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Bleg: Mathematics textbooks for high school self-study August 16, 2009

Posted by Akhil Mathew in blegs, General, math education.
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Inspired by Martin’s latest entry, and the general difficulty of teaching oneself mathematics, I’m going to shamelessly copy imitate the the idea of a recent Ben Webster post about undergraduate summer reading to ask our readers:

What books would you recommend for self-study by a serious high school student interested in mathematics, physics, or computer science?

You can assume familiarity with, at least, reasonably rigorous multivariable calculus and linear algebra.  Nevertheless, please do not suggest the proof of Fermat’s Last Theorem.  Since we’re not exclusively a math blog, let’s leave the door open to books on physics and computer science as well.

For starters, I will suggest a few of my favorites:

  • Principles of Mathematical Analysis, by Walter Rudin.   As one professor I know would say, austere beauty.   Very rigorous.  Same for the sequel Real and Complex Analysis, also by Walter Rudin.   Together these hooked me on analysis for a while.
  • Algebra, by Herstein.  Many fun exercises. 
  • Algebra, by Lang.  It’s not as well-motivated as I would like in some places, and it does a lot, but it is an excellent reference.   It also has exercises.
  • Commutative Algebra, by Eisenbud.  At first I disliked the style of this book, but I found myself coming back to it over and over.  It covers quite a bit of material and is accessible.  Eisenbud is also more fun to read than most other books on commutative algebra.
  • A Comprehensive Introduction to Differential Geometry, by Spivak. I just started reading this.  It is immensely fun to read, well-motivated, and rigorous.  I already like it much better than Calculus on Manifolds (which I think is much weaker on these attributes).  By contrast, I had difficulty with other differential geometry books such as Lang’s Differential and Riemannian Manifolds, since I could not understand the motivation behind all the mathematics, and the level was generally too high for me.


1. Michael Lugo - August 16, 2009

For those interested in discrete mathematics, I’d recommend Graham, Knuth, and Patashnik’s Concrete Mathematics, which was also mentioned at the Secret Blogging Seminar. (It might be a bit difficult for high school student — but quite a bit less difficult than some of the books you mention!)

2. Charles Siegel - August 16, 2009

I’m a big fan of Reid’s “Undergraduate Commutative Algebra” and “Undergraduate Algebraic Geometry” which are both very accessible (moreso than Eisenbud, which is rather intimidating) and work very well together. Also, Cox, Little and O’Shea’s “Ideals, Varieties and Algorithms” assumes…nothing. It doesn’t assume you know what a ring is or anything, and approaches algebraic geometry very much from the point of view of solving polynomial equations, which is helpful for someone in high school. (I wish I’d seen the book that early)

3. hilbertthm90 - August 16, 2009

Along the physics route I’d say Introduction to Electrodynamics by Griffiths. I wasn’t really interested in physics until this book.

Rotman’s Advanced Modern Algebra is essentially my only algebra resource (well, except for advanced things in commutative algebra) due to how comprehensive it is.

Although I love Rudin’s Real and Complex Analysis, I needed to learn real and complex analysis from other sources first in order to appreciate how wonderful it is.

I guess if Spivak and Rudin are allowed, then I’ll put one more out there that is amazing, but would essentially never be actually recommended to high school students. That is Bott and Tu’s Differential Forms in Algebraic Topology. Despite the name, I think you need almost no background to get through the first chunk of the book.

4. Qiaochu Yuan - August 16, 2009

I’m not a fan of Rudin’s style. I highly recommend Stein and Shakarchi’s Princeton Lecture Series instead; it covers Fourier analysis, complex analysis, and real analysis (in that order), emphasizing the connections to other subjects and concepts such as the heat equation and de-emphasizing pathological cases.

Seconding Cox, Little, and O’Shea. If you’re interested in algebraic geometry from an elementary point of view, Tate and Silverman’s Rational Points on Elliptic Curves is also worth checking out.

5. Akhil Mathew - August 17, 2009

Thanks for all the comments!

I just picked up a copy of Cox, Little, and O’Shea. I used Silverman’s The Arithmetic of Elliptic Curves last year for an independent study (while using Fulton’s Algebraic Curves to pick up necessary concepts from algebraic geometry), although I have not actually looked at Silverman-Tate. I will also take a look at the others mentioned.

Although Rudin is weak relative to say, Spivak, on motivating his material, I think he does it much better than Hartshorne or many other books. (Or perhaps I’m confounding this with the fact that the material is just easier.)

My co-bloggers probably know a lot more than I about discrete math; perhaps some of them have seen Graham-Knuth-Patashnik?

6. Anubhav Sinha - August 19, 2009

For physics, I’d highly recommend “An Introduction to Classical Mechanics” by David Morin. This book has many really interesting problems that will keep any interested physics student occupied for a very long time.

7. Vishal Lama - August 19, 2009

For a problem solving approach to high school physics, Problems in General Physics by I.E. Irodov is highly recommended.

8. Sonia - August 19, 2009

“You can assume familiarity with, at least, reasonably rigorous multivariable calculus and linear algebra.” Is that a reasonable assumption? I went to high school in India and we did rigorous nothing.
Some books I wish I’d read in high school would be:
Understanding Analysis by Stephen Abbott. (I can’t recommend this book highly enough as an introduction to Analysis, for self study or for a course.)
Topology by JM Munkres.
Graph Theory by Douglas B West
Discrete Mathematics by Lovasz, Pelikan and Vesztergombi.
Feller’s classic on Probability.
Number theory books are a dime a dozen, and I read David M Burton. Niven, Zuckerman and Montgomery is probably more popular.
Algebra is a bit trickier. I agree with the post about Herstein’s exercises being fun, but the writing is old fashioned and the notation would cause problems when they start university. And to read Lang your high school student would have to be extremely motivated.
Topics in Group Theory (Springer SUMS) by Smith and Tabachnikova is nice. I read Fraleigh (Introduction to Abstract Algebra), and found it enjoyable, but in college we used Michael Artin (Algebra). I’d recommend Artin as a self study guide. But as an undergrad textbook or as a supplement to an undergraduate course, Dummit and Foote is my personal favourite.
If you are a high school student looking for a book to teach yourself mathematics from, pick one where the treatment is elementary and there are plenty of exercises. (The Springer UTM and SUMS series are generally good.) Don’t hesitate to drop a book if it’s not fun anymore. Challenging is good, dry, technical, and uninteresting is not. Don’t focus too much on the theory, there’s plenty of time for that. But don’t read on if the book uses terms or concepts you are not familiar with. Look them up elsewhere. Seeing a lot of books that way will give you an idea of what kind of books are out there and what suits you best. But whatever you read, the important thing is to have fun, get your hands dirty and acquire a feel for the subject!

9. harrison - August 20, 2009

I don’t think there are many high-schoolers who can read Lang; I can’t read Lang. Ash’s algebra textbook is probably a more reasonable introduction, and has the advantage of being available for free online.

Another free-download which I can’t believe hasn’t been mentioned: Generatingfunctionology! Virtually everything in it can be tackled by a smart, motivated high-schooler, and it’s far and away the most fun textbook I’ve ever read.

I’m a partisan of Atiyah-Macdonald when it comes to commutative algebra (you can tell that Macdonald won the Steele Prize for exposition), but Eisenbud is also very readable.

10. hilbertthm90 - August 21, 2009

I second Atiyah-MacDonald for anyone tackling commutative algebra. One of the hardest darned books I’ve read, but very rewarding if you make it through. Nothing makes you learn commutative algebra like those exercises.

Eisenbud is great and far far easier to read, but I feel like I learn very little from it since it is so wordy. I’m never sure if I could fill in the details by myself. And more often than not, when I have to, I usually just re-look it up in Eisenbud instead of doing it myself (*bows head in shame*).

11. Akhil Mathew - August 24, 2009

Thanks for the additional comments!

I myself enjoyed Atiyah-MacDonald; I actually found it easier to read than Eisenbud, since it is more concise, and the problems were generally easier as well. Except A-M don’t cover half the material Eisenbud does. I think Bourbaki combines A-M’s brevity with (some of) Eisenbud’s comprehensiveness, though it doesn’t treat homological methods.

I also found Ash’s other textbooks useful as well, e.g. the one on complex analysis (because it covers the prime number theorem).

12. Christian - September 22, 2009

About Real Analysis, I am not a fun of Wlater Rudin’s style… I prefer Carothers, Cambridge Press. For an algebra textbook Fraleigh’s book is very good.

13. Akhil Mathew - October 5, 2009

I’m going to add one more book here that I recently found on theoretical computer science: _Introduction to the Theory of Computation_ by Michael Sipser. The style is remarkably lucid and well-motivated without being tediously overexplained, and the book has no prerequisites.

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