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Which Undergrad Subjects Should I Learn?

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By Divergence
Posted Tue Jan 03, 2006 at 09:22:52 PM PDT

I am currently a junior in high school, but I enjoy studying math independently. However, I am not exactly sure what subjects I should study to become a "pure" mathematician; for example, do I need to study multivariable calculus from, e.g., Stewart's "Calculus", or should I learn from "baby" Rudin? Do I need to study diffy Q's? etc. If anyone could, e.g., provide me with a list of subjects I should learn (up to the grad level) I would be very grateful.

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Which Undergrad Subjects Should I Learn? | 3 comments (3 topical, 0 hidden)

[new] Some other ideas (none / 0) (#1)
by mints on Thu Jan 12, 2006 at 12:04:52 AM PDT

I'd go in a slightly different direction. Instead of developing more depth in one branch of mathematics you will certainly study more of, consider exploring some branches of which you don't have any familiarity.

My strongest recommendation would by an easy abstract algebra text, such as Fraleigh's "A First Course in Abstract Algebra".

Munkres' "Topology" is wonderful. It isn't as accessible a text as the above, but the first portion, on point-set topology, is probably reasonable for self-instruction.

If you want to be super hip, work though Gauss' Disquisitiones Arithmeticae. It's a number theory text published in 1801 by arguably the greatest mathematician who ever lived.

If you're dead-set on analysis, I'd recommend Spivak's "Calculus" over Stewart's "Calculus". From what I've seen of Stewart, it isn't at all theory-oriented, focusing on the mechanics and application of calculus. Spivak is quite challenging, however, and may not be suitable for self-instruction.

If you just want to see what's out there, poke around MathWorld ( http://mathworld.wolfram.com/ ), or maybe Wikipedia's Mathematics Portal ( http://en.wikipedia.org/wiki/Portal:Mathematics ).

[new] Purity is overrated! Math is fun too. (none / 0) (#2)
by chelst on Mon Jan 16, 2006 at 08:22:23 AM PDT

As a junior in high school, I would take the time to study math at your leisure. While a systematic treatment provided in a textbook may be useful if you're looking for college credit, you may wish to consider studying math in other ways to build your appreciation of its many facets. There are popular articles and books written by Martin Gardner and others on mathematics.

In addition, the American Mathematical Society (AMS) publishes a number of books for the Student Mathematical Library that are short (100-150 small pages) and focused but allow you to explore an interesting aspect of mathematics with (often) nothing more than an understanding of a year's calculus. You may also want to visit a local university library and read the periodicals published by the Mathematical Association of America (MAA) geared towards undergraduates such as the Mathematics Magazine or the College Mathematics Journal. (You can have them delivered to your door if you join the MAA as a member (around $25-$35 for undergraduates).

If you're heart is set on moving forward rigorously, why not consider taking a course at a local university or college and receiving college credit for your studies? I would NEVER discourage self-study, but if you wish to become a mathematician, you will need to go through some hoops. A bachelors degree with a major in mathematics is probably one of them. Why not advance that goal now or over the summer? Also, you might consider linear algebra!

Good luck with your studies and don't neglect your social life!

Dov

[new] More Advice (none / 0) (#3)
by Jonny77889 on Thu Mar 13, 2008 at 04:40:35 PM PDT

Yes, linear algebra is a good subject to study since it comes up everywhere in mathematics. The first linear algebra book I used was by Howard Anton and Chris Rorres, and it's a good book. It gives a good introduction to the subject and is not hard to read. There's also one by Howard Anton (alone) that I hadn't seen--just heard about--but I would guess it's probably about the same as the previous book I mentioned.

Herstein's book "Abstract Algebra" is good and very readable and gives a good introduction to algebra. Dummitt and Foote's book is also good, but some of it may be too hard for high school students since there is some rather sophisticated material in there. But much of it is also pretty basic too. It has much more material than other algebra books that give an overall view of the subject, so it's a good reference too.

I can't think of any "must-have" DE's (mathematicians tend to call them DE's and engineers tend to call them diffy q's, according to what I've heard) books right now, but DE's would be a good subject to study.

Perhaps number theory is also a good option. Much of the material in introductory books is not too abstract and does not require much background to understand. William Leveque's book "Elementary Theory of Numbers" might be a good start. Dover has an edition of it, so it shouldn't cost much to get a copy. I hadn't examined it extensively, but it appears that George Andrew's book is pretty good, too. Dover publishes an edition as well.

If you can take college courses for credit, that's a good option. One such course I would recommend is a course on mathematicial reasoning and proof. Knowledge of this is essential for a mathematician! In fact, one of my friends who is a math professor has a mathematicially-talented son in high school who just took (or is now taking? I'm not quite sure) such a course at Austin Peay State University.

Hope this helps.

Jonny Groves

Which Undergrad Subjects Should I Learn? | 3 comments (3 topical, 0 hidden)

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