I've a not so profound background in math (just rusty high-school knowledge) and want to brush it up a little through self-study.
I want to build a decent general foundation so I'm well prepared for more advanced topics. I don't know what exactly yet but probably some discrete math (comp. science), probability/statistics, maybe quantum-mechanics, ... .

I will go through high-school mathematics (emphasis on algebra/trigonometry):
- ck12.org
- khanacademy.org
- ... .

I'll continue with MIT's 18.0x OCW courses:
- Calculus 1 en 2 (single and multi variable calculus)
- Differential equations
- Linear algebra

Am I missing some more general topics or good resources? I've read about 'The Princeton Companion to Mathematics' being good for someone with my background?

The Princeton Companion is not for this level of brush-up. It's the sort of book that you would use to explore new topics that you might not have seen before.

If you're interested in getting deeper into math, you'll probably want to find a book that helps you to transition from computation into proof-based mathematics. Proofs are things that you don't really see in the high school math level. I like "How to Think Like a Mathematician" as a much gentler introduction than the more classic "A Transition to Advanced Mathematics" (and much less expensive).

There's another book on the market called "How to Prove It" but I've not read it so I can't say how good/bad it is. It seems to have decent reviews.

khanacademy is fine to learn stuff, but there are no question sheets.

Doing questions/exercises is essential for quick progress in low-level maths.

In some of MIT's modules there are example sheets & solutions, which is good.

Basically, I would start by going and getting a basic book off amazon on algebra and/or geometry. Maybe a standard book on it, or just a questions and answer book.

Not sure what you mean by "algebra" though, as algebra is quite broad.

But yeah, work through a million questions, get used to things... don't be afraid to ask stuff on internet forums (like on 2+2) and also it's good to ask people you know if they're good enough and are willing to teach you a thing or two.

khan academy has a lot of math quizzes.. click on the "practise" tab

I'm not a math guy but this seems a bit narrow. I don't know how you'd get a good general overview, because there are dozens or hundreds of mathematical fields, often with overlap, but I know calculus and ODEs and I wouldn't say I "know math" even remotely.

It seems like abstract algebra is a good starting place since people who do "know math" seem to reference it a lot - set theory and number theory maybe? College-level geometry would probably be nice. Mathematical logic? Complex analysis?

I've learned some things in data structures, algorithms, numerical analysis, combinatorics, and computability theory, mostly as a result of reading about computer programming. That stuff is awesome to me but I don't even know how you'd get there from a "general mathematics" background. If you want computer math I recommend those topics though, I doubt you can do much with computers unless you have at least data structures, algorithms, and computability theory - but for "more discrete" tasks combinatorics is a must, and for "more continuous" tasks you really need numerical analysis. Might even want to study some more programming-related stuff too, knowing how floating-point types are implemented would be awfully valuable if you ever want to do math-related code.

As a starting point I would recommend starting courses on:

1. Calculus
2. Linear anlysis (a good book is "Linear Algebra Done Right, Second Edition" by Sheldon Axler)

Then maybe introduce abstract algebra.

Thanks for the responses:

I'll use 'How to Think Like a Mathematician' by Kevin Houston and skip 'The Princeton Companion' for now. Proofs seem to be very important and indeed in high-school we focused on computations.

That's just the high-school algebra I see often mentioned together with trigonometry as a prereq. for a course in calculus.

I've already refreshed some high-school stuff. I think the free ebooks from CK12.org are good? Problems with exercercises or questions I can post on several fora.

The 4 MIT OCW courses are 'Scholar' math-courses, specifically designed for self-study. I've already gone through some lectures on Calculus I and I think it's all pretty good organised (recitation videos, worked examples, exams (+solutions), ... ).
I also read when you're a Math Major (General Option) those 4 courses are required while the rest is flexible.
This is probably indeed far from a 'decent general background' but I guess I need to start somewhere.

good luck

Feels like we do it for somewhat similar reasons etc. so this might be interesting. I have a slightly different overall plan but there might be some helpfull info regardless:
http://forumserver.twoplustwo.com/47...ughts-1206315/

Thanks. I think the information in your thread is going to be very helpful once I finish my 4 OCW Scholar courses. Right now I just finished Unit 1 (Differentiation) of Calculus 1, still a long way to go.

http://www.udacity.com/
They have free courses for computer science and they just added a statistics 101 course, which has programming involved. It's a cool concept. They show short, well made video's, which are then followed up by questions you can answer to show you understood.

Unfortunately, it's been crazy easy so far. They are still adding the units for the statistics, so I'm assuming it will get more relevant (it's supposed to be university level).