Who Mourns the Tenth Heegner Number?

I’m the urban spaceman, baby; I’ve got speed.
I’ve got everything I need.

− Neil Innes, “I’m the Urban Spaceman” (Bonzo Dog Doo-Dah Band)

There’s an episode¹ of a science-fiction television series in which space travelers land on a planet peopled by their own descendants. The descendants explain that the travelers will try to leave the planet and fail, accidentally stranding themselves several centuries in the past. Armed with this knowledge, the travelers can try to thwart their destiny; but are they willing to try if their successful escape would doom their descendants, leaving the travelers with the memory of descendants who, thanks to their escape, never were?

This is science fiction, but it’s also math. More specifically, it’s proof by contradiction. As Ben Blum-Smith recently wrote on Twitter: “Sufficiently long contradiction proofs *make me sad*! When you stick with the mathematical characters long enough, you start to get attached, and then they disappear, never to have existed in the first place.”

Cartoon by Ben Orlin. You can buy his book “Math With Bad Drawings”.

This will be an essay about things that seem to exist but which, when you study them deeply enough, turn out not to exist after all.

The Magic of Nine

The kind of magic that grabs (and enchants) me is fantasy-literature magic, not stage-magic, but this month I’ll make an exception to talk about a new math trick inspired by Art Benjamin’s fun-packed book “The Magic of Math: Solving for x and Figuring Out Why”.

After you read this essay, you’ll be able to compute the answer to questions like “What’s the remainder when you divide 123456789 by 9?” or “What’s the sum of the digits when you multiply 123456789 by 9?” without setting pencil to paper, purely through the power of your mind. But before I explain that, I’d better address the question many of you are asking: why would anyone want to learn how to do this sort of thing? (Short answers: to stay sane, to balance their checkbook, and/or to become fame-ish.)

But first, a few words about numbers, rapidity, and mental math.

I HATE NUMBERS¹

Introducing “Thirdsday”

I’m pleased to announce that the greatest holiday in mathematics is almost upon us: the jubilant festival known as THIRDSDAY!

Huh?

Thirdsday is that magical day on which we celebrate the wonder and mystery of the fraction 1/3.

How come I haven’t heard of it before?

Don’t feel bad; I didn’t know about it either until it was discovered a couple of months ago. Or was it invented? Math can be so ambiguous that way.

Stance and distance in popular writing about math

Here’s something you’ll never see in popular writing about musicians:

“Music. For most of us, the mere word conjures up memories of metronomes and endless scales, the student’s never-ending fear of playing a wrong note (or the right note at the wrong time), and the frowns of teachers from whom a curt ‘Good’ was the highest expression of praise. And yet there are some people who just can’t get enough of making music; they practice hour after hour, honing their skills and punishing their bodies, long after the stage of life when there are parents and teachers forcing them to do it. What strange quirk of character compels this behavior?”

As I said, nobody writes about musicians that way. And yet, when the subject is a mathematician, writers sometimes come up with passages like this:

“You and math – one of the greatest love/hate relationships of all time. What is it about the subject that excites us yet sends a chilling tingle down our spine at the same time? How can it be so precise, yet so fickle? We may never know the answers to these questions, but we do know that math is ubiquitous, though some of us may try to hide from it.”

This is from the introduction to a 2010 profile of mathematics editor Vickie Kearn, which I recently saw on the blog-site of Princeton University Press, where Ms. Kearn has worked as senior editor for many years. The intro continues in a similar vein. “While math may sometimes cause us to cry tears of despair, it has caused Vickie to cry tears of joy.”

Between the World and the Mind

The wizard’s-cap graphic that appears at the top of my blog as part of the logo is a piece of an infinite mathematical surface called the pseudosphere.

I don’t study the pseudosphere in my research, and I can’t say I have a lot of intuition about it; in fact I don’t especially like the thing. So why did I choose it to visually represent what this blog is about?

ChipChip: A new sort of sorting

A uniquely French way to express contempt for someone is to call them an “espèce d’espèce” (see Endnote #1); literally, “a sort of a sort”.  This month I’m going to tell you about a sort of a sort (or rather, a sort of sorting) that, from a practical standpoint, merits this degree of contempt: the procedure is ambiguous, is annoyingly slow, and doesn’t always sort things correctly. Yet there’s an unresolved mathematical mystery arising from the way that the procedure works better than it has any right to.

But first, a puzzle:

A New Game with Infinity

Aleph-null bottles of beer on the wall,
Aleph-null bottles of beer!
Take one down, pass it around,
Aleph-null bottles of beer on the wall!

— Math nerd drinking song

You may already know the standard story about infinite sets like {1,2,3,…} and {2,3,4,…}. Even though the second set seems to be smaller (it’s missing one of the elements in the first set), Cantor taught us that the two sets are the same size (in the sense that there’s a one-to-one correspondence between them). The two sets have the same “number” of elements (namely aleph-null), and aleph-null minus one equals aleph-null. For many students, that anomaly takes some getting used to.

Cartoon by Ben Orlin, now the author of https://mathwithbaddrawings.com/2018/05/23/math-with-bad-drawings-the-book/. (I urge you to buy a copy rather than steal one, since there are only finitely many copies.)

But there’s a perfectly respectable mathematical sense in which the two sets do not have the same number of elements. With a suitable notion of what it means to “count” the elements of an infinite set of numbers, different from Cantor’s, the size of {2,3,4,…} is smaller than the size of {1,2,3,…}; in fact, it has one fewer element. Likewise, in this alternative way of measuring how big sets of numbers are, the set {1,3,5,…} is slightly bigger than the set {2,4,6,…}. How much bigger? Half an element! (Though see Endnote #2.)