To return to my second major theme: my Dearly Beloved told me that I must explain that trick where one adds up the digits of a number and finds out from that whether it’s divisible by 9. I wanted to anyway, but a request like that is irresistible. The answer can be given quickly — and several of my hopefully faithful readers did, in comments, last Friday — but I’d like to take the long way around because I do that and because it lets a lot of other interesting divisibility properties show themselves.
We use ten numerals and the place where we write them to express all the counting numbers out there. We put one of the numerals, such as `2′, in a place which denotes whether we mean to say two tens, or two hundreds, or two millions. That’s a clever tool, and not one inherent to the idea of numbers. We could as easily use different symbols for different magnitudes; the only familiar example of this (in the west) is Roman numerals, where we use I, X, C, and M for increasing powers of ten, and then notice we aren’t really quite sure what to do past M.
The Romans were not very sure either, and individual variations developed when someone found they needed to express an M of M very often. The system has fewer numerals, symbols representing numbers, than ours does, with V and L and D the only additional numerals reasonably common. By the Middle Ages some symbols were improvised to allow for extremely large numbers such as the hundred thousands, and some extra symbols were pulled in for numbers such as 7 or 40, but they have faded to the point of obscurity. This is a numbering system which runs out when the numbers get too large, which seems impossibly limited at first glance. But we haven’t changed much from these times: while we have a numbering system that can, in principle, work with arbitrarily big or tiny numbers, in practice we only use a small range of them. When we turn over arithmetic to computers, in fact, we accept numbering systems which have limits on how big (positive or negative) a number may be, or how close to zero one may work. We accept those limits because of their convenience and are only sometimes annoyed to find, for example, that the spreadsheet trying to calculate a bill has decided we want 0.9999999 of a penny.
We can look at an ordinary, Arabic-style, number — I like 2,038 for its mild in-joke nature — and its meaning is obvious. It’s the obvious things that most repay careful attention. 2,038 is the sum of two units of a thousand, and zero units of a hundred, and three units of ten, and eight units of … a unit. Each unit is a pleasantly uniform ten times the next-smaller unit and there aren’t any exceptions to this rule. The meanings of all these units are what will let us build rules where we add up digits to a number and learn something about what divides into it.
That bit about each unit being ten times what the next-smaller one is makes the system easy to work with. That isn’t an inherent rule of number systems where the place has meaning, and I’d like to describe some cases where that relationship between units is different, just to make this choice stand out better.
One example is the calendar: we have one number which represents the count of days within the current month (at the moment of publication, 14, in my time zone), another which represents the months within the current year (10, for me, which we relabel as October if we wish to avoid confusion, or which we leave as 10 if we want to spend the first third of each month with room for ambiguity about whether we wrote the date in the American or the European style), and the years since the start of the current calendar era (2011). The important thing is that increasing the months column by one represents a block of 31 or 30, or in peculiar cases 28, or in more peculiar cases 29, days. An increase in the years column represents a block of 365 or 366 days. For a further curiosity, in this scheme, it makes no sense to put a 14 in the months column, and it would make no sense to put the 2011 in the days or the months columns.
Another example became anachronistic with the surrender of all major currencies to decimalization. While the change away from pound-shilling-penny (or their equivalent in other currencies) had great advantages in accounting, the old pattern of a pound being twenty shillings and a shilling being twelve pennies made it possible to form perfectly complicated arithmetic problems in simply adding together two sums of money. (I don’t want to suggest I think people of the past did things this way just to make things difficult for themselves: the pound-shilling-penny system lets one express many fractions of a pound using whole numbers; for example, one-third of a pound could be written as 8 shillings, 6 pennies, while we have to either throw in a 1/3 today or round off that fraction of a penny. Of course, the English also just minted angels, worth a third of a pound, and marks, worth two-thirds, and left the pound to be a unit of account with no coins in the amount minted or bills printed, which also avoids the problems of a third of a pound.) Here again a unit of shillings is a different number of pennies than a unit of pounds is a number of shillings; and, like the calendar, some numbers are meaningful only in the pounds column, some numbers meaningful only for pounds and pennies, and only a few meaningful in all possible places.
The Mayan Long Count calendar nearly avoids the frustrations in each place being a different multiple of the next-lower base. It represents (if the present tense is correct; does a calendar system still exist if the rules of it are reasonably well-known but it isn’t in common use?) dates with a set of five numerals. One numeral, the k’in, counts the number of days; the next, the winal, counts blocks of twenty k’in. The next, the tun, counts blocks of eighteen winal; and then the k’atun is twenty winal, and the b’ak’tun is twenty k’atun. The exception to the twenty-of-the-smaller-unit measure seems likely to be a concession to astronomy: eighteen blocks of twenty days each is a fair fit to the length of the year. The system runs out after a large enough number of days are counted, but as mentioned, we accept that for the computers on which we calculate most everything there are some numbers too large to deal with, and some numbers too tiny to deal with, and we carry on mostly by working with numbers that don’t go into those difficult regions. We could use the Long Count scheme as a way of tallying the counting numbers, and could make a reasonable extension into rational numbers, although that group-of-18 exception makes it more difficult than the normal number system.
So making the place of a numeral carry meaning is a generally good idea, and we have a system in which the relationship between one place and the next follows a simple pattern. This sets the background.