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Books that:

  1. has an introduction to proof, logic, and topics like sets and groups. Books

  2. can prepare you for rigorous calculus texts like Spivak and Apostol.

Question 1. I've been looking at the books by Gelfand (Algebra, Trigonometry, Functions and Graphs, The Method of Coordinates). I'm not sure if they cover all of the precalculus curriculum, though.

Question 2. What would be some good calculus books other than Spivak and Apostol?

Ansh.23
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    I'm not sure there is such a thing as a fully rigorous pre-calculus book, since pre-calculus typically covers topics like exponentials and trigonometric functions that technically require power series, integrals, limits etc. to define properly. – Jair Taylor Jun 09 '17 at 16:14
  • @JairTaylor Are there any that emphasize problem solving, introduce proofs, and prepare you for rigorous calculus books like Spivak and Apostol? – Ansh.23 Jun 09 '17 at 16:16
  • I don't know of any good resources for this off the top of my head. You could look at this thread. – Jair Taylor Jun 09 '17 at 16:19
  • Are you inetrested in number theory or abstract algebra? – Vidyanshu Mishra Jun 09 '17 at 16:22
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    @Ansh.23, In my opinion Apostol's calculus (the two volume works) is self-contained in the sense that he never gets ahead of himself in the books. I recall that he spends some pages introducing the necessary pre-calculus stuff like summation, trigonometry, naive set theory, function theory, math induction... – Yes Jun 09 '17 at 16:24
  • @VidyanshuMishra I'm interested in number theory, but that isn't really going to help me prepare for calculus and stuff, right? – Ansh.23 Jun 09 '17 at 16:25
  • @EricClapton Thanks, I'll look over those again. Do I need to be experienced in proof before I start Apostol though? – Ansh.23 Jun 09 '17 at 16:26
  • Yes it does not. But I was talking about a bigger picture, calculus is just like stairs, it make you a machine while abstract maths make you a think tank – Vidyanshu Mishra Jun 09 '17 at 16:27
  • @Ansh.23, Oh learning proof techniques or ideas would not be a normal topic under the name "pre-calculus" :). You said you are fond of number theory. Then go for some introductory number theory books; you can learn how to do a proof naturally in elementary number theory! In elementary number theory you know already many familiar concepts; this can make you focus on the structure of proofs. :) – Yes Jun 09 '17 at 16:28
  • @EricClapton That's not what I really meant by the title of my question anyways. Can I still start Apostol though? – Ansh.23 Jun 09 '17 at 16:29
  • @Ansh.23, If time allows it won't hurt to go read Apostol! You know math is not a chemistry experiment that can lead to disastrous consequences when one gets ahead of himself :). We at most do thought experiments, which are safe enough. Go read Apostol; stop to identify the possible problems if stuck. Go solve the problems that are possibly things more basic. Then go check Apostol again. This could be a back-and-forth process. – Yes Jun 09 '17 at 16:33
  • @Ansh.23, As far as Apostol is concerned, it is good. I am also currently reading that. Though I would say that order of topics is quite filthy in that book – Vidyanshu Mishra Jun 09 '17 at 16:35
  • @Ansh.23 Those books don't cover everything one would want to know at the precalculus level. For example, exponential and logarithmic functions are not discussed. Nor are geometry and vectors. As I said in answer to your other question, Lang's Basic Mathematics covers what you need, though the individual topics in Gelfand's books are given more in-depth treatment there. If you want a sense of what topics need to be covered in a typical precalculus course, look at Lang's table of contents. – user49640 Jun 09 '17 at 18:39
  • Spivak's calculus is not rigorous at all imo. –  May 08 '18 at 06:18

1 Answers1

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IMHO, you are not gonna find a truly amazing pre-calculus book. Just read pre-calculus topics from random sources. You can consider these books:

$1$ George F. Simmons-Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry

$2$ Michael Sullivan III -Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

For a bit higher level, consider these.

Gilbert Strang-Introduction to linear algebra.

Amann, Herbert, Escher, Joachim- Analysis $1$.

Vidyanshu Mishra
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