Base two numerals can be awkward to compute with, and Roman numerals are even more awkward. But awkward is not the same as bad. Today I’ll tell you about a truly awful numeral system—possibly the very worst ever proposed as a way of doing arithmetic—but I claim that even such an abominable system is useful for something.

EQUAL TIME FOR THE FAR LEFT

First let’s talk about a funny way of extending decimal notation that lets you write negative numbers without using a minus sign. In this system −1 is written as …999, which should remind you of the infamous decimal 0.999…, except that where the dots in 0.999… mean “forever rightward” the dots in …999 mean “forever leftward”.

In ordinary math, the infinite decimal .999… is defined to be the limit of the terminating decimals .9, .99, .999, …; that is, it’s defined to be the real number that the fractions 9/10, 99/100, 999/1000, … approach in the ordinary sense. And that limit is most definitely 1, not some real number that’s a tiny bit less than 1. This is not an approximate truth; it’s a 100% accurate, rigorously established mathematical fact. It’s a part of how the real number system works, and it’s a feature, not a bug.

But if you’ve read my essay “Marvelous Arithmetics of Distance”, you already know that there are number systems in which things that look like rational numbers behave differently¹, and you won’t be too surprised to learn that there’s another new number-game to play. It’s the q-deformed real number “game” of Sophie Morier-Genoud and Valentin Ovsienko, and in the context of their work, it becomes true (in an admittedly somewhat arcane sense) that 9/10, 99/100, 999/1000, etc. do not approach 1 but rather approach something smaller. Except that it’s not the numbers themselves that behave in this ill-bred way; it’s their avatars in the q-deformed world, avatars that Morier-Genoud and Ovsienko write as [9/10]_q , [99/100]_q , [999/1000]_q , etc.

In this essay, without going too deeply into the underlying theory, we’ll take a concrete look at the q-deformations of the numbers 1/2, 2/3, 3/4, etc. and of the numbers 2/1, 3/2, 4/3, etc. In the ordinary real number system, all these fractions approach 1 as the numerators and denominators get big, regardless of whether the bigger number is on top of the fraction or on the bottom. But looking at these rational numbers through the spectacles that Morier-Genoud and Ovsienko have given us, we’ll see that we get two different limits according to whether the numerator is bigger than the denominator or vice versa. In particular, we’ll find that, while the q-deformations [2/1]_q, [3/2]_q, [4/3]_q, etc. approach [1]_q, the q-deformations [1/2]_q, [2/3]_q, [3/4]_q, etc. approach [1]_q’s evil (or maybe not so evil) twin.