Several readers commented on how my "What you should know by 18" list was pretty light on the math and science (though I hope some of the addendums in the comments helped with that). One of the issues there is that I had a hard time coming up with a good idea of what the minimum a civilized and functional person should know in these fields would be. It really was meant to be a pretty basic list. (A high school student could do all the reading on that list in under a year, I'm pretty confident.)
The other problem, however, is that although I deal with data analysis on a daily basis as part of my job, I've honestly forgotten most of the higher math I ever knew. I went through calculus in high school (I used the Saxon textbooks and did moderately well) but that's ten years behind me now. On a daily basis, I mostly just end up summing and averaging large record sets. Once in a while I do a little regression analysis: but that's just by using the crutch of trend() and forecast() functions in Excel, not through any solid remembrance of the math involved.
But this is something I've been feeling increasingly guilty about lately. And as my projects seem to get more and more mathematically intensive, one of these days it's going to come back to bite me.
So for those of you with serious mathematical chops: Can anyone recommend a good refresher book on higher math? On a practical level, it wouldn't hurt me to read up more on the kinds of business data analysis that I do on a regular basis, but at a more basic level I feel like I never fully "got" calculus and some of the other elements of higher math, although I gained the ability to get through problem sets with relative facility. I tend to be the sort of person who needs to nail the general theory in order to feel like I really understand what I'm doing at the particular level. So I'm thinking I may need something that takes something of a top-down approach. I still have my high school calculus text book sitting around, but I'm kind of wondering if I need to take a different approach.
Suggestions welcome.
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I'm a big fan of most of the math books published by Dover. They have reprints of many classic math/engineering texts. They are also pretty cheap. If you live near a college bookstore, you can probably peruse some there.
Also some people have an almost religous hatred of the Shaums outline series (they are hit-or-miss), but I've found them useful primers. I've worn the cover off of their linear algebra book.
Forgive the radical mathematician for saying so, but people ought to learn how to prove stuff before they're 18 (not just geometric proofs; I mean, you ought to prove the quadratic formula so that you understand it, and you ought to prove some basic tenets of arithmetic, and properties of the real numbers, and so on). Otherwise all of math is just magic tricks taken on authority.
Unfortunately, I don't know any good teaching resources. I learned this sort of stuff, first by reading Gödel, Escher, Bach in high school, but that's more a philosophy-of-mind book with its own (quasi-materialist) viewpoint; and also I started college at 17 and took a proof-intensive course right off the bat.
I mean, don't take me for being condescending, because math is hard. It's just that nobody ever thinks to really teach it before upper-level college courses.
Math is the reason I went to law school.
I haven't used it, but someone recommended a book the other day.
All the Mathematics You Missed But Need to Know for Graduate School
A few thoughts:
(1) Avoid high-school texts, which are invariably of vastly lower quality than college texts. I'd avoid "intro-level" college texts as well, and look for books written for the upper-division math classes. The Springer-Verlag "yellow books" are of reliably high quality.
(2) If you want calculus specifically, an idiosyncratic but interesting take on it can be found here:
http://www.math.wisc.edu/~keisler/keislercalc1.pdf
(The link is to a 24 MB pdf.)
You might also consider a text in real analysis, which will frequently start with Lebesgue integration, instead of Riemann, and give a good theoretical background to the field.
(3) A few upper-division texts that I've quite liked, with a leaning toward the more theoretical side of math: Halmos' Finite-Dimensional Vector Spaces, Munkres' Topology, Herstein's Topics in Algebra, and Kunen's Set Theory (some of these are quite hard).
(4) Another thing to consider is doing some practice with "problem-solving" math. The AMC 12 test (formerly the AHSME) is an excellent test, and the problems on it (a) don't require anything beyond basic trig and (b) are often quite difficult and encourage good creative mathematical thinking. Very few people even with a solid high school math background can get a perfect score on an AMC 12. If you find those too easy, you could try the AIME or even the USAMO tests. For details on all, see here:
http://www.unl.edu/amc/e-exams/e6-amc12/amc12.html
(5) Another possibility would be to read a good history of math, which gives a general context for the developments. I like both Boyer's A History of Mathematics and Kline's three-volume Mathematical Thought From Ancient to Modern Times.
Thanks, Philosopher. That should definately give me some directions to poke around.
I downloaded Keisler's book and added Boyer's History of Mathematics to my Amazon cart. We'll see where that gets me.
A quick read of Keisler's introduction looks interesting, but now I guess it's time to buckle down to another day of getting by in the business world.
Thanks also, Charles. I hadn't realized that Dover had so many cheap mathematics texts. (Cheap being being a virtue at times compared to $100+ standard texts.) Following your lead (and Philosopher's suggestion of looking into real analysis) I added Shilov's "Elementary Real and Complex Analysis" to my cart as well.
Some years ago we got an Amazon visa card, and put all our household spending through it, so as to net ourselves a $25 gift certificate every couple months. Forget airline miles, it's the best.
And between those three resources, I should have all the self study I can handle on the subject for the next 6-12 months.
"... added Boyer's History of Mathematics to my Amazon cart ..."
Wow, Darwin. Sounds like some stimulating reading you've got ahead of you.
;)
Everyone's entitled to his own interests, right Jay and Rick?
Let's see, Darwin.
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HOT
Math is good...
LOL!
Dear Darwin: "Mathematitian's Delight", by Sawyer, published by Pelican and available used, might help. It's not a refresher; it's intended to dispel fear of mathematics, but it gives clear explanations of mathematical concepts which I found invaluable in clarifying my grasp of them. Much more useful than just memorizing rules and drilling on them in exercises.
Otherwise I would check "Calculus for Dummies." No, I haven't read it, but I've had very good luck with the "For Dummies" series.
Math is another very important field. It always has been, but as science progresses and technology proliferates it's more relevant than ever.
When I took calculus at the UCLA Extension many years ago (I got a "D"), I chatted with the teacher, who mentioned that they were getting a large number of professionals, including a lot of physicians, having to take strenuous math classes in mid-career. A lot of them, especially the physicians, are taking math they never took before, because when they went to medical school it was not necessary.
Regards,
LogEyed Roman
Regarding mathematics, I'm attempting to overcome my dislike of math which was cultivated on my part by an otherwise really good teacher of math in my high school. He taught algebra, geometry, trigonometry, and the beginnings of analytical geometry from a visual point of view (I'm a verbal/auditory type myself), and also taught it as a collection of processes of calculations. I was more interested in logic than calculation in those days, so I found things rather dry. Ergo, my knowledge of math presently extends no further than basic algebra.
I'm trying to change that now. Here's how I'm doing so:
Basically, I've come to the conclusion that mathematics is a process of calculation, reasoning, and heuristic. So I'm working at developing skills in each of those processes.
For calculation, I'm coming fast to the conclusion that rapid, automatized mental calculation is both 1) a right-brain phenomenon, and 2) something that can be developed by most humans. The Art of Calculation, a Dover book, is a good place to start. Dead Reckoning, a book of algorithms for mentally deriving arithmetic functions (addition, subtraction, multiplication, and division), roots (square, cube, etc) and powers, logarythmic and trigonometric functions, also looks promising. Vedic Mathematics, although claiming to be derived from the Atharva Veda looks to be nonetheless useful. I've also got some ideas as to how to assist the process of automatizing these mental processes.
For reasoning, I'm looking into Boole's The Laws of Thought. I'm also thinking that there are several good e-texts available on the internet for assisting in the process of constructing proofs. I'll see if I can retrieve them from my (currently dead) laptop.
For heuristic, I recommend How to Solve It, by Polya. It's a good compendium of techniques for solving mathematical problems. His other, more formal, works on the subject look interesting as well.
That done, I suppose the best thing to do is to look into the three main branches of mathematics, algebra, geometry/topology, and analytics. For the last, I can recommend Calculus Made Simple by Thompson.
More later
Bernard,
I'll have to take a look for The Art of Calculation. I've gotten terribly lazy about mental arithmetic, perhaps in part because I spend much of the day using Excel and so don't need to do it. But it seems like something one ought to be able to do a bit of.
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