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I wasn't the best student in high school but I always saw the mathematics as an interesting subject, now I have to choose a degree and I'm really considering pure mathematics.

I have something like 6 months of free time, so maybe it'll be a good idea to prepare for the course. Any book recommendations?

I would like to improve mathematical thinking for my future studies and general math books (Calculus and Algebra books will be great too)

Thanks.

Bruno
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  • This will help you in all mathematical courses, it prepares you for all mathematics, as opposed to say, calculus or linear algebra books. – Git Gud Jan 23 '15 at 23:49
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    It would be helpful if you could give a few more details about which country you're in, what kind of university you will attend, and what your background is. – Jamie Radcliffe Jan 23 '15 at 23:50
  • I'm from Brazil, as I said I wasn't the best student in high school so I really don't have a strong base, I'm in the average. I'm thinking about getting a pre-calculus book to improve any lack of knowledge. – Bruno Jan 24 '15 at 00:01
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    I would actually recommend against the precalculus book. Any skills that you lack from precalculus you will see again in calculus and you will master them there. The most beneficial thing for you would be to push past what you already know and to challenge yourself with new material. Not only will you learn new things, but this will get you in the habit of reading and doing mathematics independently. – Mike Pierce Jan 24 '15 at 00:11
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    Besides Spivak's book, mentioned below, I would recommend Stark's Introduction to Number Theory. I disagree with logic and set theory as an initial topic. – user208259 Jan 24 '15 at 03:16

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I recommend What is Mathematics? by Richard Courant and Herbert Robbins, and Set Theory and Metric Spaces by Irving Kaplansky.

Take your time with Courant's book first. Kaplansky's book is more advanced; for now you can use it as a secondary source.

kobe
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  • I'm thinking the Kaplansky book is a bit of a stretch. I would recommend some experience with proofs before jumping into that one. – Mike Pierce Jan 23 '15 at 23:56
  • @mapierce271 the Kaplansky reference was meant to be secondary, after dealing with Courant's book. I'll add that note later. – kobe Jan 24 '15 at 00:01
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I personally would recommend Spivak's Calculus book. I am not sure what texts you will be using for your first year classes, but for me this book was really enjoyable and helped bridge the gap between high school math and university math. It has detailed proofs, and will introduce you to more sophisticated ways to prove theorems than whatever you would have studied in high school. For me first year calculus using this book really put me on firm ground to jump into other areas of math.

If you would also like a nice introduction to algebra (group theory in this case), I also enjoyed Armstrong's Groups and Symmetry book. It will give you a small taste of what more advanced mathematics will be all about, and will be very different from the comforts of calculus.

Sergio Da Silva
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    I agree with Spivak's book. It's an excellent introduction to mathematics, not just calculus. It also comes with an answer book, which is good but also requires discipline. – user208259 Jan 24 '15 at 03:14
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This does not necessarily answer your question, but just to get you inspired and to get a feel of what pure math is about: Fermat's last theorem by Simon Singh is a great read. It is probably because of that book that I went on to study math.

If you need to look for books to study you can also look at the course description of the first year courses at the university you wish to enroll at. There it is probably stated which books these courses will use. If not, try e-mailing the professors with that question. I am sure they will appreciate your enthousiasm!

Joachim
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Read Prasalov's "Numbers and Figures" published by American Mathematical Society. It is just 81 pages, and has 12 chapters and many of them independent.

P Vanchinathan
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