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I've exhausted the usual High School curriculum for Mathematics and i need some recommendations. I want to start a serious course in Mathematics at a local college, but that's simply too long to wait, i'd like to get started now. So right now i'm just looking for some fantastic books and some fantastic tips, any recommendations are appreciated.

hardmath
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nicholasg
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    Try "Fearless Symmetry" - it has a nice approach to some group theory/number theory. – Ben Sheller Apr 14 '16 at 16:06
  • "A Drunkard's Walk" is one that I liked a lot – Ben Grossmann Apr 14 '16 at 16:12
  • Do you want to start studying a specific topic, like differential equations, or number theory? Or do you want interesting survey books? Or popularizations of advanced math? These are all different. – Colin McLarty Apr 14 '16 at 16:34
  • I've done a slight amount with PDEs but ODEs aren't in my skillset. Definitely something i want to improve upon and gain more knowledge about. – nicholasg Apr 14 '16 at 17:52

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There are a lot of lectures on You Tube on diverse topics. I would suggest starting there. Watch the first couple of lectures, and decide if you want to take a deeper dive.

Francis Su's lectures for Real Analysis are fantastic.

I don't know if I would recommend you pick up a college level book (i.e. Baby Rudin for Real Analysis) without someone to explain the proofs with a little more detail.

Doug M
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  • I have seen tonnes of the Youtube lectures, they are very helpful but before i delve into the deeper Mathematics i need to learn how to read and understand mathematical proofs. – nicholasg Apr 14 '16 at 17:56
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If you are considering pursuing a mathematics-related degree once you begin college, being able to understand and do formal proofs is critical. A good primer on proof techniques is "How to Read and Do Proofs," by Daniel Solow.

For a more general book, Ian Stewart is a fantastic author. "Concepts of Modern Mathematics," while an older book now, is great at explaining a variety of different areas of mathematics. It might not go into a lot of detail, but it can help you find what really interests you.

Number theory as a whole is a really fascinating and instructive area of mathematics. One thing that is so great about it is that you can go as deep as you want; there are extremely basic concepts (greatest common divisors, linear congruences) to unbelievably complex areas (Riemann Hypothesis). Number theory also fits well with groups and rings, which are key algebraic structures in abstract algebra. The textbook I used in my first number theory/abstract algebra course was "An Introduction to Abstract Algebra," by Olympia Nicodemi. It isn't the most rigorous book for either topic in any way, but it is good at introducing basic concepts of both number theory and abstract algebra.

Ethan Hunt
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  • I'm currently googling the book on reading proofs you suggested above, thank you for your suggestions and response, i appreciate all the time you put into helping me. Thank you. – nicholasg Apr 14 '16 at 17:54
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I think you need a book to teach you proofs. I'd recommend Chartrand - Mathematical Proofs: A Transition to Advanced Mathematics. I think you should be able to find a PDF online

Alex Mathers
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  • Sounds good, i'm currently searching every book i get recommended in this thread, thanks for the input. – nicholasg Apr 14 '16 at 17:55