7.1: What is a “number?”

Before we do anything with bases, let’s talk about the concept of number, generally. The question “what is a number?" sounds like the dumbest question I could possibly ask you. Yet I predict that unless you’ve studied this material before, you have a whole bunch of tangled thoughts in your head regarding what “numbers" are, and those tangled thoughts are of two kinds. Some of them are about numbers per se. Others are about base-10 numbers. If you’re like most people, you think of these two sets of concepts as equally “primary," to the point where a number seems to be a base-10 number. It’s hard to conceive of it in any other way. It’s this prejudice that I want to expose and root out at the beginning.

Most people, if I asked them to name a number, would come up with something like “seventeen." This much is correct. But if I asked them what their mental image was of the number “seventeen," they would immediately form the following unalterable picture:

17

To them, the number “seventeen" is intrinsically a two-character-long entity: the digit 1 followed by the digit 7. That is the number. If I were to tell them that there are other, equally valid ways of representing the number seventeen — using more, less, or the same number of digits — they’d be very confused. Yet this is in fact the case. And the only reason that the particular two-digit image “17" is so baked into our brains is that we were hard-wired from an early age to think in decimal numbers. We cranked through our times tables and did all our carrying and borrowing in base 10, and in the process we built up an incredible amount of inertia that is hard to overcome. A big part of your job this chapter will be to “unlearn" this dependence on decimal numbers, so that you can work with numbers in other bases, particularly those used in the design of computers.

When you think of a number, I want you to try to erase the sequence of digits from your mind. Think of a number as what is is: a quantity. Here’s what the number seventeen really looks like:

It’s just an amount. There are more circles in that picture than in some pictures, and less than in others. But in no way is it “two digits," nor do the particular digits “1" and “7" come into play any more or less than any other digits.

Let’s keep thinking about this. Consider this number, which I’ll label “A":

(A)

Now let’s add another circle to it, creating a different number I’ll call “B":

(B)

And finally, we’ll do it one more time to get “C":

(C)

(Look carefully at those images and convince yourself that I added one circle each time.)

When going from A to B, I added one circle. When going from B to C, I also added one circle. Now I ask you: was going from B to C any more “significant" than going from A to B? Did anything qualitatively different happen?

The answer is obviously no. Adding a circle is adding a circle; there’s nothing more to it than that. But if you had been writing these numbers out as base-10 representations, like you’re used to doing, you might have thought differently. You’d have gone from:

(A) 8

to

(B) 9

to

(C) 10

When going from B to C, your “odometer" wrapped around. You had to go from a one-digit number to a two-digit number, simply because you ran out of room in one digit. This can lead to the illusion that something fundamentally different happens when you go from B to C. This is completely an illusion. Nothing different happens to the number just because the way we write it down changes.

Human beings have a curious habit of thinking that odometer changes are significant. When the temperature breaks 100, it suddenly feels “more hotter" than it did when it merely rose from 98 to 99. When the Dow Jones Industrial Average first reached 10,000, and when Pete Rose eclipsed 4,000 career hits, and when the year 2000 dawned, we tended to think that something truly important had taken place. But as we’ll see, the point at which these milestones occur is utterly and even laughably aribitrary: it simply has to do with what number we’ve chosen as our base. And we quite honestly could have chosen any number at all.