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In his book "Men of Mathematics", Eric Temple Bell repeatedly makes the point that a student of mathematics must read the classics.

My question is what are some classic books in mathematics ( Dictionary definition : judged over a period of time to be of the highest quality and outstanding of its kind.) that can be used by a high school/undergraduate student to start the study of higher mathematics?

Some subjects I would like reference in particular, otherwise state any book you consider a classic, are:

1) Analysis 2) Abstract algebra 3) Linear Algebra 4) Number theory 5) Combinatorics and Graph theory, etc.

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closed as primarily opinion-based by Morgan Rodgers, 2012ssohn, JonMark Perry, zhoraster, Rohan Dec 13 '16 at 6:36

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ "Principles of Mathematical Analysis" by Walter Rudin should be included. It's a classic and is still being used regularly. $\endgroup$ – Namaste Dec 12 '16 at 17:35
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    $\begingroup$ Your question has now become a bit too broad: you added eight additional disciplines after asking only about such texts in three disciplines Indeed, for each topic, you should search the questions here for highly recommended texts. $\endgroup$ – Namaste Dec 12 '16 at 17:40
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    $\begingroup$ @amWhy, Who can make it community wiki post?? $\endgroup$ – I am Back Dec 12 '16 at 17:47
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    $\begingroup$ Answerers can do so for individual answers; perhaps askers can do the same (may depend on rep), but if so, when an asker checks off on it, it makes all answers community wiki, too. $\endgroup$ – Namaste Dec 12 '16 at 17:50
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    $\begingroup$ I upvote such questions whenever I find that the question has got unnecessary downvotes. +1 $\endgroup$ – I am Back Dec 12 '16 at 18:00
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  1. Feller's first volume for Probability Theory.
  2. Arnold's ODE for differential equations.
  3. Cartan's Elementary Theory of Analytic Functions of One or Several Complex Variables for Complex Analysis
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  • $\begingroup$ Nice post, Artem, and I appreciate your answering on behalf of the community wiki $\endgroup$ – Namaste Dec 12 '16 at 18:36
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I don’t agree with Bell on this point: one may well learn better and more easily from a book that is not generally considered a classic. For example, most people have never even heard of John Greever’s modified Moore method textbook Theory and Examples of Point-Set Topology, but for me it was the ideal introduction to the field. That said, I can nevertheless name a few examples.

For someone of my generation I.N. Herstein’s Topics in Algebra is a classic introduction to abstract algebra. The first volume of William J. LeVeque’s two-volume Topics in Number Theory is a classic at the higher end of the undergraduate level; Underwood Dudley’s Elementary Number Theory is a classic at the lower end.

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In the early '70s, I used two teaching books that I consider ''classic'':

Foundations of modern analysis of J. Dieudonné (at least in Europe).

Algebra of S. Mac Lane and G. Birkoff

At a different level, I think that an ''evergreen'' is:

Methods of Mathematical physics of R. Courant and D. Hilbert.

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    $\begingroup$ I had to learn analysis from Dieudonne as a high-school student, and it put me off the subject for years. It may be a classic, but it lacks motivation at almost every step. It's just "Here's another definition. And here are three theorems that follow from it. And now another definition..." Ugh. $\endgroup$ – John Hughes Dec 12 '16 at 22:26
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The Mathematical Association of America (MAA) has got a rich collection of classic books under Doclani Mathematical Expositions. I would suggest you following:

$1$. Mathematical Gems Series ($3$ Volumes) By Ross Honsburger.

$2$. Linear Algebra problem book By Paul R Halmos.

$3$. Euler: Master of us all By William Dunham.

Some other texts:

$1$ Introduction to Commutative Algebra by Atiyah and MacDonald.

$2$ A book of abstract algebra by Pinter.

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It's a bit more advanced than the topics you asked about, but Milnor's Morse Theory and Milnor and Stasheff's Characteristic Classes are astoundingly good. (There's a pattern here: Milnor's Lectures on the h-Cobordism Theorem is pretty good too!)

At a somewhat lower level, I find Spivak's Calculus (which many might argue is an introductory analysis book) pretty darned wonderful.

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  • $\begingroup$ I don't actually have recommendations for classics in other areas, but I suppose it's possible that OP has discerned that I really do and am just keeping them secret. Hard to imagine why I'd do so, but... $\endgroup$ – John Hughes Dec 12 '16 at 22:27
  • $\begingroup$ My main point is that although the question was closed as too broad, it was reopened; if folks don't think it's too broad of a question, then any answer should address the entire question. I'm just saying, this is precisely the reason that the entire post, at the very least answerers, should mark the answer a community wiki answer. $\endgroup$ – Namaste Dec 12 '16 at 22:41
  • $\begingroup$ I answered before it was closed. I'd be quite happy seeing it become a community wiki thing, although I don't have a lot of interest in learning how to make that happen. To quote someone wise in the ways of MSE: "please be nice not only to users you already know, but especially to newcomers to our site." $\endgroup$ – John Hughes Dec 12 '16 at 23:20
  • $\begingroup$ Well, you can do so yourself, as two other answerers have done. Click on the "edit" link to the left, between "cite" and "flag". Your answer will appear in your answer template, the window where you can edit it. Below that window, on the right, is a little box labeled "community wiki" which you can click to "wiki-fy" your answer. $\endgroup$ – Namaste Dec 12 '16 at 23:36