What Bad Bases Are Good For |

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”.

Here’s how negative integers work in this system:

−1 = …999,<br>−2 = …998,<br>−3 = …997,<br>−9 = …991,<br>−10 = …990,<br>−11 = …989,<br>−99 = …901,<br>−100 = …900,<br>−101 = …899,

But what does this mean?

Since the ordinary decimal 99 means 90 + 1 or equivalently 1 + 90, and 999 means 900 + 90 + 1 or equivalently 1 + 90 + 900, and so on, it’s tempting to say that if …999 signifies anything at all, it should signify the infinite sum 9 + 90 + 900 + …. Setting x = 9 + 90 + 900 + …and multiplying that equation by 10, we get 10x = 90 + 900 + 9000 + …. Now line up those two equations this way:

When we subtract one from the other, we get x − 10x = 9 + 0 + 0 + 0 + …, or −9x = 9, with solution x = −1.

You might complain that 9 + 90 + 900 + …doesn’t really equal −1, and you’d be right under the normal rules of mathematics. But in some sense 9 + 90 + 900 + … is trying to equal −1. It turns out that the voice telling us that 9 + 90 + 900 + … is meaningless (or infinite) and the voice telling us that 9 + 90 + 900 + … is equal to −1 are both speaking the truth; it’s just that one voice is speaking to us from the world of ordinary arithmetic while the other is speaking to us from the world of 10-adic arithmetic.

(In case you’re thinking that this is all hopelessly arcane, you should know that many computers use a base-two version of this, in which −1 is …111, −2 is …110, −3 is …101, and so on, except that now "…" corresponds to a long but finite string of 1s rather than infinitely many of them. This system for writing negative integers is called twos complement, and the corresponding system for base ten is called tens complement.)

The 10-adic way of writing integers gives you a way to do ordinary arithmetic, for instance, to confirm that −2 plus +3 equals +1.1 But I hope you’ll agree that it’s a blemish in this system that many perfectly finite ordinary integers (namely the negative integers) require infinitely many digits. So, given that our perverse goal today is to find bad systems for doing arithmetic, let’s establish one hallmark of a Bad System: It should have infinite representations of some finite integers.

WHERE YOU GO IS WHO YOU ARE

If you’re suspicious of formulas like 9 + 90 + 900 + … = −1, here’s a different way to think about the 10-adic system for representing integers: instead of viewing …999 as a compressed way of writing an infinite sum, you can think of it as a kind of photo-reel of the scenery you see when you start at the number −1 and repeatedly apply a certain “jumping rule”. More specifically, from the integer n you jump to the integer (n−r)/10, where r is chosen to be the unique digit between 0 and 9 inclusive that makes n−r a multiple of 10.2

I’ll call r the residue; as a good tourist you’re expected to take a photo of the current residue (click!) each time you make a jump. Here’s a simple example:

If you start with n = 17, the only r between 0 and 9 that makes 17−r a multiple of 10 is r = 7, so the residue is 7 (click!) and you jump from 17 to (17−7)/10 = 1.

Now you’re at 1, and the only r between 0 and 9 that makes 1−r a multiple of 10 is r = 1, so the residue is 1 (click!) and you jump from 1 to (1−1)/10 = 0.

Now you’re at 0, and the only r between 0 and 9 that makes 0−r a multiple of 10 is r = 0, so the residue is 0 (click!) and you jump from 0 to (0−0)/10 = 0.

So you’re stuck at 0 forever, and the residue will be 0 forever onward.

The residue-sequence goes 7,1,0,0,0,…, and writing this backwards we get …017, which is the way we write 17 with leading 0’s.

Now let’s try an example with a negative number.

If you start with n = −1, the only r between 0 and 9 with the property that n−r is a multiple of 10 is r = 9; so the residue is 9 (click!) and you jump from −1 to (−1−9)/10, which is −1.

So your jump hasn’t taken you anywhere new; you’re still at −1, so when you jump again, you do the same exact thing, seeing the residue 9 again (click!), and so on, forever.

Writing that endless sequence of 9’s backwards, you get the beginningless sequence …999, which is the 10-adic representation of −1.

So you don’t need to think of an infinite string of digits as representing an infinite sum that may or may not converge to some number n; you can think of it as a record of what scenery you’ll see when you start...