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I have recently started getting back into mathematics as a consequence of trying to get a better grasp on machine learning. I started studying from a machine learning book but found the language extremely convoluted and abstruse. Fortunately, the book referenced Gilbert Strang's book on linear algebra.

Now, I wish to get back into mathematics, gain a better understanding and possibly a degree somewhere down the line. So, I am trying to relearn linear algebra and calculus (for now). I was trying to find some books for study and a lot of people recommended Micheal Spivak's Calculus which I managed to get a copy.

Now, here's my dilemma. From what I can recollect about studying calculus in school is that there were a lot of topics (general equations for circles, parabolas, hyperbola; limits and continuity, trigonometry, arithmetic geometric harmonic progression, complex numbers etc), that were taught before starting calculus. Spivak's book does have limits and continuity (about 50 pages) as a topic but I was skeptical about how much it could actually cover so I found Introduction to Real Analysis as a sort of precursor book to read before reading Spivk's Calculus. The first 6 chapters and chapter 9 (infinite series) of this book are familiar to me based on previous coursework.

Now my main question is should I first study limits and functions from this book before moving onto Spivak's calculus or will Spivak's calculus suffice? I am trying to rebuild a string foundation so I am looking for books with in-depth explanations (I like how Prof. Strang's book discusses concepts with the reader)

I would also appreciate advice regarding purchase of a single pre-calculus book that covers all topics or different pre-calculus books for each separate topic. I think rigorously practicing each topic would be best.

Thank you.

Ben
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Bittu
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  • For linear algebra, Elementary Linear Algebra by Keith Mathews has a lot more worked examples and, in comparison to the "machine learning book" you cited, it doesn't so quickly and so often jump off into more specialized and advanced topics while also discussing the basics. FYI, Mathews' book has been on the internet since at least the mid 1990s, and I used to refer students to it in the late 1990s. The "precursor book" you cite is definitely NOT a precursor to Spivak, but instead is what one might study AFTER Spivak. (continued) – Dave L. Renfro May 18 '21 at 06:48
  • Regarding "purchase of a single pre-calculus book", David Cohen's College Algebra is probably the best of the "modern era" (e.g. past 30-40 years or so) college algebra texts I know of. I don't know about the first 3 editions (1986, 1989, 1992), but I have the 1996 4th edition and it's more than sufficient for your precalculus needs (and can be obtained fairly cheaply). – Dave L. Renfro May 18 '21 at 06:52
  • Thank you for the response. I will look up the books you mentioned. Also, would you recommend I start with the Real Analysis book after I am done with Spivak's Calculus or should I rather focus on a different topic like multivariate calculus instead and come back to this book later because it's related to a slightly different branch of mathematics? – Bittu May 18 '21 at 21:51
  • I think getting a background in (elementary) multivariable calculus is much more relevant to your future needs. Indeed, this is probably true for most anyone continuing in math, regardless of later intended specialization in either some math-intensive non-math field or applied math or pure math. However, by the time you finish with Spivak, or even the first half, I imagine that your awareness of your mathematical needs will be much greater than they are now, and my advice is to not worry about that aspect now. – Dave L. Renfro May 19 '21 at 13:37
  • @DaveL.Renfro Hi, what would you consider some of the best U.S. precalculus textbooks of previous eras, and how do they compare to the one by Cohen? – Anonymous Aug 23 '21 at 22:47
  • @Anonymous: Between the mid 1960s and mid 1980s is the easiest period for this, as Modern Introductory Analysis by Mary P. Dolciani and others ("others" varied a little over the years) was easily the best. Various versions of the book continued later, but if you get a copy, then you will want one written before about 1980 or so -- before modifications due to calculators were made. Calculators began making their way into high school students' hands around 1975-76, by the way. (continued) – Dave L. Renfro Aug 24 '21 at 07:36
  • Off-hand, I don't know what might be such a book for around the mid 1950s to mid 1960s, but before the mid 1950s it was usually the case that the 3 subjects -- college algebra, trigonometry, analytic geometry -- were separately taught in most every university, and at a higher level than was later taught in high schools. Better students usually got the topics from college algebra and trigonometry in high school, and they began with analytic geometry in the fall of their first year followed by the first calculus semester in spring. I think this was mostly from maybe early 1920s to mid 1950s. – Dave L. Renfro Aug 24 '21 at 07:45
  • @Anonymous: Rather than trying to name some of the best considered texts (with a nod to those on the slightly more advanced/honors level) for before the mid 1950s, I'll just point to some previous lists I've posted. For (rather advanced) college algebra, see this answer. For trigonometry, see this answer. (continued) – Dave L. Renfro Aug 24 '21 at 07:59
  • For analytic geometry, see the books mentioned in this answer and in this comment. For honors calculus, see this answer. – Dave L. Renfro Aug 24 '21 at 07:59
  • @DaveL.Renfro I'm looking at the 1970 edition of Dolciani's book now, and there is a very heavy emphasis on logical formalism. For example, the "axioms of equality" state in essence that $=$ is an equivalence relation. There is constant mention in later places of the use of the "transitive property" of equality. All of this strikes me as somewhat fallacious. Firstly, because you need an axiom that says that if $a = b$, then every property true of $a$ is also true of $b$. Without that, you can't get anywhere. Secondly, it sets an impossibly high bar for rigour that I doubt can be (cont'd) – Anonymous Aug 24 '21 at 08:03
  • maintained. There would have been a simple alternative, namely, to say that "$a = b$" means that $a$ and $b$ denote the same object. More generally, just looking at the first couple of chapters, the level of dissection of the proofs seems close to unbearable. Do you think these are fair statements, and do you think they are reflective of the content of the book more broadly? (Admittedly, there are some axioms related to $=$ of the type described, but they are insufficient as they only address behaviour with respect to $+$, $\times$, etc.). – Anonymous Aug 24 '21 at 08:06
  • @Anonymous: The book was written during the new math era (late 1950s to early-mid 1970s) when basic set theory and logic notions were part of the curriculum. As for the rigor of the logical formalism, this was very thoroughly argued and discussed back then regarding what to say and what not to say. Pointers to the literature can be found using this answer and this answer and this answer. (going off-line for a few hours) – Dave L. Renfro Aug 24 '21 at 08:25
  • I'm aware this kind of thing was typical of the period. I'm wondering if you feel this might make the book less effective than, say, the book by Cohen. – Anonymous Aug 24 '21 at 08:26
  • @Anonymous: I think it largely depends on the person and also their later intentions. Given the widely offered current (beginning roughly mid 1960s, but mostly getting off the ground in the mid-late 1970s) U.S. 2nd college year courses in "how to prove things" (also called "transition to advanced mathematics"), it's not much of an issue, and Cohen is probably better. But again, this is very individual and I wouldn't try to choose unless I knew very specifically what the needs were (e.g. later study in math? going into physics? not a STEM college major?). – Dave L. Renfro Aug 24 '21 at 08:32
  • @DaveL.Renfro Thanks for your replies. – Anonymous Aug 24 '21 at 08:49
  • Does this answer your question? Study Order for high school math – user33704 Oct 08 '23 at 00:38

2 Answers2

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Compared to most standard undergraduate calculus textbooks, Calculus by Spivak is unusually rigorous and has many challenging problems. (It's also fun!)

However, the prerequisites for Spivak are no different than the prerequisites for those standard books.

You'll want some exposure to typical high school pre-calculus topics such as basic algebra techniques for solving equations, functions and their graphs (including lines, parabolas, and other polynomials) exponent arithmetic, triangle geometry, the pythagorean theorem, the quadratic formula, trigonometric functions and the unit circle, and so forth.

It sounds like you've been through that stuff before, so you should be fine. As you mention, pick up a pre-calc textbook so you can look things up when you need a refresher (though, you might be fine with just an internet connection). You don't need to remember all that stuff perfectly to get started. Just brush up when you run into something you don't remember.

Certainly, Spivak will be easier if you've already been through a more standard calculus class before, but that's not necessary.

One of the goals of the book is to rebuild what you know about real numbers and functions "from the ground up", meaning, familiarity with the above concepts is important, but Spivak aims to replace any shakier hand wavy pre-calc notions with much firmer, honest, foundations.

I suspect that reading the Real Analysis book you mention in preparation for Spivak is not only unnecessary, but backwards. Typically one would tackle Analysis after Spivak, not before. Spivak is often described as a stealth introduction to Analysis.

A few other Spivak tips based on my own experience (I'm a bit over halfway through)

  • Get a copy of the Answer Book.
  • Try to do all of the problems. They are the meat of the book, and contain many important results.
  • There are errors. Some are simple typos. Others are more substantial. MSE can be helpful when you're not sure if there's an error or not.
  • Try not to be discouraged when you get stuck on some problem or proof. Totally normal. If totally stumped, it's fine to look up the answer or search for a hint. In those cases, maybe make a note to return to the stumper at some later date to try it again and see if you can do it on your own.
Ben
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Although Spivak's calculus is a great rigorous introduction to calculus, it is important to note that it is still a calculus book. Analysis, or advanced calculus, is a theoretical, proof-based course in calculus. In order to gain the best possible understanding of the subject, I recommend studying calculus before analysis.

As for learning precalculus, Khan Academy is a great supplement!

I also recommend looking into http://www.mecmath.net/trig/Trigonometry.pdf
for a comprehensive guide to trigonometry and this book (https://sites.math.washington.edu/~m120/TheBook/precalTB2019.pdf)for precalculus. There are thousands of worksheets online that you can use to practice the material you learn in these resources.

Noah Champagne
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