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What are some good undergraduate level books, particularly good introductions to (Real and Complex) Analysis, Linear Algebra, Algebra or Differential/Integral Equations (but books in any undergraduate level topic would also be much appreciated)?

EDIT: More topics (Affine, Euclidian, Hyperbolic, Descriptive & Diferential Geometry, Probability and Statistics, Numerical Mathematics, Distributions and Partial Equations, Topology, Algebraic Topology, Mathematical Logic etc)

Please post only one book per answer so that people can easily vote the books up/down and we get a nice sorted list. If possible post a link to the book itself (if it is freely available online) or to its amazon or google books page.

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  • $\begingroup$ This is borderline, but I think it is a legitimate question of interest to math instructors. I think its better as a community wiki though. $\endgroup$
    – David Zureick-Brown
    Commented Oct 16, 2009 at 17:28
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    $\begingroup$ It's no longer possible to add useful answers to this question (as there are too many!) and it's unclear whether this question would be "allowed" by modern standards -- far too broad. As it's been popping back to the front page fairly frequently, we've decided to close it. $\endgroup$
    – Kim Morrison
    Commented Jul 11, 2010 at 13:30
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    $\begingroup$ See discussion on meta: tea.mathoverflow.net/discussion/499/… (and remember to vote this comment up, so it is visible to others) $\endgroup$
    – Victor Protsak
    Commented Jul 14, 2010 at 10:34
  • $\begingroup$ I don't have enough rep to rate anything and I only skimmed through the meta discussion but essentially what the mods did is bad for the website... Most websites start with one thing and then change their purpose by the will of the users... Otherwise they fail miserably as soon as some competition shows up... So the final judgment for closure "The system was created for people looking for precise answers to precise questions. Big list questions were an emergent phenomenon" is generally a flawed mindset. $\endgroup$
    – person
    Commented Jul 16, 2010 at 23:19
  • $\begingroup$ I asked the question because I am an undergraduate student and want to learn more about math... A lot of great books were recommended here and I guess what was suggested so far is more then enough for me to read... So at this point I don't really care about the policies at MathOverflow but my friendly advice is given in the preceding comment... And NO I'm not gonna create a new account for meta just for one post ¬_¬ $\endgroup$
    – person
    Commented Jul 16, 2010 at 23:22

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Algebraic Topology by Hatcher (available online here).

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    $\begingroup$ If you like print books (easier to carry around, scribble in, etc.), Hatcher's book sells for $37, which seems pretty reasonable. $\endgroup$
    – Michael Lugo
    Commented Oct 16, 2009 at 20:12
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Generatingfunctionology by Wilf is fun, free, requires very little in the way of prerequisites, and is as good an introduction to the methods of analytic combinatorics as could be asked for. It's long been one of my favorite textbooks.

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    $\begingroup$ The free edition is the second (early 1990s?); there's a third edition (2005), which as of now is not free. The third edition doesn't differ that much from the second one, though. For a more advanced book that massively expands upon Wilf, I recommend Flajolet and Sedgewick's <i>Analytic Combinatorics</i>, published 2009 (also available <a href="algo.inria.fr/flajolet/Publications/AnaCombi/… online!</a>) -- but this is really a graduate-level text. $\endgroup$
    – Michael Lugo
    Commented Oct 17, 2009 at 5:11
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Spivak, Calculus

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    $\begingroup$ I agree, with the caveat that the subject of the book is usually called "Elementary Real Analysis" these days. That said, when I first read this book I loved it so much that it made me want to be a mathematician. It's both rigorous AND intuitive, in a way that both qualities complement one another. $\endgroup$
    – John Goodrick
    Commented Oct 16, 2009 at 21:46
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Concrete Mathematics, Graham, Knuth and Patashnik. Extremely useful, very good exercises, and a sense of humor that appeals to me.

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    $\begingroup$ I finally gave in and bought this book last week, after realizing that at any given moment over the last few years I was more likely to have it checked out of the library than not. $\endgroup$
    – Michael Lugo
    Commented Oct 17, 2009 at 5:04
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A basic undergrad algebra book which I feel is not as well known as it should be is Michael Artin's Algebra. I have it in soft cover so I hope it's actually the one in this Amazon link. Anyway it's beautifully written, provides context and motivation and is just a pleasure to read or browse. How often do you find a basic text written by a world-class expert?

"Always study from the masters".

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    $\begingroup$ As far as I know there is only one book by Michael Artin with that title. I have the hardcover and it looks like the one you link to. Apparently Artin is working on a new edition. $\endgroup$
    – Michael Lugo
    Commented Nov 1, 2009 at 17:15
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    $\begingroup$ That book is so much better than Dummit and Foote for undergraduates. D&F is also useless at the graduate level, where much better texts like Lang blow it out of the water. $\endgroup$
    – Harry Gindi
    Commented Nov 30, 2009 at 12:16
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    $\begingroup$ Artin is going to be rough going for undergraduates who are not well versed in basic geometry and linear algebra,fpqc.But you can't help but love the infectious passion with which Artin weaves his craft in front of the students.He loves algebra and he's trying to prosyletize his students to it. A book with a similar geometric bent,level and also by a master that students will probably find easier going is E.B.Vinberg's A COURSE IN ALGEBRA. But Artin's book is very good and it's good news for all of us that Artin is revising it. $\endgroup$
    – The Mathemagician
    Commented Mar 27, 2010 at 22:01
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For undergraduate level topology (mostly point set topology) I recommend "Topology" by Munkres. I learned topology from this book as an undergrad and I remember this being one of my favorite books at the time.

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  • $\begingroup$ I felt the same, it is great to read as an undergrad (I recommend students to try to do most of the proofs mentally instead of reading them). $\endgroup$
    – Jose Brox
    Commented Dec 21, 2009 at 15:04
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Milnor, Topology from the differentiable viewpoint.

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  • $\begingroup$ A variant that many prefer over Milnor would be Guillemin and Pollack's Differential Topology. $\endgroup$
    – Ryan Budney
    Commented Nov 6, 2009 at 21:26
  • $\begingroup$ @Ryan. It's a matter of taste, of course. Milnor does much less material that Guillemin and Pollack, but reading it was an amazing experience for me. Guillemin and Pollack is a very good book, but I never got nearly as much from it. $\endgroup$
    – Ilya Grigoriev
    Commented Jan 23, 2010 at 7:09
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    $\begingroup$ No no no!! This book is awful as an undergraduate text! It's a great reference for someone who already knows the material, but the proofs skip many "simple" steps, and the author makes no attempt to explain the concepts from a intuitive point of view! Our professor assigned this book for the undergraduate course in Topology at SUNY Stony Brook, and at the time it was of absolutely no help to me whatsoever. $\endgroup$
    – BlueRaja
    Commented Jul 7, 2010 at 4:31
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Serre, A Course in Arithmetic.

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The Chicago undergraduate mathematics bibliography is a nice annotated list of books.

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  • $\begingroup$ Our own Pete Clark is one of the authors of that list way back when he was an undergraduate there,I believe,Michael. $\endgroup$
    – The Mathemagician
    Commented Mar 18, 2010 at 20:49
  • $\begingroup$ Pete's online CV has him graduating from Chicago in '98, so you're right (unless two mathematicians named Pete Clark came from Chicago in the same year!) $\endgroup$
    – Michael Lugo
    Commented Mar 18, 2010 at 21:28
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    $\begingroup$ +1000 It was a GREAT idea to develop a database of critical reviews sorted by subject and difficulty level, a few of them by multiple people. Even if the list itself is somewhat obsolete, one gets a good feel of relative strengths and weaknesses of "canonical texts" ca 1998. $\endgroup$
    – Victor Protsak
    Commented May 24, 2010 at 6:20
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Ordinary Differential Equations by Vladimir I. Arnold

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Needham, Visual Complex Analysis. I read this while in high school, and it's simply beautiful. I recommend this book as a supplement to any first course in complex analysis (a different book should probably be used for the main textbook since Needham's is very pretty, very engaging, but not very rigorous).

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  • $\begingroup$ Excellent book. $\endgroup$
    – John D. Cook
    Commented Jul 10, 2010 at 21:07
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Miles Reid, Undergraduate Algebraic Geometry.

ps - anyone who thinks one can teach an undergraduate class out of Hatcher's Algebraic Topology (which is a great book) at more than 10 universities in the US is sadly deluded. Ditto for a few more things I've seen here.

pps - somewhere between a third and half of the math majors here could handle abstract algebra out of Artin. It would be great for those that could, but we're not going to ditch half our students.

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    $\begingroup$ Very astute comments! Many answers to "best textbook" questions are way off the deep (macho) end. $\endgroup$
    – Victor Protsak
    Commented May 24, 2010 at 5:48
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Not exactly one of the topics in the question, but I particularly liked Silverman and Tate's Rational Points on Elliptic Curves.

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Differential Geometry of Curves and Surfaces, by Manfredo Do Carmo is an excellent introductory book.

https://www.amazon.com/Differential-Geometry-Curves-Surfaces-Manfredo/dp/0132125897

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The Princeton Lectures in Analysis by Stein and Shakarchi are great introductions to Fourier, complex, and real analysis (in that order!).

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Complex Analysis, by Lars Ahlfors

https://en.wikipedia.org/wiki/Lars_Ahlfors

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    $\begingroup$ It is a complete mystery to me why people are still using this monstrosity. Actually,it's not-it's because Ahlfors was a God at Harvard and they're afraid of being struck down by lightening using anything else.I can think of at least a half a dozen texts now that are better then this one. $\endgroup$
    – The Mathemagician
    Commented Mar 18, 2010 at 20:42
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    $\begingroup$ I'm not a huge fan of this book, either... $\endgroup$
    – Qiaochu Yuan
    Commented May 25, 2010 at 6:32
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    $\begingroup$ 'Elementary Theory of Analytic Functions of One or Several Complex Variables' by Cartan is a far superior text. $\endgroup$
    – Saikat Biswas
    Commented May 10, 2011 at 23:51
  • $\begingroup$ @TheMathemagician, which books would you recommend instead of Ahlfors? $\endgroup$
    – FNH
    Commented Dec 24, 2017 at 23:41
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    $\begingroup$ @FawzyHegab The most up-to-date,complete and user-friendly text on the subject that exists is COMPLEX ANALYSIS by Theodore Gamelin. Requiring only a background in nonrigorous calculus,the book brings one from the basics of complex numbers through analytic functions through differential and integral complex calculus to graduate level topics like Julia and Mandelbrot sets and Runge's theorem. It also includes many applications to geometry and physics, such as Euclidean mappings and fluid dynamics. It also has a several discussions of Riemann surfaces.at different levels. An outstanding book. $\endgroup$
    – The Mathemagician
    Commented Dec 25, 2017 at 6:07
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Spivak, A Comprehensive Introduction to Differential Geometry. There is a nice geometrical philosophy and plenty of motivation.

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    $\begingroup$ Especially the newer version which is typeset in LaTeX! The typesetting of the older version is still charming in its own ugly way, though. Also, I'm not sure if Spivak should count as an undergraduate level book... $\endgroup$
    – Kevin H. Lin
    Commented Dec 28, 2009 at 15:24
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Real and Complex Analysis by W. Rudin is a beautiful and extremely well written book which presents the fundamentals of real and complex analysis highlighting the interactions between different results and ideas.

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    $\begingroup$ I'm all for "adult rudin"... but undergraduate? I don't see it. Have you seen it used at the undergraduate level? $\endgroup$
    – Michael Hoffman
    Commented Oct 30, 2009 at 13:28
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    $\begingroup$ Actually, Rudin has an undergraduate-level book also, the "small Rudin". I learned from it, and it was fine. I loathe the big Rudin, though, even for graduate level. I never managed to learn anything from it; I especially hated the way every proof refers to a million previous results as "Lemma 12.1.8" without mentioning what they actually are. As a result, reading anything required flipping through the whole book after every line just to know what he is talking about. $\endgroup$
    – Ilya Grigoriev
    Commented Jan 23, 2010 at 7:16
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    $\begingroup$ I don't like either Rudin particularly,to be honest.Adult Rudin tries to put too much into one book.Folland is the same level and is just so much more pleasant to read. $\endgroup$
    – The Mathemagician
    Commented Mar 18, 2010 at 20:44
  • $\begingroup$ Adult Rudin is used for (3rd year) undergraduates at Cambridge. $\endgroup$
    – user85798
    Commented Jun 24, 2014 at 21:59
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Guillemin and Pollack, "Differential Topology"

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  • $\begingroup$ Even with the tons of good introductions to differentiable manifolds available these days,Deane-this is STILL one of the very best and well worth the money. $\endgroup$
    – The Mathemagician
    Commented Mar 27, 2010 at 21:51
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Dummit and Foote's Abstract Algebra is an excellent book for learning group theory, ring theory, and module theory. There's also a section on basic algebraic geometry and homological algebra.

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  • $\begingroup$ Galois theory is also covered quite extensively. The Algebraic Geometry section doesn't discuss Projective varieties, so I don't recommend it for AlGeo. However, the problems are ample so also counter-examples. $\endgroup$
    – Abhishek Parab
    Commented May 26, 2010 at 4:17
  • $\begingroup$ From what I've read so far it really is a great book, and the fact that it covers such a major part of basic abstract algebra makes it even greater. $\endgroup$
    – Pandora
    Commented Jul 10, 2010 at 21:40
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I didn't see any suggested books from the great Russian school of mathematics, here is a brief list of superb, well written, example oriented books:

  • Elements of the Theory of Functions and Functional Analysis by A. N. Kolmogorov and S. V. Fomin
  • Theory of Functions of a Real Variable by I. P. Natanson (this I think, it's hard to find)
  • Theory of Functions of a Complex Variable, Second Edition (3 vol. set) by A. I. Markushevich
  • Elements of Functional Analysis by L.A. Lusternik and V.J. Sobolev
  • Problems in Mathematical Analysis by B. Demidovich
  • Calcul intégral et differentiel (2 vol. set) by N. Piskounov. In English should be something like Differential and Integral Calculus
  • A Course of Mathematical Analysis (2 vol. set ) S. Nikols'skii
  • Differentialrechnung und Integralrechnung (3 vol. set), by Gregor M. Fichtenholz. Unfortunately, there is no English translation of this book, only the German translation that it's mentioned. I think this was THE calculus book on Russia. THIS BOOK SHOULD BE TRANSLATED INTO ENGLISH, and I suspect that there is no copyright, it appears around 1959 I think.
  • Mathematical Analysis (2 vol. set) by Vladimir A. Zorich. This is a very recent book from a great mathematician which is in Moscow university. The book is based upon is lecture.

I have also found a Spanish translation of a book written by S. Banach about "Differential and integral calculus". It is very good as an undergraduate book. The spanish translation (for those who want to search) is Calculo diferencial e integral

Books that undergraduate should not touch (in my humble opinion) are books written in bourbaki's style.

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Ireland and Rosen, A Classical Introduction to Modern Number Theory is a great second course in number theory. In spite of being part of "Graduate Texts in Mathematics" series and unlike Rudin's Real and Complex Analysis (see a comment above), this is a book at the undergraduate level. It only presupposes undergraduate algebra as in Herstein Topics in Algebra or M. Artin's Algebra, undergraduate analysis like in Rudin's Principles of Mathematical Analysis and basic number theory. In fact it recalls or proves many of the necessary results in each of those fields. A Classical Introduction to Modern Number Theory bridges the gap between basic number theory (that covers modular arithmetic, Fermat's little theorem and QR) and books like Lang's Algebraic Number Theory or Cassels and Fröhlich.

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    $\begingroup$ GREAT book. The best number theory book for honor students. $\endgroup$
    – The Mathemagician
    Commented Jul 7, 2010 at 5:42
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As an undergraduate, I loved Shafarevich's book Basic notions of algebra. This is not a textbook, but gives small beautiful tastes of a broad choice of topics in algebra, emphasizing connections with other fields.

I found it very stimulating, in the sense that every example or overview of some topic in this book made me want to learn more details about it. In fact I became interested in algebraic geometry because of this book.

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  • $\begingroup$ The absolute best of scores of algebra books that I've ever read. $\endgroup$
    – Victor Protsak
    Commented May 24, 2010 at 5:39
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Topics in Algebra by I. N. Herstein. A new edition will be coming out this year.

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  • $\begingroup$ Herstein is long dead. Would love to know who did the revision and I hope it didn't get watered down in the process. $\endgroup$
    – The Mathemagician
    Commented May 26, 2010 at 2:16
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I am surprised no one has mentioned Halmos' Naive Set Theory or Finite-Dimensional Vector Spaces or Rudin's Principles of Mathematcal Analysis. There's also Sheldon Axler's Linear Algebra Done Right and Royden's Real Analysis.

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    $\begingroup$ I believe we are supposed to put just one book per answer. It makes it easier to wote up. $\endgroup$
    – GMRA
    Commented Oct 17, 2009 at 18:18
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    $\begingroup$ Sheldon Axler (not Steven), for what it's worth. $\endgroup$
    – Tom Leinster
    Commented Oct 24, 2009 at 2:48
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For discrete mathematics, I would recommend Van Lint-Wilson's "A Course in Combinatorics" as a good introductory text. It consists of 38 (in my edition) chapters that give (often largely self-contained) introductions to various areas of the field. Although it doesn't go nearly as in depth as, say, Stanley's "Enumerative Combinatorics" or a text focused solely on graph theory, I found it excellent for giving a broad overview and indicating to me where I wanted to explore deeper.

My one caveat would be that some chapters require background in either linear algebra or basic group theory, though those are easily skippable due to the structure of the book.

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Basic Algebra by Jacobson.

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    $\begingroup$ Volume I is a great guide for the advanced undergraduate, but I think that volume II is beyond what all but the most sophisticated undergraduates can deal with. $\endgroup$
    – David E Speyer
    Commented Oct 30, 2009 at 13:05
  • $\begingroup$ I agree: volume II is harder. $\endgroup$
    – lhf
    Commented Oct 30, 2009 at 18:05
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General geometry: Coxeter, Introduction to Geometry.

Not so much a textbook as a collection of essays (in particular, it doesn't have exercises), but all of the essays are instructive and enlightening.

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Ok, this is not a single book, but I have often found books from the Springer Undergraduate Mathematics Series (SUMS) to be excellent. Here is a list of titles.

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Another one I like is "An introduction to Lie algebras." by Erdmann and Wildon.

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