the rules or still too broad. I'm just looking for 1 or 2 examples, anything that would help me have a little more guidance and make it a bit easier to learn on my own, I'm certainly not looking for definitive answers, let alone debates).

I have already posted a similar topic about Geometry and have received some excellent recommendations. However, the introduction to one of the books says I need good, solid knowledge of Pre-Calculus and so that is what I am looking for right now. Even though this topic is usually taught in high schools, I'm looking for college level books, since I doubt high school books are as rigorous as I want them to be.

For reference, one of the ones I know about and am going to use for this project as a whole is Euclid's Elements, so I am looking for other books with that level of rigor (though additional explanations are always helpful, since Euclid's Elements is very rigorous but doesn't elaborate much on explanations).

Thanks a lot!

  • $\begingroup$ See the references I posted at Preparing For University and Advanced Mathematics (most of which are at other web pages I posted links to). $\endgroup$ – Dave L. Renfro Oct 6 '16 at 15:39
  • $\begingroup$ Thanks a lot for your answer! I had actually already taken a look at Gelfand's Algebra, but I thought his approach was too practical. I already know how to use all of those things as tools, what I really wanted was a very rigorous approach, building literally everything from the axioms themselves. Right now I am learning to prove all of the basic properties of addition, multiplication, etc. and a - b = the difference between a and b has just been defined for natural numbers. Gelfand, for example, just straight jumps into teaching how to add negative numbers, which is not what I'm looking for. $\endgroup$ – José Guedes Oct 6 '16 at 17:38
  • $\begingroup$ If you really want RIGOR, try to get the MEGSSS books I mentioned in my comments at Logic and set theory textbook for high school. Interestingly, it seems someone (not me) has deleted some possibly useful information I posted about those books. A phrase search for "When I purchased a few of the Elements of Mathematics books in 1988" brings up the comment, but it isn't available when you visit the webpages. And google's Cached option doesn't seem to work either. $\endgroup$ – Dave L. Renfro Oct 6 '16 at 22:08
  • $\begingroup$ I can see the comment you are referring to at the bottom of the page! These books sound very interesting, but I couldn't find them anywhere. Do you know if they are available for downloading? $\endgroup$ – José Guedes Oct 7 '16 at 4:41
  • $\begingroup$ Humm...I did not realize that I actually posted an answer to this question! I was in a hurry yesterday when I wrote my MEGSSS comment and didn't scroll down the page far enough to notice I had an answer. I don't know how to obtain the books beyond what I did (write the publishers), but I did it in 1988 . . . $\endgroup$ – Dave L. Renfro Oct 7 '16 at 14:00

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