All the work they made them do was rigorous.
— Exodus 1:14
As I write this, it’s April 5, midway through the eight-day festival of Passover. During this holiday, we Jews air our grievances against the ancient Pharaoh who enslaved and oppressed us, and celebrate the feats of strength with which the Almighty delivered us from bondage — wait a minute, I think I’m mixing up Passover with Festivus. In any case, on the first two nights of Passover, Jews tell the story of the Exodus from Egypt before dinner, reading from individual copies of a book called the haggadah (“the telling”), and after dinner they sing a few extra songs, one of which is called “Ekhad Mi-Yodeah?”, or “Who Knows One?” It’s a cumulative song, with each verse containing one more line than the verse before, and in my family the game is to try to get through the increasingly long middle part of each verse without taking a breath. That’s tricky when you get to the last verse, whose English translation is
Who knows thirteen? I know thirteen!
Thirteen are the attributes of God;
Twelve are the tribes of Israel;
Eleven are the stars in Joseph’s dream;
Ten are the commandments;
Nine are the months to childbirth;
Eight are the days to circumcision;
Seven are the days of the week;
Six are the order of the Mishnah;
Five are the books of the Torah;
Four are the mothers of Israel;
Three are the fathers of Israel;
Two are the tablets of the covenant;
One is our God in heaven and earth.
But what does it mean to know a number? Continue reading
Mr. Jabez Wilson laughed heavily. “Well, I never!” said he. “I thought at first that you had done something clever, but I see that there was nothing in it after all.”
“I begin to think, Watson,” said Holmes, “that I make a mistake in explaining.”
— Arthur Conan Doyle, “The Red-Headed League”
When I was a kid living in the Long Island suburbs, I sometimes got called a math genius. I didn’t think the label was apt, but I didn’t mind it; being put in the genius box came with some pretty good perks. For one thing, the kids who thought I was a genius at math had lower expectations of me when it came to other things, so to the extent that I was endowed with a normal measure of certain traits (like a healthy sense of humor), people were impressed by how normal I was — and to the extent that I had deficits of athleticism or common sense, people tended to be forgiving: “What do you expect? He’s a genius!” (See Endnote #1, though.)
There were other kids I heard about, most of them living in New York City and attending Stuyvesant High School and the Bronx High School of Science, whose mathematical achievements were described in awed tones on the high school math grapevine, and I envied their endowments and their successes, but I didn’t think they were geniuses either. They knew lots of things that I didn’t know — things I wanted to know — but they didn’t know things in a different way than I did. The things those kids knew weren’t facts, but rather habits of thought: the kind of habits that won you top honors in math competitions and science fairs.
I got acquainted with some of those kids in my junior and senior years of high school, and they were awesomely good at solving math problems, but there was no uncanny aspect of their performance that seemed worth explaining away by appealing to some mystical inborn attribute like “genius”.
The first time I encountered dimensional analysis, I didn’t recognize it for the magic wand that it is; it seemed to be just a form of mathematical hygiene — useful for avoiding mistakes, but not much else. To get a sense of what I mean by hygiene, consider a question that came up in my own household (arising from the incident described in Endnote 1): how do we convert 6 feet per second to units of miles per hour? We know that there are 60 × 60 = 3600 seconds in an hour and 5280 feet in a mile, but suppose we aren’t sure what to do with those numbers to get our answer; we suspect we should multiply 6 by 3600 / 5280 or by 5280 / 3600, but we’re not sure which. To keep ourselves on the right path, we use something called the factor-label method: we attach units to those numbers and we multiply three fractions that carry those units along in their numerators and denominators, with the three fractions representing the assertions “The speed is 6 feet per second”, “3600 seconds equals 1 hour”, and “1 mile equals 5280 feet”:
This gives an answer with units of miles per hour, after we’ve cancelled sec with sec and ft with ft. If we had tried to combine the fractions the wrong way, multiplying where we should have divided or vice versa, the appearance of unfamiliar units in the answer (like square-foot-hours per square mile per second) would have hopefully alerted us to our mistake.
But dimensional analysis can do much more for us. In many cases, it gives us a way to figure out the answer to a physics problem without actually applying the laws of physics. Bertrand Russell once wrote: “The method of ‘postulating’ what we want has many advantages; they are the same as the advantages of theft over honest toil.” In many ways, dimensional analysis gives us the same bargain: it offers us a way to cheat the universe at its own game.
Here’s how Roald Dahl describes what happened one morning when the Queen of England invited a large and unexpected guest (“The BFG” ) to have breakfast with her:
There was a frantic scurry among the Palace servants when orders were received from the Queen that a twenty-four-foot giant must be seated at breakfast with Her Majesty in the Great Ballroom within the next half hour. The butler, an imposing personage named Mr. Tibbs, was in supreme command of all the Palace servants and he did the best he could in the short time available. A man does not rise to become the Queen’s butler unless he is gifted with extraordinary ingenuity, adaptability, versatility, dexterity, cunning, sophistication, sagacity, discretion, and a host of other talents that neither you nor I possess. Mr. Tibbs had them all.
The redoubtable Tibbs may have possessed dexterity and sagacity, but his grasp of allometry was sadly deficient. Allometry is the part of biology that studies how the size of a creature relates to other aspects of the creature’s life. Mr. Tibbs, noting that the Big Friendly Giant was four times the height of an ordinary man, decided that the BFG needed a quadruple-sized meal:
Everything, Mr. Tibbs told himself, must be multiplied by four. Two breakfast eggs must become eight. Four rashers of bacon must become sixteen. Three pieces of toast must become twelve, and so on. These calculations about food were immediately passed on to Monsieur Papillion, the royal chef.
To Tibbs’ consternation, eight eggs weren’t enough to put a dent in the giant’s appetite; even seventy-two eggs weren’t enough to satisfy him.
What was Tibbs’ mistake? And how many eggs should he have offered the BFG?
Two primal pleasures from my years of childhood (and maybe my years of infancy too, though how would I know?) are feelings I’ll call coziness and spaciousness. The first is the feeling I’d get at night pulling the blankets up over my head; the second is the feeling I’d get standing on a beach staring out at the ocean. These pleasures seem like opposites, but sometimes you can have both at once — for instance, when beneath those blankets you have a flashlight and a book full of magic and adventure. I’ve learned that the adventure called mathematics has its own ways of combining the love of the large with the delight of the small, and I’m going to tell you about one of my favorite combinations of these two opposite pleasures: the roots of unity. Continue reading
Customer: “But this receipt proves that I bought the phone less than two weeks ago!”
Manager: “I understand, sir. But you can only get a full refund if you return it within fourteen days, and you’re one day late.”
This surreal exchange isn’t from The Twilight Zone. It took place, with me starring as the hapless Customer, in a perfectly ordinary suburb called Watertown, Massachusetts, and this insidious mixture of social and mathematical ills could visit your town too. The social sickness is a familiar one: big companies like Verizon Wireless can screw you over any way they like, and if they screw over enough people in enough different ways, then the people who got screwed over in any particular way will be too dispersed to find one another and take action. The mathematical malaise? Fencepost error.
[This is the text of a presentation I made on October 7, 2017 at the kick-off event for Global Math Week, held at the Courant Institute of Mathematical Sciences in New York City. Earlier in the day, James Tanton gave his usual brilliant presentation on Exploding Dots, so in my talk I was able to assume that the audience knew what Exploding Dots is about; they also recognized my riff on Tanton’s signature line “I’m going to tell you a story that isn’t true”, as well as the significance of the word “Kapow!” (and its variants) in the Exploding Dots story. You might want to visit YouTube and sample Tanton’s Exploding Dots videos to get a feel for what it’s all about. For the full Exploding Dots spiel, try https://vimeo.com/204368634. There are two videos of this talk: there’s https://www.youtube.com/watch?v=x8wcbONcGHk (the one made by the Global Math Week folks back on October 7) and there’s https://www.youtube.com/watch?v=G8vVDrxcIi8 (my PowerPoint slides, narrated by me). The latter has better visuals, and it has captions, but it’s missing the audience reactions.]
I’m going to tell you a story that’s as true as I know how to make it. I don’t have a background in ethnomathematics, or the history of mathematics, or the history of math education, so please forgive any mistakes, omissions, distortions, or mispronunciations (and let me know about them, but not now!). The story I’m going to tell is as old as civilization, or at least as old as money — because as long as currency has existed in different denominations, there’s always been a need to make change, and to find systems for making change efficiently and accurately. The story I’m about to tell involves many parts of the world over the course of many centuries. And in many ways it’s a story about sand.