I would strongly suggest Jerry Shurman's Calculus and Multivariable Calculus texts, which are available for free on his website:
people.reed.edu/~jerry/
He posts answers to the exercises on his website as the course progresses each year. This year there are no answers because he's not teaching the course; he's on sabbatical.
IMO the theorems in Apostol are tricky reading for most first learning Calculus.. Although probably for the HN crowd are appropriate. But I can't argue with the content or preparation it provides.
I personally prefer Stewart as I think it is appropriate for a wider range of ability, while still being excellent.
Ha, yeah. Apostol definitely was a bit painful at first. But it felt like deriving calculus helped me learn better and remember it longer than more rote methods.
Apostol is quite dry and unmotivated. A grad student friend of mine recently wrote a textbook to replace it, and has been testing it on the freshman honors calculus course here.
"Mathematics: Its Contents, Methods and Meaning", seconded! I bought it a couple of years ago when I first read about it on HN and have to say that it is a highly readable account of mathematics.
Thank you for your feedback, psykotic. In our defense, the submission form was open to everyone. We simply published the entries we received. We may do a follow up post that incorporates other suggestions, if we receive enough of them.
I appreciate the effort to put together the list but the site looks very spammy to me.
All the book links are affiliate links to Amazon. There are two adsense blocks one of which is right under the title (and is quite annoying). Most comments about the books don't explain why the book is good, and don't give much more information than what I can find in Amazon reviews.
EDIT: I wrote "site" but I actually meant "post", sorry about the confusion.
Yes, this particular post could seem spammy, but I have been reading this blog for quite a while and generally find the content to be relevant and interesting.
Even with this particular post, I personally have no problem with affiliate links and this list is an interesting compilation which the readers of the blog helped to create.
Sadly the list is missing my favorite math book, "Linear Algebra Done Right" by Sheldon Axler. It's an absolute must if you are learning linear algebra and want a deep understanding of it.
Rather than make you grind through mechanical operations on matrices, as most books do, this book takes a coordinate-free algebraic approach and does an amazing number of things with it, cutting directly to the chief insights. There are ninja proofs in this book, 4 lines long, showing some deep and useful thing about linear algebra that other books would spend pages proving in a very verbose and uninsightful way.
I would have preferred a list with more editorial input than "listed in the order received." As a math novice, I'd be much more interested to learn what Salman Khan's favorite math book is than Joe First-Post's.
I own this book, and I use it occasionally, but it really requires the right audience.
Firstly, it has several parts. Some of these are historical, philosophical, or otherwise interesting to a general audience, but some of them are technical (I would put most of pages 157-729, nearly 60% of the book, in this category).
For people that are interested in mathematics, but do not have a fairly extensive background (I would estimate 2-3 courses beyond the calculus series), these technical sections are probably not very useful (and certainly not entertaining).
For people (like me), who are still pursuing an education in mathematics, I would say this book is indispensable. It gives a great overview of individual branches of mathematics, including fairly rigorous explanations of important results and conjectures.
The exposition portions of the book are good, but don't warrant the high price of the book on their own. Overall, though, it is a tremendous reference.
Edit: I forgot to mention that the folks who write the articles are all outstanding mathematicians - many of them Fields medalists or other award winners. That makes for some very good reading.
I agree with everything you said and would only add that if you have studied lots (for example, are a graduate student in mathematics), then it will probably be frustratingly shallow on anything you know reasonably well (or not mention it at all, if it's specialized), but it is still an amazing resource to get started in fields of math that are not your own.
This highlights one of the weaknesses of the list -- the apparent lack of an editor. Whereas shou4577's comments explains the book's strengths and target audience, the original submission's description was so concise as to be near useless ("Coherent overview of all of pure mathematics.") This isn't intended as a slam of the original suggester, but to point out that the blog listing is only a starting point and in some regards less useful than the topic book lists some Amazon reviewers compile.
Gilbert Strang, Linear Algebra and Its Application, THIRD EDITION. Very undergrad, somewhere in between first and second semester level, and oriented to matrices (opposite of Axler), but very readable and useful.
I found "A Logical Approach to Discrete Math" by Gries and Schneider to be delightful. It teaches discrete math using formal logic as the basis of the whole system.
I found Concrete Mathematics to be a bit dense and not very suited for self learners like me. Also it does not cover the basics like proofs, counting etc.
I've read good reviews of Susana Epp's Discrete Mathematics With Applications, and I liked what I read in a preview. It's expensive, though, so I'm waiting for the international edition (about 1/3 the cost), which is due out next month.
