Today we’ll talk about some paradoxical things, like the logarithm of zero, and the maximum element of a set of real numbers that doesn’t contain any real numbers at all. More importantly, we’ll see how mathematicians try to wrap their heads around such enigmas.
All today’s logarithms will be base ten logarithms; so the logarithm of 100 is 2 (because 100 is 102) and the logarithm of 1/1000 is −3 (because 1/1000 is 10−3)). The logarithm of 0 would have to be an x that satisfies the equation 10x = 0. Since there’s no such number, we could just say “log 0 is undefined” and walk away, with our consciences clear and our complacency unruffled.
BUT WE AREN’T GOING TO DO THAT, ARE WE?
I thought my earlier essay on .999… did a pretty good of explaining why I (along with 99.999…% of mathematicians) say that it equals 1, until I asked some of my students what they got out of it; then I got a humbling jolt of pedagogical reality. The students agreed that .999… is the limit of the sequence .9, .99, .999, etc., and they also agreed that the limit of that sequence is 1. So you might think that they would have agreed that .999… equals 1, but no: they couldn’t swallow that conclusion.
I’ve decided that part of what’s going on is that my students arrive at college with a number-sense that’s so deeply grounded in their experience with terminating and non-terminating decimals, in worksheet after worksheet, that these concrete representations have taken on an independent reality for them. At that point, it does little good to tell them “.999… doesn’t mean anything till we assign it a meaning” or “We’re going to define .999… as a limit”, because they already “know” what .999… is: a dot followed by infinitely many 9s! No attempt to redefine .999… can shake loose their sense of what it already means to them.
So today I’m going to come at the problem of .999… from a totally different direction. Continue reading
I have trouble with three-dimensional space. Yes, I do live in it, and I get by without hurting myself too badly too often, but honestly, I miss a lot of what’s going on. There’s no reason for you to care about my problem, except that it touches on the issue of “What does it take to be a mathematician?”, and the details of my limitations might serve as a useful antidote to the idea that math ability is always linked to spatial intuition. But I’ve got an ulterior motive for these confessions: I need some help, and I’m hoping one of you can provide it.
BERKELEY, CALIFORNIA, 1983
I remember my zeroth day of grad school, the day before the start of classes. I was attending a workshop on how to be a good teacher — something the university required all incoming Ph.D. students to do, since most of us would serve as Teaching Assistants for at least one semester. The seasoned TA leading our orientation wanted us to experience the sort of Aha! moment that good teachers instigate, so she led us through the famous “handcuffs puzzle” in which two people tied together by cords must extricate themselves from one another. Continue reading
Pi, that most celebrated of mathematical constants, leads a curiously double life. On the one hand, we have numerical formulas for pi, like Leibniz’s formula π = 4 × (1/1 − 1/3 + 1/5 − 1/7 + …); imagining a world in which this expression converges to a value other than 3.14… is as hard as imagining a world in which 2+2 doesn’t equal 4. On the other hand, we have a geometric definition of pi as the ratio of the circumference of a circle to that circle’s diameter, and this definition of pi lets us imagine that pi is a physical constant like the speed of light — that it could have a different value in an alternative universe that’s built using a different kind of geometry. Could there be worlds in which geometrical pi equals 3.24…, say, and in which the more open-minded scientists and mathematicians speculate about other worlds in which pi has some crazy value like 3.14…? Continue reading
Ingredients of today’s mathematical stew include beans, boats, never-ending chess-games, a composer who’s into aperiodic percussion, an ABBA from Scandinavia that’s not the famous pop group, a Jewish camp song, and a way to calculate 02 − 12 − 22 + 32 − 42 + 52 + 62 − 72 using calculus. Oh, and did I mention beans?
During college, I learned about the traditional Jewish tenet that if you perform some voluntary religious observance three times in a row, you’re obliged to keep doing it forever — that through force of repetition, what was formerly a mere custom becomes as binding as a commandment (or, some say, you’ve effectively made a vow to do it forever, even if you didn’t intend to). The word for this phenomenon is chazakah, or “strengthening”.
You probably haven’t heard of David C. Kelly; he doesn’t write best-sellers or give TED talks, or study the center of the galaxy or the human genome or the social impact of algorithms. But he’s inspired and nurtured hundreds of people who’ve done these things and much more. The vehicle of this inspiration is a summer program that that Allyn Jackson has called “a national treasure” and that for the past forty years has been quietly shaping American mathematics. Some people call it “Yellow Pig Camp“, but many of its alums (including yours truly) simply call it “Hampshire”. It’s the Hampshire College Summer Studies in Mathematics program, or HCSSiM for short, founded by Kelly in 1971.
Hampshire doesn’t teach students how to be better at high school math. It leapfrogs over AP Calculus and jumps directly to college- and graduate-level topics: graph theory, cellular automata, non-orientable surfaces, etc. Continue reading
I don’t especially like people who talk a lot about themselves, but I have a soft spot for sentences that do. Case in point: the self-referential sentence “This sentence is false.” I really like that one. You may think you’ve seen this sentence before, but in fact you’ve seen other sentences, each made of the exact same words in the exact same order. I admit it’s hard to tell apart all those sentences, each separately subverting its own lonely self!
(If you think that two sentences that have the same words in the same order must really be the same sentence, consider the following two-sentence passage: The other sentence in this passage is lying! The other sentence in this passage is lying! The two sentences have the same words in the same order, but they’re saying very different things. And now that I’ve written down those two mutually-referential sentences, I have to say I don’t like them at all. They’re kind of shrill and unpleasant, and they remind me too much of current events.) Continue reading