# Which calculus text should a 36-year-old use for self-study?

I am 36 years old, and have forgotten a lot of math from high school, of which I only took up to Algebra 2. However I am teaching myself mathematics and am now, completely fascinated with the logic and beauty of it, but pitiably I am only now beginning to end Algebra 1. I've a long way to go yet, and have the Saxon method at home (Alg.1, Alg.2, Advanced Mathematics (trig and precalc), and Saxon Calculus). So far I like it, as it is what I am accustomed to, having used it in high school.

But my question is if a calc text like Spivak or Apostol would be better than the Saxon Calc. I have found that Saxon is not mentioned in the same circles as Apostol or Spivak.

• I wouldn't recommend just sticking with Calculus! Learn some proofs and basic number theory, set theory, topology, convex geometry, etc. There are many interesting topics that are accessible to learn without even knowing Calculus. As far as your question is concerned I would recommend Spivak or Adams-Essex – Samuel Reid Apr 5 '12 at 3:08
• The quality of Spivak's book is best appreciated by someone who has already done calculus at a more mechanical level. – André Nicolas Apr 5 '12 at 3:20
• I agree with the other two - Spivak is the way to go. – Joe Apr 5 '12 at 3:33
• What do you guys think about Stewart's Calculus? – Jashin_212 Apr 5 '12 at 19:03
• @Jashin_212 : I think it has a ton of exercises with complete,detailed solutions and that's its' one saving grace that makes it useful to instructors. To students-not so much since it gives a very weak presentation of calculus' basic theory. It's really a book about undergraduate problem solving rather then a calculus book-it's almost like the subject matter is irrelevant to the exercises. – Mathemagician1234 Apr 8 '12 at 5:54

My answer is not specific to calculus, but learning any subject from books:

Take any book that has a reasonable reputation. Start reading. You will sooner or later get stuck. Try reading some/figuring out what happens. If still stuck, get another book. Repeat. If this process begins to cycle a lot, then just switch to a different subject, read it for a while, come back later to the subject you start from.

This is what appears to work for me. I find it easier to learn about a subject when I know related this (even not directly related). Maybe it'll work for you.

(Regarding calculus text: a book I like is Hardy's "Course of Pure Mathematics")

• Why is there a downvote: is personal advice wrong-headed, or Hardy's book is not liked, or because I did not constraint my answer to the implicit constraint that one should use only one reading source? I appreciate feedback how to make my answers more helpful. – Boris Bukh Apr 9 '12 at 13:08
• Hey! I hope tou don't think I gave your suggestion a thumbs down. No no no, I appreciate all feedback I can get, it helps ,\me be more rounded out. I, in fact, like your suggestion, now if my budget will allow it... – Mario Piper Apr 10 '12 at 1:46
• Libraries are your friend! (Especially for older, more classic texts.) I fondly remember checking out math books from San Francisco Public Library every week or so. – Boris Bukh Apr 10 '12 at 7:12
• True, so true. I have checked out a number of books regarding modern physics, obviously written in layman's terms. And everyone has something different. – Mario Piper Apr 10 '12 at 19:47
• For what it's worth to posterity, the third edition of Hardy is available gratis at Project Gutenberg: gutenberg.org/ebooks/38769 – Andrew D. Hwang Jul 22 '13 at 14:11

Reading your situation, I can offer some advice from someone who is 33 years old and currently self-studying Apostol. I cannot speak to Spivak directly, but I imagine the things I have to say about Apostol will apply more or less equally to Spivak.

Apostol is a wonderful book, but would not be my choice as an introduction to either calculus or proof-based mathematics. My background differs somewhat from yours in that I took a full year of Calculus in high school, and have an engineering degree so took classes on ODE, linear algebra, and statistics (amongst some other courses that made significant use of mathematical techniques). That was years ago, so I've forgotten many things, but with that background I find Apostol challenging at times, and it certainly requires a lot of motivation to keep moving forward. Without someone to guide you, there are lots of places in a rigorous treatment of Calculus to get hopelessly confused. If you look at my profile you can see the questions (an expanding list) that I have asked from problems in Apostol to get a flavor for the sorts of exercises that the book contains. I think it would be very difficult to undertake the book without a good background in proving mathematical statements rigorously, and a strong notion of the ideas of Calculus. The theorems and definitions would just be too obtuse if you didn't already understand what was going. Apostol is not the place to learn these things.

That said, I would echo one of the comments that there is math to learn other than Calculus. Not to say Calculus shouldn't be on your agenda, just that there are beautiful and accessible areas of math that might appeal to you and are better for developing mathematical technique. I strongly recommend looking into books on elementary number theory (preferably something with complete solutions) or discrete math. These areas are completely accessible to you right now, provide a forum in which to apply some of the basic mathematical techniques you are learning, will really help you develop your skills with manipulating equations and "thinking mathematically," and will start you on the road to understanding the difference between solving equations at the algebra level and writing proofs.

As for actual book recommendations, a very straightforward number theory book with solutions is Jones & Jones. It's not a great book at all, but it has full solutions and is targeted at undergraduates. A very ambitious book might be Concrete Mathematics, much of which will be too advanced at the moment, but is a book that can grow with you. It contains wonderful discussions of very concrete (as the title might suggest) problems, and methods for working with sums and equations which are invaluable as you gain maturity.

Hope that helps.

Added: Also since you seem to be interested in mathematics generally, I would highly recommend taking the time to read some math history and "popular" math books. Math has a wonderfully rich and entertaining history. These can give great insight into problems that have been pivotal to the development of mathematics, and give glimpses to the sorts of things mathematicians are interested in. Also, many "pop" math books are about theorems or concepts that have fascinated mathematically minded people for centuries, so they are likely to fascinate you. Finally, they can be read right now while you are still developing your basic math skills, and can provide lots of motivation and inspiration.

Specific books I'd recommend are E.T. Bell's "Men of Mathematics" for math history, some of recent books on the Riemann hypothesis are good reads about an open problem (du Sautoy, The Music of the Primes and Derbyshire's "Prime Obsession" were both good), and I found "Symmetry and the Monster" by Ronan really inspiring (if you want to get a glimpse of what "modern" algebra is about).

• Thank you for that comment. I am, I guess, afraid that if I choose the wrong book, I'll sacrifice some important aspects of what I should know. I have a book on number theory by Pommersheim/Marks/Flapan called Number Theory, A lively Introduction with Proofs, Applications, and Stories. If you suggest getting into number theory, would you suggest that I complete Algebra 2 and Trigonometry first, or should I begin right away? In just a little while I will be learning to complete the square and derive the quadratic equation. Thanks again. About to run out of characters. – Mario Piper Apr 5 '12 at 19:26
• Mario again: when I wrote about completing the square and deriving the quadratic equation, it was a comment of where I am currently at, just to clarify. And I noticed the book I have on number theory does not have any solutions. I was given the book by a math teacher at the high school I work at, but never really looked at it yet, so I guess it won't be all that helpful. But the Jones and Jones looks interesting. More later. – Mario Piper Apr 5 '12 at 20:23
• @MarioPiper I wouldn't worry about choosing the "wrong" book at this point. Some books will cover some topics more thoroughly than others, but learning math is not a linear process: you learn and relearn things as you go. Even if a book doesn't cover some point in depth, you'll revisit it again in a later book. There's no reason not to try out reading some number theory (or any other area that you find interesting) books and see how they go. It's good to see the math you're learning in one area applied elsewhere, and will improve your skills more rapidly than practice alone. – user23784 Apr 6 '12 at 0:14
• Thank you for that insight. I will take that advice to heart. – Mario Piper Apr 6 '12 at 22:43
• @MarioPiper Hey, I also just added a paragraph to my answer regarding "other" math reading that you may want to consider. – user23784 Apr 8 '12 at 5:30

I think one's first exposure to calculus-no matter how gifted or ambitious the student is-should be a physically and geometrically motivated approach that illustrates most of important applications of calculus. Sadly, many people think that means a "pencil-pushing" or "cookbook" approach where things are done sloppily and with no careful explanation of underlying theory. That's simply not true. You can certainly do calculus non-rigorously while still doing it carefully enough to give students the broad picture of the underlying theory. I think in the case of a late beginner like you, this is even more important since your mathematical instincts are either undeveloped or so rusty as to be near useless. You'll need this combination of intuition and careful rigor to truly learn the material correctly.

The best example of this kind of book, to me, is Gilbert Strang's Calculus. Strang's emphasis is clearly on applications and it has more applications then just about any other calculus text-including many kinds of differential equations in physics (mechanics), chemistry (first and second order kinetics), biology (modeling heart rythum) and economics and a basic introduction to probability. But Strang doesn't avoid a proof when it's called for and the book has many pictures to soften the blows of these careful proofs. This would be my first choice for a high school student just starting out with calculus and I think it'll work very well for someone in your situation.

• Thanks! I'll look into it. I am a person who likes the essence of something, be it espresso, unsweetened chocolate, or other such culinary delights. In mathematics, I tend to dislike pictures, but rather, I find great delight in a page full of equations. I don't know why, but that is what appeals to me. It'll be nice to check out Strang's calculus, though... – Mario Piper Apr 10 '12 at 9:44

I think Saxon's books can be good for fairly weak high school students who also do not intend to progress further than precalculus or a first semester calculus course. However, I do not think Saxon's books are appropriate for anyone who aspires to reach Spivak's calculus level at some future time. My suggestion would be to look at the first two references I cited yesterday at Preparing For University and Advanced Mathematics (the 4 Gelfand School Outreach Program books and Mary P. Dolciani's book), followed by Mathemagician1234's suggestion of Gilbert Strang's book.

• Thank you for your response to my question. I do have another question resulting from yours. What makes Saxon so bad in your opinion. I ask that, not in disrespect, but curiosity. Have you gone through them, or seen other students fall behind because of them? Just curious. – Mario Piper Apr 6 '12 at 22:41
• @Mario Piper: This is a strange situation for me! Usually when a discussion of Saxon's books comes up (internet discussion groups, real life discussions with colleagues, etc.), I'm on the side that sees some good in them. However, my usual argument is that they can be good for weak students whose teachers teachers are also weak (in math) or inexperienced. Such students and teachers, in my opinion, are well served by the texts' no-nonsense and sterile (from a mathematical enrichment perspective) approach that uses a strong sterile learning approach. continued – Dave L. Renfro Apr 9 '12 at 14:09
• @Mario Piper: Given that you're just ending with algebra 1, I don't think it matters much if you continue with Saxon. If you like Saxon, I think Saxon's algebra 2 text would be fine. However, once you start getting into trigonometry and precalculus, I recommend looking at other texts, even if you continue with Saxon. As for whether to continue into Saxon's calculus or another calculus when the time comes, it would be best to wait until that time comes, since then you'll probably know a lot more about your goals and capabilities (mathematically and how much time you're willing to spend). – Dave L. Renfro Apr 9 '12 at 14:17
• Thanks!!! I can see your dilemma. I do like the sterile, if you will, approach so far. I'm not big on pictures, although I am a photographer, go figure. Anyway, do you have any suggestions on what Algebra 2 book, as well as a trig and precalc books, would be helpful? Thanks! – Mario Piper Apr 10 '12 at 1:42
• @Mario Piper: First, I would suggest getting the (inexpensive) "Gelfand School" books. See math.berkeley.edu/~wu/Gelfand.pdf for a review of 3 of these books, as well as mathforum.org/kb/message.jspa?messageID=6654316 for a comment of mine about one of the problems in the algebra book. However, you'll want something more comprehensive, with more exercises, than the Gelfand books for your main study purposes. For this, Stewart's Algebra and Trigonometry amazon.com/dp/1439047308 would definitely work, but it's very expensive. – Dave L. Renfro Apr 10 '12 at 14:42
1. I love Saxon's books. I'm actually at (or near by 1-2pts) the top of my college CalcII class, and know that Saxon's books have prepared me incredibly well.

2. As far as a recommendation for learning Calc, I suggest using http://www.khanacademy.org/.

Based on your background, I would try to go simple, simple, simple, easy, first. Some suggestions: *Continue the Saxon program (it is working!) *Schaum's Outline *Granville, Smith, and Longley

Remember you can always go deeper later. Richard Feynman first learned calculus from "The Practical Man's Guide" not from a ballbuster. Perfecting something easy will serve you much better than going through something too hard and not really learning the basics well (or even giving up).

Don't jump into the Spivak, Courant, Apastol, Hardy, etc. that are often recommended here by people who are super accelerated and gifted (and theory oriented). Note, that this isn't saying that you can't ever look at those books. But baby steps and progression. Don't start a running program with a marathon.

P.s. I realize this is an old question, but the Q&A serves searchers, not just the original writer.

I look back so fondly on my AP Calculus text by Best and Penner. With the red and green covers. Of course I also had a really good teacher. Yes, I was able to get a $$5$$ on the AP Calculus BC. This probably helps contribute to my fond memories.

But I really felt that the authors managed to communicate a deep understanding of the subject. It was my impression that the two volumes were masterfully written.

That being said, I have since looked at (what seems like) plenty of other calculus texts, and have found that most have something to offer.

Also, to echo one of the comments, why limit yourself to calculus?

One last word: one of my favorite professors at Berkeley was Hung-hsi Wu, who once said (something to the effect that) if you don't understand something you're reading, put it down.

Keep a good lifestyle also helps. The start may be unexpectedly difficult, but bit by bit, you will accumulate enough to finally start a normal learning process.