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The books required to become a Pure Mathematician

  • #1

Main Question or Discussion Point

Hi all,

So I'm finishing my first year in college and I did Calculus(based on Stewart). However, I don't feel I learned a lot during this year. There were not many proofs, the only proofs I studied was induction and contradiction(and some proofs in Stewart).

I honestly do not like the book at all, except for it's exercises. So now I feel I should take calculus from the beginning in a proof-wise manner(not sure if this makes sense) and move on to self-learning more higher level math before next year.

Basically for next year, I will have Advanced Calculus, Linear Algebra, Real Analysis and Introductory Algebra, in that order.

Now I'm been searching a lot for books and a lot have come which really peeked my interest. I literally want to buy them all(probably the cheap used ones or the cheap ones which are of bad quality print) but I;m not sure what books I have to get to get a complete undergraduate mathematics knowledge.

So here are the books I already own:
- Stewart Calculus 4e (duh)
- Spivak Calculus
- Apostol Calculus V1 and V2.

I bought Apostol and Spivak because they were highly recommended but never got the chance to work through each of them. They look hard but they seem to be extremely enlightening, specially Apostol which goes really in depth with theorems and proofs.

So I'm thinking going through these 2 books for now but then I need books which will cover the topics I'll do next year.

Here's what I found during my search:
- Naive Set Theory by Halmos
- What is Mathematics by Courant (I actually already ordered it)
- Differential and Integral Calculus V1 and V2 by Courant
- Introduction to Calculus and Analysis V1 and V2 by Courant
- Principles of Analysis by Rudin
- Real and Complex Analysis by Rudin (not sure which one to choose among the Rudin books)
- Linear Algebra done right by Axler
- Linear Algebra by Hoffman
- Linear Algebra by Strang (got the videos online)
- Introductory Real Analysis by Kolmogorov
- Mathematical Analysis by Apostol
- Functional Analysis by Rudin

So the list above is what I found but I'm pretty sure some books from one author may cover mostly the same stuff as another author, but I don't know, thus this post.

If you have more suggestion of books please post them here.

Thanks :)

Answers and Replies

  • #2
From my experience, Naive Set Theory, Principles of Analysis Rudin (defintely this is the one to do first and even then it might be too difficult), and Linear Algebra Strang are all good. These are only the ones I've had experience with. I'd say limit it to those intially and if Rudin is too difficult then use Intro to Analysis Rosenlicht which is good and cheap. Number theory might be another good option. When you eventually get to Abstract Algebra, Dummit and Foote is good. Rudin's Real and Complex Analysis and Functional Analysis book are graduate texts.
  • #3
Science Advisor
Homework Helper
Before buying a whole bunch of books, I suggest working through amy one of: Spivak or Apostol or Courant, or Courant and John, working lots of problems. Then you will have a lot of knowledge to help you choose further books. You will also have learned how hard it is to be a mathematician, and have a better idea of whether you really want to do it.

As someone once said about learning to be a magician: "if you can do one trick well, you are already a magician". In the same way, if you have mastered one of those books, you are already on your way to being a mathematician.

Best wishes to you!
  • #4
Science Advisor
Homework Helper
By the way your list of books contains lots of books that cover the same ground of calculus and analysis, and linear algebra. You have nothing at all on abstract algebra, number theory, differential equations, geometry or topology. Have you looked at my thread on becoming a mathematician?
  • #5
By the way your list of books contains lots of books that cover the same ground of calculus and analysis, and linear algebra. You have nothing at all on abstract algebra, number theory, differential equations, geometry or topology. Have you looked at my thread on becoming a mathematician?
Hi mathwonk. I actually got most of these books from your thread :) But the thing is exactly like what you said. several books contain more or less the same stuff and I want to narrow them down.

Here's an idea of my curriculum for my math major. I need a book(or more, whichever is better) on each of these. I realized that college math is not easy. The thing is, I feel that math cannot be covered just in classes. More should be done by myself. I got a B in calculus but that was because I did very bad for my first semester(failed all my class tests) and stopped going to lectures all together for the second semester. But I finally learned to love my math again and pulled out a B in my calc. class(which is a whole year class covering Calc. 1, 2, some basic linear algebra and some basic multivariable)

Here's my curriculum for the next 2 years. What would you suggest as books for all of it? It's not like I'll buy all at once, but I want them to be here when I look for them, and I think it could be helpful for others as well. :)

Second Year:
This is what math major should do. There is Differential Equations and Fourier Methods as well, but I don't think math majors can take them. So basically I would say I'm restricted to these 4 but if I can self-learn DEs and Fourier Methods by myself, I would do so(provided I know where to start).

Differentiable functions, independence of order of repeated derivatives, chain rule, Taylor's theorem, maxima and minima, Lagrange multipliers. Curves and surfaces in three dimensions, change of coordinates, spherical and cylindrical coordinates. Line integrals, surface integrals. Stokes' theorem. Green's theorem, divergence theorem.

Matrices, Gauss reduction, invertibility. Vector spaces, linear independence, spans, bases, row space, column space, null space. Linear maps. Eigenvectors and eigenvalues with applications. Inner product spaces, orthogonality.

REAL ANALYSIS Sequences, subsequences, Cauchy sequences, completeness of the real numbers. Series: convergence, absolute convergence and tests for convergence. Continuity and differentiability of functions. Taylor series and indeterminate forms. Sequences and series of functions, uniform convergence, power series.

Further linear algebra: equivalence relations, the quotient of a vector space, the homomorphism theorem for vector spaces, direct sums, projections, nilpotent linear transformations, invariant subspaces, change of basis, the minimum polynomial of a linear transformation, unique factorization for polynomials, the primary decomposition theorem, the Cayley-Hamilton theorem, diagonalization. Group theory: subgroups, cosets, Lagrange's theorem, normal subgroups, quotients, homomorphisms of groups, abelian groups, cyclic groups, symmetric groups, dihedral groups, group actions, Caley's Theorem, Sylow's Theorems. Ring theory: rings, subrings, ideals, quotients, homomorphisms of rings, commutative rings with identity, integral domains, the ring of integers modulo n, polynomial rings, Euclidean domains, unique factorization domains. Field theory: subfields, constructions, finite fields, vector spaces over finite fields.

Third Year:
For third year I'll have to choose 4 among these:

I'm thinking Algebra, Metric Spaces(or Logic and Computation, since I'm a CS major as well, but I'm not sure if it will benefit me more than metric spaces), Complex Analysis and not sure for the last one.

An introductory course of modern abstract algebra involving the following concepts: algebraic operations; magmas and unitary magmas; semigroups; monoids; closure operators; equivalence relations; categories; isomorphism; initial and terminal objects; algebras, homomorphisms, isomorphisms; subalgebras; products; quotient algebras; canonical factorizations of homomorphisms; free algebras. Various classical-algebraic constructions for groups, rings, fields,
and vector spaces, seen as examples of these concepts, will be described in tutorials.

An introduction to the theory of complex functions with applications.

The propositional and predicate calculi: their syntax, semantics and metatheory. Resolution theorem proving.

An introduction to metric spaces and their topology, with applications.

A selection from lattices and order, congruences, Boolean algebra, representation theory, naive set theory, universal algebra. (Please note that this module is not a prerequisite for entry to the Honours course in Algebra.)

A selection from the implicit function theorem and inverse mapping theorem, Lebesgue integral, Fourier analysis, Hilbert spaces, Lebesgue and Sobolev spaces, Fractals and approximation theory. (Please note that this module is not a prerequisite for entry to the Honours course in Functional Analysis.)
  • #6
Gold Member
should include Artin's algebra
  • #7
As mathwonk said, you should start with one of the three/four books on Calculus : Spivak, Apostol or Courant/Courant and John. I personally chose Spivak because I liked the way he presents things more. If possible, try out the three books and see for yourself which one you prefer. After having a good background in Calculus 1 and 2, you can choose to continue the Calculus sequence with Calc III (Multivariable Calculus), or jump straight to Linear Algebra/ODE's. I'll personally go for Calc III. Complete the other two courses (LA, ODEs) and you should have a good "basic" mathematical background. You should also take a PDE course. You can choose to take different courses : Algebra, Real/Complex Analysis, Diff Geometry, Topology, etc. It mainly depends on your interests/goal.

Now for the books, you should look at the mathematical books listing and see for yourself. Spivak, Lang, Lee, Arnol'd are all good authors.

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