I’m a mild fan of dozenal, also known as base twelve or duodecimal (which is a very base-ten way of naming base twelve, and which Wikipedia helpfully points out should not be confused with the Dewey Decimal system). I haven’t memorized my times tables in dozenal; like most people, I find decimal sufficient for daily life (and much easier to use with existing technology), which is precisely why imminent global change to dozenal is unlikely.
Still, I find it interesting to wonder what arithmetic would be like if we’d ended up with a more sensible base system, like dozenal, or if we one day did manage to switch to it. The particular subject of this article is: What if, instead of adding a couple of symbols to the familiar decimal ones, we designed new symbols specifically for dozenal? And what if, instead of choosing arbitrary symbols, we built them out of components that each conveyed a particular meaning?
(My proposal is basically equivalent to one made four years ago by Peter Sorrenson. The main difference is that his were designed with a seven-segment display in mind, whereas mine were designed with hand-writability in mind, and with some reference to the Hindu-Arabic numeral tradition.)
Anyway, the goal is to build twelve symbols from various components that each mean something. For example, each odd numeral could have this component:
Each numeral that is one more than a multiple of three could have this component:
while each numeral one less than a multiple of three has this component:
This would be enough for base six (apparently known as senary or heximal), but for dozenal we need to be able to distinguish between the numerals for one and seven, or for two and eight, for example. In order to do so, we have one more component, which appears in numerals that are one or two less than a multiple of four:
This is just enough to distinguish all twelve numerals from zero to eleven, but there’s one problem: the numeral for zero has no components at all. To solve this problem, I fall back to the familiar-to-me Western Arabic branch of the Hindu-Arabic numeral tradition, and use this symbol for zero:
One of the virtues of this system is that you can get a long way towards adding two numerals just by superimposing them. In some cases, this is all you need to do. For example, + = . (Actually, the right arm of the is curled around to coincide with the when both are present.) And + = .
But this isn’t quite the whole story. For example, if both the numerals you’re adding contain , then the result will contain instead; if both contain , the result will contain ; and if one contains and the other contains , then the result will contain neither. Also, two s add up to give a , and two s cancel each other out.
Those rules for addition might sound complicated, but they’re actually simpler than the (that is, three dozen and nine, or forty-five) arbitrary rules for adding non-zero decimal numerals that you’ve internalized. If you’d used these dozenal numerals since your early childhood, perhaps you’d be more likely to use visual intuition to assist in adding numbers. (This is baseless speculation; I have no expertise in cognitive science.)
The rules for multiplication are a bit more complicated, but still probably simpler than their decimal equivalents. An interesting consequence of them is that no perfect square can ever have either or in its last digit. That actually only leaves four options for the last digit, one of which is zero. And we can go even further: if a non-zero perfect square ends in an even numer of zeros, then its last non-zero digit must be , , or ; if it ends in an odd number of zeros, then its last non-zero digit must be .