How to start with mathematics? [closed]

I fell in love with mathematics a bit too late when I've already taken decisions regarding my future, career-wise. Now I would like to learn math on my own but I'm a bit confused as where to start. My knowledge of mathematics is comparable to that of a 15-16 year old highschool freshman. I would like to know how would you (if you were in my position) start your learning adventure on your own. There are a lot of resources online, of that I'm sure but I would like to follow a path. I'm pretty sure I can learn mathematics on my own and been thinking about this decision for almost a year.

Thank you!

I'm also an older person trying to teach myself math. I took math till high school and since then I have taught myself to beginning-graduate level in one area (analysis) and bits and pieces of other areas at an undergraduate level. My suggestions,

• Time is limited, don't waste it on popular books. Study high-school and undergraduate textbooks and do the problems. It may be a good idea to go to an older generation of textbooks since they are less chatty than the current ones. The Chicago Undergradute Mathematics Bibliography is useful. But don't obsess too much on which book to read. At this level it is hard to get things very wrong. Pick one book and persevere with it.
• Work in sprints. It takes time to get into a productive frame of mind. So trying to do math for a little while each day does not lead to any progress and results in frustration. Instead, dedicate a week of your vacation or at least a whole weekend for learning something new. Between these sprints revise what you have learned by doing more problems.
• Get it right the first time. While studying a text if you feel that you don't understand something fully, stop and work on your difficulty. Look at other books. Ask for help on forums like this. Go and brush up on the prerequisites if need be. But never leave a gap. Don't skip more than one exercise in ten.
• Be patient. A good undergraduate training takes 4 years of work with teachers and peers. My guess is it will take at least a dozen years to achieve the same level of mathematical maturity working part-time on your own. But at the end your understanding will be better than an undergraduate's since you will have had the benefit of greater intellectual maturity and more time to reflect.
• Is it worth it? Math is no doubt beautiful. But the time you give to it comes at the cost of your career, personal life and other interests. Are you willing to make the necessary sacrifices? If you are then it might be a good idea to make the sacrifices upfront and go back to school to learn math. The chances of fulfillment would be much higher.

I hope this is useful, I am just describing what worked well for me in the past. I recommend using real physical books (for example, I used Mathematics for the Million - Lancelot Hogben at this stage and I would recommend this particular book also because I am very fond of it).

Also you can a very good grip of the basic mathematical concepts by programming them (e.g. algorithms to convert between bases, factoring quadratics, finding square roots by newtons method when you get to calculus) - if you are just programming for programmings sake you will probably not learn very much mathematics (or at least, most of that which you do learn is by accident), but if you do it as an act of detailed explanation it can be an excellent process to find much clearer understanding of whatever it is you're learning about (and as a bonus you can run them to automatically solve problems that you already know how to do!).

In finding a path it would be useful for you to make a list of the more advanced theorems or applications which fascinate you (you will have to choose your own but I mean things like why are the orbits of planets in newtons universe ellipses, why you cannot square the circle or how raytracing works etc.) - then find out which setting they live in, then study the building blocks needed to get started working on the background - it doesn't even matter if you give up after a couple months extremely hard work by discovering that you cannot possibly study this stuff yet because you will still have got a lot out of it.

Khan Academy has excellent short videos you can take at your own pace, from basic arithmetic to advanced calculus. He has a very conversational style as well.

http://www.khanacademy.org/