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I am currently an undergrad student and recently changed my major to mathematics (after taking my first proof based math course). Unfortunately, now that I am taking several upper division math courses, I feel that my mathematical foundations are a bit weak and atrophied. I am considering taking some time off to try to study math on my own to improve my foundations but I'm not really sure what the best texts would be to do this. I know this is pretty vague, but basically I'm looking for textbook or other resource recommendations that would provide a rigorous, proof based approach to basically all of math that a typical student would learn from elementary school through about the first year of undergraduate study. Any help would be appreciated, thanks.

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    $\begingroup$ Look at the pre-requisites for the courses you intend to take and study those. You should be able to look at the books the pre-requisite courses recommend/require. You can also talk to your professors or a TA or a some kind of counselor who would be able to give you targeted advice and with whom you'd be able to have a deeper conversation. $\endgroup$ Mar 13 '19 at 3:51
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    $\begingroup$ Self study is much harder than taking classes. $\endgroup$
    – John Douma
    Mar 13 '19 at 3:56
  • $\begingroup$ You can always read your old textbooks again. I sometimes find that reviewing old material refreshes my memory better than looking for a new source. $\endgroup$ Mar 13 '19 at 5:01
  • $\begingroup$ @JohnDouma Depends on the teaching and the textbooks available. A really "good" book (i.e. fitting your individual requirements) can complement classes and even be a viable substitute. But in most cases, I agree with you. $\endgroup$ Mar 13 '19 at 6:00
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    $\begingroup$ Talk to your professors. Get their advice. Holding office hours is what they get the big bucks for. And students like you, with a genuine interest in learning mathematics, is what they live for. $\endgroup$ Mar 13 '19 at 13:11
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I would suggest Foundations of Analysis by Edmund Landau, it starts from the obvious $a=a$ and Peano axioms and goes upto proving theorems on real and complex number system. As for calculus Schaum's outline of Advanced calculus by Murray Spiegle, there is also a similar book( Schaum's outline) on Modern Algebra, that proves almost every theorem, including commutative property-which at first may not seem to need a proof, on Algebra and Arithmetic, though I can't remember who wrote it.

As for the last comment: I did not study such maths at University, being a self thought, but I'm much better than my peers who studied maths. Beside mathematics is more fun if you study it by yourself. And believe me, you wont forget the math if you are self thought.

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For any foundational topic of mathematics, there is a multitude of elementary textbooks. You can easily find them via Google, on Amazon, in your faliclity's library, etc.

Picking one over the other is a matter of taste. They differ in the order of topics, the detailedness of explanations, amount of exercises and solutions thereof, and - most importantly - style. Just as with educators (teachers, professors, ...), finding a textbook that meets your personal style is crucial for the speed of your progress.

I recommend you get recommendations from fellow students or friends, or just try different books. After the first chapter, you should be able to decide whether or not it suits you.

I hope this answer is the least opinion-based possible.

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I recommend the Art of Problem Solving website; their books are very helpful as well in terms of re-understanding fundamentals from prealgebra to calculus II.

I also recommend the youtube channels 3blue1brown and blackpenredpen, along with The Organic Chemistry Tutor for longer reviews. 3blue1brown has really helped me grasp the conceptual aspects of calculus. blackpenredpen explains more advanced topics such as erf/erfi in contexts of simpler concepts. The Organic Chemistry Tutor makes long videos that thoroughly cover the topic given (he's also really nice at teaching math-related topics in science).

P.S I'm just a sophomore in high school taking AP Calculus BC so these recommendations are probably biased towards my limited area of study, but they were really helpful for me so I hope they can help you too.

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I completely agree with the advice in the comments. Look at course syllabi for the classes, both upper-level and foundational, that you're interested in – different professors who teach the same class often teach from different textbooks, and you can "shop" textbooks to see the different ways the same material is presented. Oftentimes professors write their own notes/informal textbooks to teach from, which are great resources and condense the material to the absolute essentials.

Self-study is definitely harder than taking a class, so prioritize finding a textbook with good exercises if you choose to go down this route. I would also look into enrolling in an independent study or reading program at your university, where your self-study is guided by a graduate student or faculty mentor.

To answer your question directly, these textbooks were the canonical resources that professors at my university recommended and relied on heavily:

  • Principles of Mathematical Analysis by Rudin ("baby Rudin") for calculus and beginning analysis. A great place to start nailing down the foundations of proof-writing.
  • Real and Complex Analysis by Rudin ("papa Rudin") for real and complex analysis
  • Complex Analysis by Stein and Shakarchi, a personal favorite of mine to understand complex analysis.
  • Abstract Algebra by Dummit and Foote, the be-all, end-all resource for algebra. This book seems to be used universally in undergrad/beginning graduate algebra courses, so self-studying this text will prepare you if you choose to take algebra classes. Lots of good exercises!
  • Linear Algebra Done Right by Axler and Linear Algebra Done Wrong by Treil. The "right"/"wrong" designation comes from how the material is ordered, rather than major pedagogical differences.

Here are textbooks in undergraduate/beginning graduate elective topics that I've run into (the bias towards algebra is pretty strong):

  • Topology: A first course by Munkres. Like the Dummit and Foote, this seems to be the be-all, end-all reference.
  • Algebraic Topology by Hatcher.
  • Representation Theory: A first course by Fulton and Harris.
  • Rational Points on Elliptic Curves by Silverman and Tate.
  • Anything written by David Cox (Galois Theory, Primes of the form $x^2+ny^2$, Ideals, Varieties, and Algorithms). His textbooks are well-explained and full of great exercises.
  • Random Walk and the Heat Equation by Greg Lawler (notes). Covers random walks, statistics, harmonic functions, and a bit of measure theory, which are all cool analysis-dependent/adjacent topics.

Good luck!

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