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I'm a Chem. Eng. grad. I did well in my math courses and I really like math. I've started digging for some more to learn on my own. I like applying math but I'm most satisfied when I have a deep understanding of things and a good intuition about it. Given that I have completed 3 calc. courses including multi-variable calc., a course in diff. equations, and linear algebra, what would be good topics to learn next? I'm interested in everything. I especially enjoy trying to prove things myself. I'm intrigued by abstract algebra and complex analysis but is there something I should learn first before jumping to those subjects? Thanks.

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Abstract algebra might be a nice taste of higher pure mathematics. I would hold off on complex analysis until you've learned some general topology and real analysis (unless you already have!). A common recommendation for Abstract Algebra is Dummit and Foote. –  Ben Feb 13 '13 at 16:30
    
@user21154, without pretending to know best,let me tell you I am reading IN Herstein's "Topics in Algebra" without even knowing multi-variable calculus,differential equations or linear algebra. You do not need anything but an open mind to read it.(more precisely, I am in high school). amazon.com/Topics-Algebra-I-N-Herstein/dp/0471010901 –  user60469 Feb 13 '13 at 16:52
    
Perhaps complex analysis would not give the best initial taste for math, but wouldn't it be the most useful for a Chemical engineer major to know? –  user45150 May 3 '13 at 0:58

2 Answers 2

up vote 3 down vote accepted

One recommendation is to learn how to do proofs (this will be a huge help in all math courses).

For example:

  • Problem-Solving Strategies (Problem Books in Mathematics) Arthur Engel (Author)

  • Problem Solving Through Problems Loren C. Larson (Author)

  • What Is Mathematics? An Elementary Approach to Ideas and Methods Richard Courant (Author), Herbert Robbins (Author), Ian Stewart (Editor)

  • How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library) G. Polya (Author)

  • How to Think Like a Mathematician: A Companion to Undergraduate Mathematics Kevin Houston (Author)

Also, you might want the answers here: Strategy to improve own knowledge in certain topics?

Regards

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Thanks. Oh, I forgot to mention I also took a course in mathematical logic which gave me some exposure to proofs. The link you gave me is helpful. –  user21154 Feb 16 '13 at 4:39
    
@user21154: you are very welcome! Regards –  Amzoti Feb 16 '13 at 4:42
    
+1 G. Polya's is a bit old. ;-) –  Babak S. Feb 17 '13 at 14:07
    
@BabakSorouh: yes, I agree that it is dated, but I actually still find it instructive. I think it is helpful for younger mathematicians to be able to read how well regarded mathematicians think about problem solving. This is a nice write-up. I wish that MSE would somehow publish on this topic as a community effort! Regards –  Amzoti Feb 17 '13 at 17:20
    
@Amzoti: And that's why it is accepted here. ;-) –  Babak S. Feb 17 '13 at 17:31

If you are interested in the subject in general for its own sake, a wonderful book is Aigner and Ziegler's "Proofs from THE BOOK". The books by William Dunham, e.g. "The mathematical universe", "Journey through genius" are outstanding.

If you want to learn stuff to apply it, go trawl through Open Course Ware, take a look at Coursera. Look at the books from The Trillia Group and the ones they link. The lecture notes by William Chen are very readable. I'm sure searching for "lecture notes" and your specific interest(s) will turn up many more good resources.

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