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May 5th, 2007, 06:18 AM   #1
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Relearning math

For a while now I've been thinking about completely relearning all my mathematics, maybe going as far back as simple addition. I'm in my early 20s and ace college level math courses, but like you guys pointed out in my other thread, that's more thanks to algorithms than insight. I might add I was never very good at the "simpler" math, and paradoxically as it sounds, I do often feel like I lack some basics when solving complex problems. So could anyone point me to some comprehensive books or websites (not just collections of equations) towards that end? Something with a lot of historical context would be best. Thanks!
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May 5th, 2007, 06:42 AM   #2
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I don't know if roadnottaken will agree, but I would really recommend starting off by reading this great book:

Journey through genius : the great theorems of mathematics
by William Dunham


While it is not a textbook, and may not necessarily teach you much in the way of anything new, it might help you to see mathematics in a different light. Just have fun and read whatever you find that interests you in the book, and I think you might begin to understand some of the basic ideas that many people used to formulate many of the great theorems. The whole point I'm getting at is that the book talks about how people created theorems, rather than just how to use theorems and formulas, which is what most people are taught in schools these days. Understanding a little about how theorems are thought up and proven can often help you understand mathematics in general much better.
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May 5th, 2007, 01:25 PM   #3
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I would recommend:

The Art of Problem Solving: Vols. I and II
by Richard Rusczyk and Sandor Lehoczky


These books were written for middle and high school students, and, while some of the material (especially in Volume II) is rather advanced, they put a huge emphasis on getting an intuitive grasp of material. They teach general methods, rather than expecting you to memorize formula, and they either prove in the text or have you prove in an exercise virtually all of the material that they cover.

Unlike most textbooks, their problems are almost all different, and you really have to apply yourself to solve the harder problems (some are pulled from the International Mathematics Olympiad).

The books are available at artofproblemsolving.com.

Certainly, Infinity's recommendation also sounds like a good choice (I might have to get it myself :P ).
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May 6th, 2007, 02:19 AM   #4
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Thanks guys! I mentioned I'm not a high schooler anymore, and I never any problems with my coursework (I'm actually much better at the "harder" than the "easier" stuff), but so often I feel like I don't really know what I'm doing. No matter what it is, anyone can be told that when you have A, do B to it, to get C. But A, B, and C must be rooted in thousands of other variables I wish I knew. So thanks for your recommendations, I'll look into them for sure!
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May 6th, 2007, 03:53 AM   #5
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Considering your initial question about quadratic equations, I would say that you would be completely lost with the Art Of Problem Solving books. However, I have never seen them, but I believe they deal more or less with olympiad-level mathematics. This assumes the reader to master elementary high school lessons perfectly. I may be wrong, because I repeat once again that I 've never seen them. You'd better check it out or ask on the Mathlinks forum for more information. The people who wrote the books are deeply knowledgeable in general mathematics (meaning they are masters in elementary notions, where "elementary" ranges from high school to general Master-Level education. I don't know them as for their specialities are concerned). Also, they seem to be honest when it comes to recommending their books or not. They will tell you, honestly I believe, whether you can use their books for your purpose and will redirect you to better ones if necessary.

My personal recommendation would be for you to get some of those high school math books from the 60s (I don't know what you guys have got in the USA), at a time where everything came with proofs (whenever reasonably possible). Almost everything they produce now is quite crappy, and so is the level of their average colleges (I'm not talking about the elite ones).
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May 6th, 2007, 03:07 PM   #6
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You might find Art of Problem Solving's introduction series more at the level you are looking for (or, as julien pointed out, they may be able to recommend better books for you).
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May 11th, 2007, 03:13 AM   #7
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Also get The Language of Mathematics by Frank Land (ISBN 0719530725). You'll like it.
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