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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.

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closed as off-topic by rschwieb, Namaste, Xander Henderson, Claude Leibovici, Tyrone Jul 24 '18 at 10:57

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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!

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  • $\begingroup$ Both these books seem really interesting, I've already ordered them. Thank you. $\endgroup$ – John Miller Jul 24 '18 at 12:17
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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.

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  • $\begingroup$ That seems interesting, but I've already covered proof via induction, contradiction as well as basic group and ring theory in class so it might be a bit repetitive. Perhaps you know something more abstract? $\endgroup$ – John Miller Jul 23 '18 at 15:23
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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).

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  • $\begingroup$ Actually, I'm applying to Cambridge and I've already ordered A Mathematician’s Apology by Hardy, New Applications of Mathematics by C. Bondi, and The New Quantum Universe by T. Hey & P. Walters from the Reading List for undergraduates. But just going back to what I was saying I'm not against reading algebra and analysis, I just do not want to make sure it's written well and not basically a textbook for linear algebra and mathematical analysis. Introduction to Number Theory and the last one seems interesting too. $\endgroup$ – John Miller Jul 23 '18 at 16:51
  • $\begingroup$ Yes, as I recall, Cambridge recommends a bunch of non-math books about math, but no math books, to incoming undergraduates, for reasons only they know. I would ignore the advice and read actual math. If you absolutely want to avoid duplication, you could read a textbook that you know is not the primary textbook for a given course. Also, you will need to decide what your level of interest in physics is so you can make appropriate decisions about what to pursue in university. Schaum's Outlines risk being rather dull. Kleppner/Kolenkow and Purcell are more likely to pique your interest. $\endgroup$ – Dave Jul 23 '18 at 17:18
  • $\begingroup$ My interests in Physics are purely theoretical, and there are many modules in the Mathematics course at Cambridge, or in any mathematics BSc in the UK, that have theoretical physics as it overlaps heavily with applied mathematics. I also think books such as "A Mathematician’s Apology" are interesting in that universities want you to not only be able to read and understand your subject, but want to be there, and hopefully become a researcher which is, from what I have understood, the principle of the book. $\endgroup$ – John Miller Jul 23 '18 at 20:13
  • $\begingroup$ I'm not against reading books on the history of math, biographies of mathematicians, and so on. It's just odd that Cambridge recommends against students reading ahead. $\endgroup$ – Dave Jul 23 '18 at 22:24

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