Bases For Comparison

Back to the theme of divisibility of numbers. Since we have the idea of writing numbers with a small set of digits, and with the place of those digits carrying information about how big the number is, we can think about what’s implied by that information.

In the number 222, the first two is matched to blocks (hundreds) that are ten times as large as those for the second two (tens), and the second two is matched to units (tens) which are ten times as large as those for the third two (units). It is now extremely rare to have the size of those blocks differ from one place to the next; that is, a number before the initial two here we take without needing it made explicit to represent ten times that hundreds unit, and a number after the final two (and therefore after the decimal point) would represent units which are one-tenth that of the final two’s size.

It has also become extremely rare for the relationship between blocks to be anything but a factor of ten, with two exceptions which I’ll mention next paragraph. The only block other than those with common use which comes to my mind is the sixty-to-one division of hours or degrees into minutes, and then of minutes into seconds. Even there the division of degrees of arc into minutes and seconds might be obsolete, as it’s so much easier on the computer to enter a latitude and longitude with decimals instead. So blocks of ten, decimals, it is, or in the way actual people speak of such things, a number written in base ten.

There are the two exceptions. They come to us through working on computers, where having the blocks be units of two is convenient. At least, it’s convenient for the computer; for us, this base two ends up producing numbers that are tiresomely long, so we use blocks of sixteen instead. In base sixteen, none of the numbers we need to deal with are longer than they are in ordinary writing, and actually most of them turn out to be shorter. Base sixteen digits are quite easy to convert to base two, with each of these digits corresponding to a unique set of four base-two digits. We might press a claim that base two digits are just a longwinded way of writing base sixteen, but would shy away in the face of strong opposition.

Of course, for base sixteen, we need to import some new symbols, since we don’t want to try squeezing “10” or “11” or so into a space meant for a single digit, and after some fussing around with typographical tricks in math popularizations of the 1950s and 60s we settled on using the letters A through F. I remember one of Isaac Asimov’s science essays from the 1960s introducing all this different-base talk; since he wanted to use base twelve, he used “T” for ten and “E” for eleven, and I never found that part of the essay quite comfortable to read. “1T” is just a strange way to write twenty-six.

The choice of what base to use is arbitrary. Any are as logical and consistent as any other, at least as long as we stick to positive numbers and avoid base one as looking awfully suspicious. There aren’t any numbers that can be expressed in base two which can’t be expressed in base ten, or base nineteen if we wished. We can pick a base to match whatever is convenient, for calculation or for communication. Base two is convenient for calculation, particularly if we happen to be computers. Base sixteen and base ten are convenient for communication, and ten is at least familiar for calculation. The other bases are, mostly, sources of problems to give the advanced arithmetic student who’s getting a little bored to work on while the rest of the class catches up. I appreciated the bonus problems in elementary school, at least.

Some of them make appearances, though. Base eight appears sometimes as a way of getting the advantages of base sixteen — mostly, of representing base two numbers without being so very long to write — without being forced to use letters as numerals. I concede I may be biased against base eight numbers. I learned base sixteen as anything besides a novelty act first, and was unafraid of “B” as a numeral. Worse, my most recent encounter with base eight has been as a programmer, learning after much tedious error-tracing that the Javascript function “parseInt”, which says what integer a given string of letters is supposed to be, will in some cases default to the supposition the string given it is in base eight and crash to a halt rather than hesitantly venture to presume to guess that “09” might possibly in this context be understood as a proposed representation for the number nine.

Base twelve might have had its advantages. There is a connection between the base one writes numbers in and whether the decimal representations of a fraction ever ends, or whether it goes on for an infinite series of digits, and just what that connection is will be obvious shortly, for those to whom it’s not already obvious. But base twelve has more fractions which come to a stop, rather than repeating forever, than has base ten. For example, 1/3 we know is in decimal a never-ending 0.333333… in base ten, while in base twelve it’s a compact 0.3 and ends. If we like to work with exact things, and wish to avoid fractions, this is a nice property. But nobody much wants to change, and it would probably take something astonishing to make a change.

And this summons an trivia point with bases I find wonderful enough to make it worth closing on. In the astonishing events of the French Revolution and the preparation of the Metric System, committee member (and physicist, and mathematician, and one of the thesis students of Leonhard Euler) Joseph-Louis Lagrange apparently suggested that besides throwing out the old-fashioned systems of weights and measures they might also switch the nation’s numbers to base eleven — in which no fractions (other than those which are multiples of eleven, for example, one-twenty-second) would come to finite ends. They would enjoy an equal inconvenience. It has been proposed that Lagrange was being facetious. It feels like a deadpan joke to me, as well, but against the rules for facetious comments, the idea was not taken up and made horrid reality. It has also been proposed he was trying to squash a proposal to move the country to a base twelve system for all things. Perhaps he had multiple motives.