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

# Avoiding chazakah with the Prouhet-Thue-Morse sequence

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 0^{2} − 1^{2} − 2^{2} + 3^{2} − 4^{2} + 5^{2} + 6^{2} − 7^{2} 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”.

# Will ’17 be the Year of the Pig?

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

# Breaking logic with self-referential sentences

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

# Going Negative, part 2

Last month, when I gave some ideas about how to justify the law of signs, my focus was on the kind of explanation that works when kids first encounter negative numbers. But in a way I wasn’t being 100% honest, and my use of some farfetched examples (like the balloon-stealing clown) was a tip-off. I think that the *real* justifications of the law of signs — not the most pedagogically appropriate ones, but the most historically honest ones — come from the body of material the students will encounter later in their studies, long after they’ve learned, enthusiastically or reluctantly, to calculate products in the standard way. These are justifications teachers seldom talk about with their students, but I think they matter.

So this month I’ll talk about those other rationales, and try to resolve any remaining qualms you may have about the law of signs that stem from a sense of symmetry-violation. I’ll also discuss the option of chucking negative numbers entirely. (Seems extreme, but as a parent of two children, I can sympathize with this way of brokering the conflict between yes-it’s-negative and no-it’s positive. “You know what, kids? If you can’t agree on a restaurant, we won’t go out to dinner at all.”) Continue reading

# Going Negative, part 1

*Minus times minus equals plus. / The reason for this we will not discuss.*

— W. H. Auden, recalling a popular verse from his school days

Ever tried mixing together your two least favorite foods? I suspect you haven’t. Nobody mixes two noxious ingredients and expects the results to be tasty. So why should numbers behave differently in the numerical recipe called multiplication? What mystical two-wrongs-make-a-right alchemy removes the taint of negativity and makes the product of two negative numbers positive? It just don’t make no sense!

The pioneering sixteenth-century algebraist Girolamo Cardano had qualms about this alchemy, and toyed with the idea of defining the product of two negative numbers to be negative. In the intervening centuries, legions of schoolchildren have been tempted to follow that road. But the mathematical community staunchly insists that it’s the wrong road. Why?

Continue reading

# Bertrand’s Ballot Problem

Early leads don’t always lead to much. Remember Republican presidential candidate Rudy Giuliani? Back in 2008, the outspoken New Yorker polled so well that some pundits predicted he’d win the nomination, but his lead fizzled out before the convention. The same thing has happened with many other would-be presidents from both sides of the aisle over the past couple of decades. So you might be inclined to discount a candidate’s early lead in polls. In fact, this reasonable inclination was part of why FiveThirtyEight analyst Harry Enten didn’t see Donald Trump’s nomination coming; Enten figured that Trump would wind up being just another Giuliani. Oops.

So, how closely do we expect early polling in an election process to correspond with the final outcome? A classic problem in probability theory strips this real-world question down to manageable size.