Some books and tips
First I feel I should give a small "warning":
In maths, even if the book explains everything step by step as it goes along, you will have to stop and think, and some things will seem difficult at first. Even if you are very smart, you will read a maths book considerably slower than a normal pocket book.
That's really a good thing though. It means you are being forced to think wider and cleverer. Don't let that intimidate you, just don't give up.
I don't know how much mathematics you know from school and so on, but you are lucky in that there are plenty of EXCELLENT books on mathematics.
I have been comparing my books with the corriculum books of a psychology student I know. While they have some good, some bad, and some horrible books, all the math books I've used and read have been fantastic. In maths there are plenty of authors with a passion, and plenty of understanding, and there is very little room for just throwing fancy terms around, or talking nonsense.
The books in my experience differ somewhat in their focus, and in what they presume you already are familiar with. By this I mean that one book might give you lots of examples of applications in physics, and expect you to be content with getting to proofs and rigid analysis later (or not at all), while another might focus on building the theory little by little, and use purely mathematical examples.
I tend to prefer the latter kind of books, mostly because I am frustrated by the uncertainty involved in applying formulas I don't feel I know the limitations and sources of I think. This differs from person to person though.
When it comes to prerequisites, some books presume that the reader has done a lot of work with maths and will feel comfortable seeing familiar relations in the book, but most I have come across actually start more or less from scratch. My first book in abstract algebra, or the book I used in real analysis, for example doesn't require any knowledge beyond knowing what + means and things like that. They would be extremely difficult (impossible for most people) to get through if that is all you are familiar with though.
One thing you should know is that calculus, which is basically a set of methods for calculating changes in formula values as their variables increase or decrease, is considered fundamental knowledge for further studies at all the higher learning institutions I have come across.
There are literally hundreds of calculus-books, and I very much suspect there are hundreds of very good ones, but to give you an example, the book I would turn to if I were in your shoes is:
Calculus - A Complete Course
by Robert A. Adams
That one starts at the basics and builds on them to cover quite a lot of ground. It will probably last you a full year, and when you've gotten through it you will be able to get further into:
- Differantial equations
Which roughly speaking relate speeds of changes of function values to the function values themselves, and ask you to find out what the unknown function which has the described properties can/must be. Very central in modern physics.
- Analysis
Which, again roughly speaking, explores the foundation for the calculus techniques, and the nature of the continuous number line.
- Linear algebra
Which, again very roughly speaking, explores what you can deduce from knowing that some objects (such as numbers, but we could also be talking about for example formulas) are in some constant proportion to oneanother.
\begin{array}{cc}x + y = 3 \\ 2*y = 2\end{array} would be a trivial example, where the "objects" as I put it are the unknown numbers x and y.
- and a huge load of applications
, without too much trouble. Of course, there would still be a lot to learn, but you would have a foundation to build on.
A subject I like a lot is abstract algebra. It, again very roughly speaking, explores formulas and the manipulation of formulas, when the formulas need not represent numbers, and need not follow any of the rules that you are used to (such as a*b = b*a), unless such rules have been defined in the particular case. A formula in abstract algebra can perfectly well decribe something like turning a cube around in space (a might represent turning the cube upside down for example).
Abstract algebra is usually seen as building on linear algebra, and linear algebra is usually covered after or alongside calculus (there is a little linear algebra in the calculus book by Adams), so you have kind of a natural progression there.
Here are three books I have enjoyed a lot, which demand nothing more than knowledge of simple compulsory school mathematics:
Linear Algebra and its Applications
by David C. Lay
A First Course in Abstract Algebra
by John B. Fraleigh
A Companion to Analysis
by Tom W. Körner
Of course, as you can probably tell from my previous mention of these topics, getting through these books will be a handfull if you haven't done a bit of university level maths.
Some classics, which can be said to concern mostly quantities you can count to are:
Concrete Mathematics
by Ronald L. Graham, Donald E. Knuth, and Oren Patashnik
An Introduction to the Theory of Numbers
by G. H. Hardy and E. M. Wright
(This one is getting a bit dated. But of course, being that this is maths, what it teaches is still true and usable, and I certainly had fun with it)
My book selection here is affected by my interests in maths. You should find out what you are interested in. But don't miss calculus, or you'll get into trouble down the line.
Some readers of this post might scorn my crude descriptions of the mathematical subjects I have mentioned, but I think the descriptions should be sufficient to give a bit of a feel for what the subjects are about.
If your interest is in modern psysics, you will want to have (in addition to a thorough understanding of calculus) some familiarity with differantial equations, statistics, linear algebra, and eventually some topology.
I go to a university, but in practice I do self study, and I have all the while (not that this is necessarily a good idea). I would advice you to follow some progression, like those described on lecture schedules for courses at many educational facilities (available online).
It is important to work some excercises, or what you read won't stick, and you'll run into situations where you can't even figure out what you're reading about.
Conversely, if all you do is excercises, you will get stuck and not know how to solve them.
So get a bit of both.
And again: Keep some progression. Don't read "every once in a while", or a full day every two weeks, or something like that, or you're not likely to get far. You should try to keep something fresh available in your mind at all times.
The subject you are about to enter can be great fun, and really change the way you think, and expand your scope.
Good luck!
). Then I would recommend picking up a solid textbook in that field and just start reading.
