The Infinite Stairway

I’m sure you’ve counted (“One, two, three, . . . ”) on too many occasions to count. The process can be boring (counting sheep), exciting (counting your winnings at a casino), or menacing (“If you kids aren’t at the dinner table by the time I reach ten, I’ll …”). But one thing counting is not is liberating. What could be less free than the inexorable succession of the counting numbers? And yet the very regularity of counting numbers gives us the freedom to think about them in multiple ways, arriving at conclusions along delightfully varied paths.

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Teaching with Magic Paper

I know that the sentence “The year is 2022” is just a bland statement of fact, but it hits my ear like a voice-over in a trailer for a bad science fiction movie made in the 1900s. Blame Walter Cronkite; I grew up watching his TV series The Twenty-First Century (1967-1969) and came to indelibly associate the 2000’s with The Future. Now that I actually live in The Future, surrounded by many of its predicted marvels, my degree of enthrallment varies from marvel to marvel, but I never tire of the wonders of magic paper. You know the stuff I mean: you write something in one place and the paper makes copies of itself elsewhere so that people in those other places can read the words you just wrote. I’m sure you’ve all used it. I’m using it now.

Magic paper helps me with some problems that have long bedeviled classroom teachers like myself: How do you find out what’s going on inside your students’ heads in the midst of a lesson without derailing it? How do you get all your students to actively participate without having the class descend into chaos? How do you communicate with a large group of students without the conversation devolving into what math educator Henri Picciotto calls a “pseudo-interactive lecture” dominated by the teacher and the two or three most vocal students?

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What Lovelace Did: From Bombelli to Bernoulli to Babbage

I want to tell you about difference tables for polynomials, not only because they’re fun but also because they’ll give us a chance to see how polynomials played a role in the dawn of the computer age through the work of computer pioneers Charles Babbage and Ada Lovelace.

But first, where did polynomials come from?


“Thing” is a marvelously flexible word, as are similar words like “res” and “cosa” that other languages have used to signify unspecified objects. Often the word denotes a group of people who have come together for some purpose: think of the Roman Republic (the “public thing”) or the Cosa Nostra (“Our Thing”). Curiously, the English word “thing” itself seems to have traveled in the opposite direction, starting out as meaning an assembly of people and ending up as meaning, well, any-thing. Math has made its own uses of nonmathematical words for indefinite objects: in Indian and Arabic algebra, the quantity being sought was often called “the thing”. It was natural for European algebraists to borrow this usage, and indeed Renaissance algebra was sometimes referred to as “the art of the thing”. (See Endnote #1.)

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Let x Equal x

Dedicated to the memory of Herb Wilf

Mathematicians celebrate the French thinker René Descartes for inventing Cartesian coordinates.1 But we should also remember him as the person who tilted the terrain of Europe’s mathematical alphabet, using early letters of the alphabet to signify known quantities and imbuing later letters (especially x) with the pungent whiff of the Unknown. If you learned to write quadratic expressions as ax2 + bx + c instead of xa2 + ya + z (and I’m guessing you did), it’s down to Descartes.2

My topic this month is polynomials like ax2 + bx + c. In school math, you first learned about x as an unknown, a number hiding behind a mask. (“What is x? Let’s find out.”) Later you learned to view x as a variable, so that a formula like y = ax2 + bx + c is a function or rule: if you give me an x, I’ll give you a y. (“What is x? No number in particular; x ranges over all real numbers.”) I’ll touch on both points of view today, but I’ll be stressing a viewpoint that’s probably less familiar, where x is neither an unknown nor a variable, but just, well, itself. From this perspective, polynomials appear as number-like objects in and of themselves, with their own habits and mating behavior.

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Twisty Numbers for a Screwy Universe

If new kinds of numbers were like new consumer products, mathematicians would have every right to fire the marketing company that came up with the names “complex numbers” and “imaginary numbers”. I mean, what kind of sales pitch goes with that branding? “Psst: wanna buy a number? It’s really hard to understand, and best of all, it doesn’t even exist!”?

We mathematicians have nobody but ourselves to blame, since it was one of our own (René Descartes) who saddled numbers like sqrt(−1) with the term “imaginary” and another mathematician (Carl-Friedrich Gauss) who dubbed numbers like 2+sqrt(−1) “complex”. Now it’s several centuries too late for us to ask everybody to use different words. But since those centuries have given us a clearer understanding of what these new sorts of numbers are good for, I can’t help wishing that, instead of calling them “complex numbers”, we’d called them — well, I’ll come to that in a bit.

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Good Shurik Grothendieck

I used to tell people that the title character of the film Good Will Hunting didn’t strike me as very believable — not because of the self-taught janitor’s ability to do cutting-edge research, but because of his contempt for his own work. At one point in the movie, having shown his mentor a proof he’s just written, he sneers “Do you know how easy this is for me?” and sets the proof on fire with his cigarette lighter — at which point his mentor, a world-class mathematician with a Fields Medal to his name, dives onto the carpet not so much to prevent the building from burning down (buildings can be rebuilt, after all) as much as to rescue a proof that the mathematical world will cherish.

“That’s a teenager’s idea of what being a genius is like,” I would tell people.

“Oh, and are you a genius?” one woman once asked me skeptically.

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Tricks of the Trade

In the 1950s, a Scottish mathematician named C. Dudley Langford looked at a stack of six blocks his young son had assembled (see Endnote #1) and noticed something interesting that would lead him to the mathematical discovery he’s remembered for today. 

Langford noticed that between the two red blocks was one block, between the two blue blocks were two blocks, and between the two yellow blocks were three blocks. Being a mathematician, Langford immediately wondered “Could we do this with more than three colors?”

Can you figure out how to add two green blocks and arrange the eight blocks so that there will be one block between the red blocks, two blocks between the blue blocks, three blocks between the yellow blocks, and four blocks between the green blocks?

And, having succeeded with four colors, can you do it with five?

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Breaking Pi

I love working with others to discover new mathematics, but there’s a kind of research I’d love even more: helping decode a Message from an extraterrestrial civilization. The chance to do that would make me drop all my mathematical projects — though in a way it wouldn’t, since decoding the Message would almost certainly involve a lot of math.

As a teenager I was captivated by a 1973 book called Communication with Extraterrestrial Intelligence. It was edited by a not-yet-world-famous astronomer named Carl Sagan who was interested both in sending messages to the stars and in seeking messages from the stars to us. He went on to host the incredibly popular TV program “Cosmos” and to write several best-selling books, including the novel Contact about which I’ll have a lot to say later.

The reason I’m writing this particular essay this month is because almost exactly two centuries ago, the mathematician and astronomer Carl Friedrich Gauss proposed sending a message to the moon. (Gauss’ ideas about life on other worlds had a respectable pedigree in European thought; see the excellent articles by Aldersey-Williams and Dillard listed in the References.) Gauss had invented a kind of signaling device he called the heliotrope, and on March 25, 1822, he wrote a letter to the astronomer Heinrich Olbers, saying “With 100 separate mirrors, each of 16 square feet, used conjointly, one would be able to send good heliotrope-light to the moon. … This would be a discovery even greater than that of America, if we could get in touch with our neighbors on the moon.”

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The Clatter of the Primes

The trouble began, as trouble often does, with a rivalry between friends. It took place during the Big Before, when numbers and operations were new and still figuring themselves out, and none of them had any idea what a universe was or whether having one would be a good idea.

Plus said to Times “No offense, friend, but I’m just better at building numbers than you are. Starting from 1, the smallest number, I can build lots of new numbers: 1+1 is 2, 1+1+1 is 3, and so on. But look at you! 1×1 is just 1. 1×1×1? 1 again. And so on. Boring!”

Times naturally became defensive. “Now that’s just not fair. You’re using the wrong building block. Instead of 1, try 2.” And the number 2 began to twinkle. “2×2 is 4. 2×2×2 is 8. And so on. See, I get new numbers, just like you, and mine are bigger than yours!”

Plus said “I can get all those numbers, and more; it just takes me longer. But I get some numbers you can’t get. 3 is 1+1+1, but you’ll never get 3 by multiplying 2’s.”

Times, thinking quickly, retorted, “I never said I could get everything from 2’s. I also use 3 as a building block.” Then the number 3 began to twinkle. “For instance, with 2 and 3, I can get 2, 4, 8, and so on, and 3, 9, and so on. And I mix 2’s and 3’s, so I get 6 and lots of other numbers too.”

Plus said “What about 5? How do you get 5 by multiplying 2’s and 3’s?”

Times airily answered “Oh, I never said 2 and 3 would be enough! 5 is another one of my building blocks.” And the number 5 began to twinkle.

Plus asked “How many of these building blocks do you have?”

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