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Funny you should ask... I took a lot of math classes all through college, but at every step I was reaching beyond my grasp, so I never had great understanding. And then after a few years, I forgot it all. So in my early thirties, I started over.

Much more important than which text you use is your attitude, and a willingness to really walk through and understand the proof of a theorem, and a willingness to work through problems. Having said that, here's what I did:

Go through the chapter in Feynman Lectures on Physics, Volume I, where he starts with integers and goes through trigonometry until he winds up at Euler's Theorem. Do this, and you'll really understand numbers (as well as algebra and trig).

Then I went through the appendices of my college calculus textbook to pick up some algebra tricks I had never really learned. (This is a recurring theme, BTW: you learn a fundamental idea, and then there a bunch of tricks around the fundamental idea that enable you to actually solve problems. So, to really "get" math, you need to truly understand the most important fundamental ideas, and you need to learn some of the problem-solving tricks.)

From here, the school route is to press on to calculus. What's more practical is to actually learn and understand some probability and statistics. Especially Bayesian reasoning (http://yudkowsky.net/bayes/bayes.html). Understanding statistics and probability will actually improve your everyday life. But assuming you still want to press on to calculus...

You need to learn about limits. Actually work through some limit problems. And then you need to read through the definition of a derivative, and compute some derivatives by hand, computing the limits. And then you'll really understand derivatives.

(By the way, when you understand derivatives, you also understand differential equations. When people take differential equations classes, they're just learning the bag of tricks used to solve different patterns of differential equations.)

Now read through the proof of the mean-value theorem until you get it. This will enable you to understand the fundamental theorem of calculus. And so now you understand integrals. There's a bag of tricks around solving integrals which you can learn. At this point you could also start toying around with Mathematica; you now know just enough to begin appreciating how cool it is.

Once here, most math courses take a little detour and teach some numerical methods. I wouldn't sweat it too much, although it's a good trick to know that you can express a lot of different functions (e.g., y = the sine of x) as algebraic series, because it lets you approximate solutions to problems).

Now learn about vectors and simple vector algebra, which is just enabling you to generalize your understanding to multiple variables (e.g., z = x^2 + y^2). This will introduce different flavors of derivatives, as well as some different flavors of integrals. Just go get the book "Div, Grad, Curl and All That". You'll need to read a different book to read and understand the theory, but reading Div, Grad, Curl will give you an intuitive feel, which can be a big hurdle to getting multivariable calculus.

Before, during, or after your study of "Div, Grad, Curl...", you might want to learn about matrices, which is a short hand for writing systems of equations that transform one vector space into another vector space. This is worth knowing if you really want to understand 3D graphics programming.

And now you know as much math as your average physics or engineering student, although you should learn about Fourier analysis, because it's fun, and then you'll understand how your CD player works.

You could quit at this point, and you'd be in pretty good shape, but everything you've done up to now falls under the heading of "applied math". If you want to get a taste of what most mathematicians do, you'll need to look at what's called "abstract algebra". This is actually a ton of fun - just think of it as a big ol' puzzle: what if you tried doing "math" with stuff other than numbers? The most general notion is that of a set. And then you can learn about "groups", which are sets with a little more structure, if you will. And then you go on to "rings". And then "fields". For all this stuff, go get Herstein's "Topics in Algebra". It's far and away the best text.

That's as far as I got. I suspect there's a ton of other fun things out there (number theory? graph and network theory?), but I don't know anything about it.

If the original poster is a computer person, I'd recommend some changes to this list. This is "Math for physics" but "Math for Computer Science" is rather different, and might be more interesting.

I'd recomment calculus up to integration. Don't worry about integration tricks, except for integration by parts (the most important formula in mathematics) and u-substitution. All the other integration tricks are pointless crap used to fill up time in calc classes.

Vector math is useful if you like either computer graphics or physics, but is not crucial.

On the other hand, everyone should know probability, even the purest mathematicians. Just don't try to learn it out of an "Introduction to Probability and Statistics for Engineers" book, all such books should be burned. Real/functional analysis would also be useful to better understand probability.

I'd also suggest combinatorics/graph theory, and perhaps the theory of automata. That's edging towards computer science, but it is a fundamentally mathematical topic.

Also, it will be very slow going. It's not like picking up another computer language/framework; it's even harder than Haskell. I've have a Ph.D. in mathphys/num analysis, but it still takes me a long time to push through an introductory textbook in a field too far removed from my own. For instance, I sat through 4 semesters of abstract algebra (3 at the grad level), and I still don't understand it. Don't get discouraged.

Number theory is cool because it works with items that everybody understands, namely numbers. The only prerequisites you need are basic concepts like squares, even and odd numbers, and similar items. Instead of building up some giant edifice of weird symbols and concepts, like one does in calculus, you're just solving puzzles.

It will also answer 90% of the "hard" problems on the GRE, if anybody cares.

Also, I second not jumping into calculus too quick. It can be frustrating and isn't terribly useful for problems outside of Physics.

Nice list. I have been kind of doing the same thing - only over many years. One book that was incredible and disturbing was a small paperback - giving a layman's guide to Godel's Proof (maybe by Nagel&Newman?). The topic is too abstract to ever apply to a real world problem. But it is so profound that it is hard to imagine going through life not knowing it.


It is an almost totally overlooked area in any curriculum (math, CS, physics). And it is just as fundamental as algebra.

Algebraic topology isn't that incomprehensible as some think X the basic ideas are very simple and intuitive. Take a look at this short introduction http://www.inperc.com/wiki/index.php?title=Topological_Featu.... You need to know at least as much.

Wow, that's a great list, I'll just add studying statistics. It's a really interesting topic and it really opens your eyes to a whole new world.

Good luck.

Thanks for a great essay. I'm in exactly the position you describe at the beginning. I have a degree in Math, but always felt like what I was learning was just ahead of what I was understanding. I work in computers, so I get to use the reasoning skills I developed on a daily basis, but I have been wanting to start a rediscovery of Math. Thanks for the great pointers.

very interesting.

network and graph theory are definitely interesting topics. but what you've covered is most definitely a lot of interesting math. im an engineering (just a freshman), so all i get to see is integral calc. but theres definitely a lot of interesting math out there. combinatorics is also pretty nifty.

a very nice structured list. i might have to go pick up some books...

What I am concerned about is the math I learned in high- school and college, is now after 19 years forgotten. I am sure many people are in the same spot, wanting to relearn it. Where are all the websites for this? This is the one I found, and not sure it is applicable? relearnmathatmiddleage.com

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