I have currently finished AS maths (British curriculum), but the problem with the British curriculum is that it misses out the basis of maths; it misses out a focus on functions. When I look at other curriculum, I feel like there is a huge gap in my understanding of maths because I'm not as good with functions. I'm now going off to Uni to study medicine, but I want to also learn maths in my spare time, so I want to start right from the beginning and go all the way to the top: to understanding complex proofs about complex topics and to also be able to write my own proofs. HOWEVER, my MAIN focus is to learn new topics in maths before I get good at proofs. I'm asking because when I come to learn chemistry in Uni, I will not really need maths as much, but I will indeed be a much better interpreter of the universe around me if I can understand studies which are done on a high mathematical basis.
So please, can anyone guide me towards some topics to start with? And give me a rough idea of the different types of things that I would learn? Also, please recommend BOOKS (can't learn without them :)). Please help me with this because I really do love maths and want to learn it; all of it. Thank you very much in advance :)
[Note: I couldn't find an appropriate tag, so I had to put the one below.]
First of all, I like your ambition and I hope you succeed in both endeavours.
One thing to address is that mathematics is a surprisingly large subject; Most mathematicians are actually just professional in one or two different areas of mathematics, and only sufficiently proficient in the rest, much like (specialist) physicians know enough about the human body but specializes in some specific area. That is, you cannot study mathematics from the bottom to the top; One usually deals with separate subjects from the bottom to the top.
You mention you want to understand functions and proofs, which to me sounds like you want to study analysis. The best introduction to rigorous analysis according to me is Abbott's Understanding analysis, which includes (naive) set theory and the construction of the real numbers, sequential and functional limits, integration and series, and some metric topology. This book to some extent requires some basic calculus, but I think it should be possible without it. It was also one of the first books that made me comfortable with rigorous proofs, and not just hand-waving with intuitive remarks. The book can be found online, and some chapters can be omitted.
I would also recommend reading a book on linear algebra, being that it is perhaps the most useful and well-understood areas of mathematics. For this I would recommend Linear Algebra Done Wrong by Treil, since it introduces everything in the way it should have been introduced in any introductory course in linear algebra. It includes vector spaces, linear combinations and bases, linear mappings, matrices, determinants, inner product spaces, singular value decomposition, spectral theory and some tensor analysis. This also requires some basic inuition linear algebra, but everything is explained in detail.
Both books require you to do the exercises, and I would assert that mathematics can only be learned doing the exercises. These two books will give you a solid foundation to build upon.