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Sheldon Axler's "Algebra & Trigonometry" vs. "Precalculus: A Prelude."

  1. Jun 2, 2014 #1
    Hi all,

    I've been looking and searching around for a good book for some pre-calculus review, and decided on one of Sheldon Axler's books. Looking at his site and the contents of both books (links below), I'm not sure what the difference is between them, other than some re-ordering of things in the newer "Precaculus.." book. In fact, the "Algebra & Trigonometry" book seems if anything to have a bit more, which is a tad confusing. Maybe the newer "Precalculus.." books has corrections, better layout, etc.? Anyone use any or have any opinion on which to get? Check them out below. Besides the marketing reasons, why would anyone get the "Precalculus.." over "Algebra & Trigonometry"?




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  3. Jun 2, 2014 #2


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    I expect they will both be proof-based but the algebra/trig one will be at a lower level, for young people.

    If this is for some kind of home schooling situation, definitely don't use one of these. Youngsters shouldn't be learning from proof books, period.

    PS. I've looked at the precalculus one before but not the alg/trig, it may be different in style.
  4. Jun 2, 2014 #3


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    I have the A & T book now and used to have the 1st ed of the Pre C book (the grey one). The A and T book has 125 more pages-a lot of the difference is from Ch. 7 Systems of Equations, esp. section 7.4 Matrix Algebra, which was not in the Pre C book. To me, the books are about the same: they're both a bit too tough for me. I'm using Blitzer's 3rd ed. of Pre C, it's a lot easier for me. I might try Axler's book when I'm finished with Blitzer's. Hope this helps.
  5. Jun 2, 2014 #4


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    What do you mean by proof-based? Why is it bad? What do you mean by youngsters? Do you mean all youngsters? Some? Many? Most?

    People of all ages benefit from understanding what they are doing and why.
    Rote learning is not the best way.
  6. Jun 2, 2014 #5


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    I mean, preteens and young teenagers have short attention spans, they need short lessons with some kind of "cash value", something to hold the interest. They want to see a number line, not a successor axiom.

    And if you read the introduction to his Precalculus book, it reads very similar to the linear algebra one, that this is real math, if you're not spending an hour per page, you're going to fast, question and probe everything told to you, yadda yadda. I mean it's just not suitable for young first-time learners in general.
  7. Jun 2, 2014 #6
    I guess I should have posted a bit about myself. I am an adult student going back to finish my undergraduate degree that I put off as I worked in the IT field for many years (almost 20 years). I took calculus 1-3 before, and did very well, found it intuitive and not that hard to be honest, but I forgot most of it, and not only that, but the algebra and trig that are so important to understadning it. So this summer my plan is to review all the "pre" calculus mathematics on my own before I start school in the fall and take calculus 20+ years later. Things will come back, but I've got to work at it too. So I've been looking for a good, succinct, book that I can use to go and review all the things I've forgotten. And the Axler books I mentioned seem right on the money for me. I don't want a 1000+ page book with pretty pictures, 100 exercises, etc. I'm fine reading text like Axler's - in fact I prefer it. The math books I used 20+ years ago were less heavy and (I feel) better written, though I could have gotten lucky at the university I went to. Needless to say I didn't keep any! :(

    Anyway, I've looked again the the "Table of Contents" for both books I linked, and they seem to really be exactly the same. The order of topics is a bit different, as I was saying, and really minor stuff is shuffled around, but either will accomplish the same thing I feel. I'm probably just nitpicking. If you go to the links and look at the table of contents for both you'll see they are indeed the same. Same rigor, etc. I'm not sure why a "Precalculus" book is even needed when you have an "Algebra & Trigonometry", as that IS precalculus...I think "precalculus" is somewhat a confusing subject, and maybe because high schools in the US break down algebra I II, geometry, and trig too much. Anyway I digress...

    Another book I looked at is this:


    Lang's book linked above seems to be a good little gem, but not sure if it's enough on its own. Ideally I would just use one book. Axler or Lang. Any opinions?
    Last edited by a moderator: May 6, 2017
  8. Jun 2, 2014 #7
    Lang covers everything you need in order to succesfully study calculus and higher math. It doesn't cover more or less. So it most definitely is enough on its own.
  9. Jun 2, 2014 #8
    I looked and the only thing I saw maybe lacking a tad was the trig section, but I could be wrong just going on the table of contents from Amazon's "Look Inside".

    I forgot to mention that another gem seems to be this:

    Last edited by a moderator: May 6, 2017
  10. Jun 3, 2014 #9


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    It sounds like you know what you want. Of the Axler books you mentioned, I'd choose the precalculus one; it's probably the one he wrote first and will have a nicely abstract approach.
  11. Jun 5, 2014 #10
    While it may be true that I may know what I want, that doesn't make it easier to navigate dozens of books to find a good one. This has proven to be more taxing than I expected. I'm actually surprised at how bad most books have become, content watered down, 1000 examples, etc. Axler's books seem refreshing, but perhaps there are even better ones. Like Lang's "Basic Mathematics". So I'm curious and looking for opinions. While 1-2 books are good, too many (for me at least) leads to confusion, strange overlap, etc.

    In the process I'm also looking for a decent Calculus book to review from. There again seem to be the 1000+ page ones with dozens of examples, not enough theory, etc. Then there are ones that are on the other end - theory and not meant for review. The best I've seen so far, and please chime in, in terms of striking a good balance in terms of theory and application is George Simmons' "Calculus with Analytic Geometry" 2nd ed.


    MIT uses it, but I haven't seen much talk about it. A lot of talk about Apostol (Caltech uses it) and Spivak on this forum, but they seem a bit on the theory end for review. Thoughts?
    Last edited by a moderator: May 6, 2017
  12. Jun 5, 2014 #11
    Simmons is an excellent writer. I like all his books. Unfortunately, I have never had the chance of reading his calculus book. But I have no doubt it's very good.

    If you're not much into theory, then Spivak and Apostol are not good choice. I think Simmons will be best for you.
  13. Jun 5, 2014 #12
    I very much like theory, but wonder if Apostol or Spivak are the best choices when going back to review Calculus. When I took it, I found it easy and got top grades, but that was a while ago. I need something to kick me back into things. One thing I do know from experience, is that starting with the wrong book can be OK for a little while. You may do well and get top grades in Calc I-III, Linear Algebra, Diff Eq, etc. But if you are coming from a more "computational approach - i.e. engineering math courses", you will have trouble when later taking more advanced math courses which are more theoretical in nature. So for me, even now, the best way to start is with more words, less pictures, more imagination, more proof-based, etc. But there has to be a balance, and I'm not sure how balanced Apotol or Spivak are.
  14. Jun 5, 2014 #13
    Apostol really has a good balance. It's more theoretical than other books, but it's also practical. If you're already familiar with some calculus (perhaps long long ago), and if you're not adverse to theory, then try Apostol.

    Spivak would probably have too much theory for now.
  15. Jun 6, 2014 #14
    OK, maybe I should start a new thread as this is diverging into Calculus books, as I got the original topic out of the way. But I'll try it here first and hope people chime in :)

    I've looked at the 2 schools I need to choose between for attending starting this fall, and what books they use for their calculus curriculum.

    School 1 uses Thomas' "Calculus Early Transcendentals (12th Edition)":


    School 2 uses "Stewart's Calculus: Concepts and Contexts (4th Edition)":


    These seem similar to me from browsing them on Amazon's site, but don't know enough details about either.

    So the question is, which additional calculus book would best supplement either the Thomas or Stewart books? I'd like to supplement to better be prepared for the later courses, which will be more theoretical and harder. I'm not sure a smooth transition from either the Thomas or Stewart book will be easy without another book to supplement.

    So given all this, maybe it's easier now to recommend either the Apostol, Simmons, or Spivak books? Which combo would work best?

    Thomas/Stewart + Apostol
    Thomas/Stewart + Simmons
    Thomas/Stewart + Spivak
    Last edited by a moderator: May 6, 2017
  16. Jun 6, 2014 #15


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    So what book did you choose for algebra/precalculus? And how long do you think it'll take to work through it? Because I assume you want to get through that before starting calculus. Also that could influence your decision on which calculus book to buy.
  17. Jun 6, 2014 #16
    I chose Axler's "Precalculus 2nd Edition". It will take me 2 months at most to get through it now that I have it and have looked through it.
  18. Jun 6, 2014 #17
    Thomas/Stewart + Apostol and Thomas/Stewart + Simmons would be overkill, I'd go with Thomas/Stewart + Spivak

    By the way, I love these beautiful books, you should check it out :

    Analysis by Its History by Ernst Hairer and Gerhard Wanner
    Geometry by Its History by Alexander Ostermann and Gerhard Wanner
  19. Jun 6, 2014 #18


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    Ok, that is what I was waiting to see. [strike]This should set you in good stead for Apostol. If you like Axler, that is my recommendation. Otherwise, come back wiser later and you'll be better equipped to choose a book to follow it with.[/strike]

    Wow, $90 for Apostol volume 1, $240 new, who are they trying to kid? I'll follow this with a new post with some recommendations.
    Last edited: Jun 6, 2014
  20. Jun 6, 2014 #19


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    Last edited by a moderator: May 6, 2017
  21. Jun 7, 2014 #20

    I haven't looked at Lang's Calc stuff yet, but it seems the recommendation was because Apostol's price was too high? How do you think they compare in content and presentation, price aside. It seems Apostol was your first choice, but curious about more of an insight in addition to the recommendation.

    To your previous post, yes I do like the Axler book, and find it easy to work with, and things coming back quickly. It might have been 20 years ago I took these things, but I guess my brain cells are still there for the most part :) So I think calculus will come back quickly as well.

    I don't intend to use Apostol, Lang, Simmons, or Spivak for review over the summer (I may if I have time), but to simply complement the Stewart or Thompson texts, as I feel they are too computational and will not prepare me best for later courses as I was saying. I have a personal example of this if I may deviate for a minute here. When I was a physics student back in the day (at Cornell), I took mostly the math department variants of Calc I-III, Linear Algebra, etc. But one semester I needed to double up and had to take 2 of the math courses in the engineering department. Similar things, but more applied instead of theoretical. I did very well, As, etc. But the following semester, when taking a Mathematical Physics course which used 2 texts and special notes (one text was this one - https://www.amazon.com/Mathematical-Physics-Eugene-Butkov/dp/0201007274), for the first time in my life with math, I had a bit of a struggle at first. It wasn't that the material was harder, but that I wasn't able to just read the books anymore. And I think some of that had to do with having taken the engineering maths courses instead of the pure math courses the semester prior. I felt I came in inadequately prepared, had to work more at it, and it was very challenging. I ended up doing well, but yeah, I still think (and this is why I'm asking here) that a more theoretical approach initially, even in beginner courses, will prepare one better for later, more advanced texts. Thoughts?

    I'm also curious, how DO people here feel about Stewart/Thompson? Same as me? I wanted to get some discussion going more in depth, besides the welcome recommendations too.
    Last edited by a moderator: May 6, 2017
  22. Jun 7, 2014 #21


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    This type of discussion doesn't interest me, the question you should be asking is not, how do they compare, you should be asking, this is what I want, which one is closer to what I want? Or better yet, are these books sufficient for my needs?

    Also realise that whatever we say about books here, someone will come along later and reply and say, "In my opinion, the new edition of book X is really great, they've totally improved it!". So I don't even want to go there.
  23. Jun 7, 2014 #22


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    What does one say when one recommends a book? Is it, this is a book I would use, or, this is a book I think you would use?

    If I always say, this is the book I would choose, this would be unfair. There are many good books out there and many people out there. One can't always say, this is the book I would use. It has to be, this is a book that could work for you.

    And how does one judge that? It's a combination of reputation and perception. You perceive what type of book the person wants and choose a book based on reputation. For example, Apostol and Lang have reputations, are both rigorous books, and seeing that the author wants a rigorous book, that is the direction that the recommendations go.

    I have used Spivak and Axler books before and seen Apostol. I have to make a decision, a book in the style of Axler. Having seen Apostol, that is in the style of Axler. But the price is insane. Lang is cheaper and recommended many times on this forum. And both volumes can be had for less than one volume of the former book. Both are rigorous calculus books.

    This should be a slam dunk. Buy it or don't, that is all.
  24. Jun 7, 2014 #23
    I agree with this very much. Great post!

    But I want to make clear that Lang and Apostol are two very different books and are meant for different audiences. Both are on the rigorous side of calculus books. But Lang is clearly intended for a first course in the subject, let's say at a HS level. Apostol is more rigorous than Lang (Lang leaves out epsilon-delta arguments which I agree do not reall belong in a first course). I would use Lang for a (honors) course in a university. Still, Apostol focuses in the exercises a lot on computations (which tend to be quite tricky in comparison with other calc books). Spivak is even more theoretical than Apostol since its exercise tend to be almost all theoretical. Spivak is more of an analysis book in my opinion.
  25. Jun 7, 2014 #24
    I understand all that is being said, but I have already said, "While it may be true that I may know what I want, that doesn't make it easier to navigate dozens of books to find a good one. " So I don't see how asking for more insight and personal opinions is a bad thing. Yes I like Axler, and it's fine. If I got Lang's "Basic Mathematics" instead, it might also be fine and l would like it just as much. And I would have no problem describing the differences between them if someone asked. That's all really. So when I saw Apostol then Lang, I was just curious for more insight, as price will not play a role for me.

    So for me, yes, I like to see how they compare. Because as you say, these are some of the ones with great reputations, but distinct differences in writing style, etc. So it goes without saying they are all good - Spivak, Apostol, Simmons, Lang, etc. A comparison for me would help. A descriptive one. Yes, I can buy them all on Amazon, play the read and return game, but why not ask others that might have used them? I don't see the harm.

    Obviously I make up my own mind in the end, but I don't mind people chiming in with different views and opinions. Again, I don't see the harm. There could however be more harm in making a recommendation based on an intuition. And if someone knew what they really wanted, then there would be no question to begin with.

    I suppose it's gotten off topic, which is why I said maybe I should start a new thread, but it's all fine with me. I gave a personal example of why I thought the way I did, and was hoping to hear from others too.

    To get back to the topic, really it comes down to what works best with Thomas/Stewart as a *supplement*. If one has experience with either Thomas, Stewart, and the others mentioned as possible supplements, then that's what I wanted to hear. To make a recommendation without knowing Thomas or Stewart would be unfair. Otherwise it's all fine with me.
  26. Jun 7, 2014 #25
    Thanks. Yes, this is more like what I'm looking for. So Lang then doesn't make sense to supplement either Thomas or Stewart, based on what you say. For me, a supplement to a Thomas or a Stewart would be the book that is different than the computational style I see in those. So back to Apostol, Simmons, and Spivak :)

    One thing though is you said you like Lang. And I'm curious why. I'm curious why people like they books they do and how they learn. And discussion about that is only positive in my opinion. Like I said above, maybe I'd like Lang's "Basic Mathematics" just as much (or more even) than Axler's "Precalculus". I had to make a decision, and went with Axler. I could have gotten both, returned, one, etc. And again, I can do that with all of these. But am curious about why people like certain books, backgrounds, etc. Again, all replies are helpful.
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