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First of all I'm sorry if this is not the right place to post this. I like math a lot. But I'm not sure if i want to do a math major in college. My question is: Can you give me a list of books that will give me the knowledge of the subjects a person doing a math major would have? I think I know all the stuff a good high school student knows. Thanks.

Martin Sleziak
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Asinomás
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    I'm not quite sure I understand what you are asking. Do you want resources which will give you an idea of what you'd learn as a math undergrad? – Alex Becker Jul 25 '12 at 02:08
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    I think this is just too broad, but we'll see what others have to say. Here's an idea: go to the math department webpage for a university — e.g., Northwestern or Michigan — and see which courses they require of undergraduates. Find webpages for those courses and see what sorts of books and assignments they use. You don't really need an expert for any of this. – Dylan Moreland Jul 25 '12 at 02:12
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    A good math major and a good mathematician has knowledge of subjects other math majors do not have. – William Jul 25 '12 at 02:14
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    Or, if you want to do it right, you could check the University of Chicago. – Alex Becker Jul 25 '12 at 02:14
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    Speaking of that school to the south, there is this annotated list of books. – Dylan Moreland Jul 25 '12 at 02:20
  • @DylanMoreland I had no idea Pete Clark went to Chicago! – Alex Becker Jul 25 '12 at 02:23
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    @Alex: What, were you expecting a shrine in my name in Eckhart Hall? Hmm, well you could at least ask Diane Herrmann about it... – Pete L. Clark Jul 25 '12 at 14:44
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    Some universities put all/most of their notes online. For example, MIT's OpenCourseWare. So, perhaps that is a good place to start? (It is free-er than buying books! But I do prefer reading books to looking at a computer screen...) – user1729 Jul 25 '12 at 16:41
  • This blog posts has some interesting list of topics and books/online materials for them: How to Become a Pure Mathematician. – Martin Sleziak Jul 26 '12 at 09:37
  • Actually I have a very different answer because I don't think it is necessary or even good to just cover as much 'relevant' content as possible. Rather from my experience with others, once you are able to devise proofs on your own that are both complete and precise, and understand exactly how a proof utilizes the structure of the objects involved, then the content itself will be easy to pick up naturally along the way. For this I would suggest that you start with a book like Spivak's Calculus and attempt to prove each theorem yourself before reading the given proof, and not just do exercises. – user21820 Jan 25 '14 at 10:06
  • Duplicates https://math.stackexchange.com/q/127814 – user1147844 May 28 '23 at 00:12

6 Answers6

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Using some of the recommendations Others gave me and the Stanford math major checklist I have made the following list: One should read all books corresponding to a subject (in order) not just one of them.. The first part is a requirement while in the second part students usually take at least 2 electives ( I give 4 examples).

Calculus:

Calculus by Michael Spivak

Calculus volumes 1 and 2 by Tom M.Apostol

Analysis

Principles of Mathematical Analysis by Walter Rudin

Real and complex analysis by Walter Rudin

Topology

Topology by James Munkres or

General Topology by Stephen Willard (harder)

Linear Algebra

Linear Algebra by Friedberg,Insel and Spence

Differential Equations:

Ordinary Differential Equations by Tenenbaum and Polland

Partial Differential equations by Lawrence C evans.

Algebra

Abstract Algebra by Dummit and Foote

Combinatorics

Introductory Combinatorics by Brualdi

Set theory:

Introduction to set theory by Hrbacek and Jech

Electives:

Algebraic Topology

Algebraic Topology: an introduction by W.S Massey

Algebraic Geometry

Undergraduate algebraic geometry by Miles Reid

Number theory:

An introduction to the theory of numbers by Hardy and Wright

Algebraic number Theory (If you also take Number theory)

Algebraic Theory of numbers by Pierre Samuel.

Asinomás
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    Im going to try to read all of these books in the following three years and ill let you know how it works out for me. – Asinomás Jul 25 '12 at 14:27
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    You would be lucky to read all of them in next three years. –  Jul 25 '12 at 14:39
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    +1. Excellent list except for "big Rudin" included-that's way too tough for most undergraduates. I'd also recommend Vinberg's A COURSE IN ALGEBRA instead of Dummit/Foote. – Mathemagician1234 Jul 25 '12 at 20:50
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    @CarryonSmiling How did the reading work out for you? – seeker Jul 11 '16 at 22:05
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    I read everything except for differential equations, those suck. Although I'm more into computer science now. – Asinomás Jul 12 '16 at 02:39
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    im almost done with my undergraduate studies now. I think this list would do well with a functional analysis book. Also, set theory is optional in my opinion. – Asinomás Mar 05 '18 at 19:34
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Here's one possible list.

Principles of Mathematical Analysis by Walter Rudin

Topology by James Munkres

Linear Algebra by Friedberg, Insel, and Spence

Abstract Algebra by Dummit and Foote

Ink
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    Some do; I personally didn't (not the OP). If you'd like a combinatorics book though, I recommend Introductory Combinatorics by Brualdi. Lots of information with good exposition and examples. – chris Jul 25 '12 at 03:46
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    Ah. A doable list. Not only that, but if you do only these books but thoroughly, then you will actually be more than well prepared for most grad programs. – Matt Jul 25 '12 at 17:30
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This is a blog which describes on how to be a pure mathematician. You can go through it and find out what all opportunities you have in various fields of mathematics.

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Here is another useful list.

This is a link to the Mathematics Programs offered at the University of Toronto (St. George):

http://www.artsandscience.utoronto.ca/ofr/archived/1213calendar/crs_mat.htm

A course number with a Y indicates a full year course (72 hrs of lecture) and a course number with H indicates a half year course (36 hrs of lecture):

First Year

MAT157Y1 - Analysis I Text: Calculus by Spivak. Used in the past: Principles of Mathematical Analysis by Rudin.

If you have never been exposed to abstract mathematics Spivak is probably better to go with. UofT has been teaching from Spivak's for awhile now.

MAT240H1 & Mat247H1: Linear Algebra I & II Text: Linear Algebra by Friedberg et al. Used in the past: Linear Algebra Done Right by Axler.

Second Year

MAT257Y1 - Analysis II

Text - Analysis on Manifolds by Munkres Used in th past: Calculus on Manifolds by Spivak

Go with Munkres on this one. Spivak is barely a little over 100 pages in length! So you can imagine how terse it is.

MAT267H1 - Advanced Ordinary Differential Equations Text - Differential Equations, Dynamical Systems, & Introduction to Chaos by Hirsch et al. & Elementary Differential Equations by Boyce and DiPrima

Third Year

MAT347Y1 - Groups, Rings, & Fields Text: Abstract Algebra by Dummit and Foote

MAT354H1 - Complex Analysis I Text: Complex Analysis by Stein & Shakarchi. Used in the past: Real and Complex Analysis by Rudin

MAT315H1 - Introduction to Number Theory Text: An Introduction to the Theory of Numbers by Niven. Used in the past: A Friendly Introduction to Number Theory by Silverman.

MAT344H1 - Introduction to Combinatorics Text: Applied Combinatorics by Tucker

MAT327H1 - Introduction to Topology Text: Topology by Munkres.

MAT357H1 - Real Analysis I Text: Real Mathematical Analysis by Pugh. Used in the past: Real and Complex Analysis by Rudin.

MAT363H1 - Introduction to Differential Geometry Text: Elementary Differential Geometry by Pressley.

Fourth Year

A lot of these courses are cross listed so they're actually graduate courses. Check here for texts and references:

http://www.math.toronto.edu/cms/tentative-2012-2013-graduate-courses-descriptions/

Hope this helps!

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Chicago undergraduate mathematics bibliography

ELEMENTARY

This includes “high school topics” and first-year calculus. Contents

INTERMEDIATE

Roughly, general rather than specialized texts in higher mathematics. I would not hesitate to recommend any book here to honors second-years, but they might not find easy going in some of them.

TO BE CONTINUED

Community
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I'm a bit unsure about this question, and its intent. But it is always important to have an idea of some ways to continue one's education.

One of my favorite, though undermentioned, resources is the Mathematics Autodidact's Guide, published by the AMS. It's a short pdf (linked here).

But FWIW, here is a list of the undergraduate math classes and their books I took and used, respectively, as an undergrad (this doesn't account for my self-study or the research bits that I did, but every budding mathematician must distinguish himself from the rest in some way or another):

Calculus (3 semesters):
Calculus in One and Several Variables by Salas, Hille, and Etgen
Vector Calculus by Marsden

Linear Algebra (2 semesters):
Carlen and Carvalho's terrible, terrible book
Linear Algebra by Apostol
Topics in Algebra by Herstein

Algebra (3 semesters):
Topics in Algebra by Herstein
Abstract Algebra by Dummit and Foote

Real Analysis (2 semesters): Intro to Real Analysis by Rosenlicht (great, though few know it)
Real Analysis by Bartle (this is intense, but flawed in that it doesn't do Lebesgue)
Advanced Calculus of Several Variables by Edwards (this was done with Bartle in one semester)

DE (2 semesters):
One of the Ordinary Differential Equations by Marsden (boring)
Calculus of Variations by Gelfand and Fomin

Probability (1 semester, thank god):
Intro to Probability by Hogg and Tanis

Combinatorics (1 semester):
Discrete Mathematics by Grimaldi

Graph Theory (1 semester):
Graph Theory by West (a great book)

Number Theory (2 semesters):
Elementary Number Theory by Rosen (doesn't require algebra)
Introduction to Modern Number Theory by Ireland and Rosen (different Rosen, famous book)
Davenport's Multiplicative Number Theory

Complex Analysis (2 semesters): Stein and Shakarchi's Complex book (part of their series on analysis) Conway's Functions of One Complex Variable

And then there were some electives in problem solving (using, e.g. Larson's Problem-Solving through Problems), game theory (Conway and Berlekamp's Winning Ways with your Mathematical Plays), additive number theory, etc. Find what interests you and follow it, I suppose.

davidlowryduda
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    Oh - I should note that there is a fundamental flaw in my undergraduate education: I never took a dedicated topology class. It wasn't until I went to grad school that I learned topology 'for real.' – davidlowryduda Jul 25 '12 at 16:37
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    @Chuck: I said exactly one of those books was terrible. Every book has strengths and weaknesses. Except for Carlen and Carvalho, which has only weaknesses. – davidlowryduda Jul 25 '12 at 18:27