I'm an international student about to go into my last year of high school, and I haven't found many mathematical books that interest me. I'm looking for a mixture of interesting, but respected books, as it's also a way to show my passion for Mathematics in my personal statement. So far I've read Schaum's Theory & Problems with Modern Algebra, and I've started reading the Physics version. I've written papers in Machine Learning which is another field of Mathematics that I enjoy, however, I think something more fundamental might be more insightful. If you had any recommendations for a concise (Not a 378 page proof of 1+1=2) read, or a like STPMD a book that focuses' on problems as well. Thank you.
closed as off-topic by rschwieb, Namaste, Xander Henderson, Claude Leibovici, Tyrone Jul 24 '18 at 10:57
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "Seeking personal advice. Questions about choosing a course, academic program, career path, etc. are off-topic. Such questions should be directed to those employed by the institution in question, or other qualified individuals who know your specific circumstances." – rschwieb, Namaste, Xander Henderson, Claude Leibovici
I was in your position some years ago so I can pass on what I found great reading. Both are very well known classics but as no one mentioned them..
- Enjoyment of mathematics (Rademacher and Toplitz). Amazing book that includes many vignettes on different parts of mathematics. Totally recommend it. I learnt a lot of mathematics from this little book.
- Mathematics:it's content,method and meaning (The Russian giants of math i.e Gelfan'd, Kolmogorov, Aleksandrov, etc) This is one of the best books to read in my opinion. I read the first volume while in high school and it was such an amazing experience. It's a compendium (almost encyclopedic) of articles written by experts about the different branches of mathematics. I'm still reading it to this day. I think the notation is a little bit dated but not terribly so.
I think both of these books will give you a nice coverage of mathematical ideas to play with and discover. After that you can very well jump into what interested you the most, for that search here for recommended bibliography or ask a question if none has. Best of luck!
When I was in high school, the book "Nuts and bolts of proofs" helped me gain insight into what proof writing would look like. It has tons of examples, and it walks you through the common proof techniques and basic logic, like proof by contradiction and proof by induction. It's also pretty short. Mathematics is more than calculation, and I think this book is a good introduction to what Math has to offer.
In a comment to Ninja Hatori's answer, you say that the books suggested by Ninja Hatori are third-year books. That's true, but in the U.S. (which I understand from your question is where you'll be attending university) most undergraduates don't have serious contact with proofs until their third year. Well-prepared students can read at least some of these books out of high school. Since you've already read a book on abstract algebra, albeit one that is not very advanced, you have enough preparation to at least try these kinds of introductory algebra or analysis books, or slightly easier ones.
You also say that you'd prefer to avoid reading something that you'll learn later in university anyway - but the basic courses in algebra and analysis are usually where people form the habits of thought that allow them to learn other areas of mathematics (in addition to being direct prerequisites for many of them), so you certainly can't avoid these things indefinitely if you want to progress in math.
You can find a large number of suggestions in the syllabus for the undergraduate program in math at Cambridge. You could look at some of the books recommended for first or second-year courses (Parts IA and IB). https://www.maths.cam.ac.uk/undergrad/course/schedules.pdf Some of the recommended books, especially those published in Britain, tend to be quite concise because Cambridge students have a large number of small courses.
That being said, I can recommend a few books that are outside the central core of the undergraduate math curriculum and would be accessible to you now, as they don't have any advanced prerequisites.
Stark, Introduction to Number Theory.
Hopcroft, Ullman, Introduction to Automata Theory, Languages, and Computation.
Feller, An Introduction to Probability Theory and Its Applications, Vol. 1.
The first edition of the second book is supposed to be significantly better than later ones (which I haven't seen).