Counting to ten: it’s simple. Obvious. We have ten fingers; counting in units of ten is the only natural way to do things. Right?

Not so fast.

Technically you can count with any base number (aka radix). If you’re not used to thinking in these terms, visualize one of those old-school clocks with little tabs that flip down to display a digit — only instead of having the digits 0-9 in the set, somebody’s taken out a few. (Or added some, but for illustrative purposes, let’s say they’ve been removed.) Each position on the clock has only 0-6. The rightmost position ticks up to 1, then 2, 3, 4, 5, 6 . . . but when it needs to display 7, it flips back around to 0 instead. Meanwhile, the position to its left now ticks up to 1. That’s base 7, wherein a quantity of 7 is recorded as 10. (The “tens unit” is now the “sevens unit;” ergo, you have one 7 and no 1s.)

If this seems like an *exceptionally* abstruse thing to bring up, even in a worldbuilding Patreon on its sixth year of often quite abstruse topics . . . well, it is and it isn’t. You see, although probably everybody reading this essay is accustomed to a base 10 (a.k.a. *decimal*) system of counting, not every culture uses that system. And in fact, traces of other systems persist even in our decimal society.

Take, for example, the opening of the Gettysburg Address. “Four score and seven years ago” — wait, *score*? Why do we have a special word for “twenty”? In a base 20 (a.k.a. *vigesimal*) system, that’s entirely normal. Vigesimal counting is common in indigenous Mesoamerican cultures; you see it at work in the Mayan calendar, for example. It also crops up in many other places around the globe, including as a vestigial relic in European languages. It’s not as common as decimal counting, but it isn’t hard to see where the idea came from: after all, we have not only ten fingers but ten toes.

If you’re reading this essay on some kind of electronic device, you’re also interacting with a *binary* system. That was a fundamental leap in the development of computing; instead of trying to engineer clunky devices with gearing that can register ten possible positions, why not make simple on/off switches? These days we make them microscopically tiny, and everything from text to music to art gets represented with ones and zeros. Some aspects of computing also work with *hexadecimal* notation, base 16 — though, because our writing system only has digits up to 9, we represent 10-15 with the letters A-F.

This isn’t just about how we perform math. Although I don’t believe there are any societies whose numeric systems were fully hexadecimal, traditional Chinese measurements had sixteen of a smaller thing equal one of a larger. Ever bought a dozen of something? That’s *duodecimal* counting at work. Traces of this system persist in Germanic languages, in the form of irregular words for 11 and 12 (compare *eleven* with, say, the Japanese *juu-ichi*, which literally translates to “ten-one”). If you read about historical time periods, you might encounter a “long hundred” as a unit, meaning 120; even now you’ll hear about items being bought by the gross, 144 — i.e. 12 times 12.

Why would we count in twelves, when we have ten fingers? Maybe because it’s useful in other respects. 10 can only be cleanly divided by 2 and 5 (and yes, by 1 and 10; I see the math nerds raising their hands at the back), whereas 12 can be divided by 2, 3, 4, and 6. That’s handy when you want to make sub-units of your base. Also, remember back when we discussed lunar calendars? There are twelve full cycles of the moon in a year (though the total number of days falls a little short of the solar year). We still have twelve months now, though not pegged to the moon, and two sets of twelve hours in the day. It’s fairly common for time measurement to be duodecimal, in many parts of the world.

And if you like the divisibility of 12, wait until you check out 60! *Sexagesimal* counting is the reason we have sixty seconds in a minute and sixty minutes in an hour — or rather, Babylonian astronomers who used sexagesimal counting are the reason. Interestingly, though, there are two ways to get at that number. For people in Mesopotamia, it was built out of six sets of ten. In Asia, however, sixty shows up as the product of five sets of twelve. You can subdivide that number all kinds of ways, which can be mathematically very useful.

Those are the most common systems, but practically everything up to twelve has cropped up in human society *somewhere*. Some of it is still hand-based, just not in the way we’re used to imagining: you can count duodecimally by using your thumb to point at each of the three bones in the fingers of the same hand, and sexagesimally if your other hand tracks how many dozens you’ve counted. *Quinary* (base 5) counting is also easy to map to a hand. If you’re the Yuki people of California, you do octal (base 8) counting in the spaces between your fingers, instead of on the fingers themselves; if you speak a Pamean language in Mexico, you do it on the knuckles of a closed fist.

There are some natural languages with *quaternary* (base 4) and *senary* (base 6) counting systems, but they’re a good deal rarer. *Ternary* (base 3) appears to be used only in computing, not in any cultural language; the same goes for *nonary* (base 9), which is kind of just a special case of ternary. If *undecimal* (base 11) counting has ever existed outside of specialized mathematics, it happened in the Polynesian cultural sphere, but this may not have been a true base 11 system. *Septenary* (base 7) I can only find employed as specialized groupings — seven days in a Western week; seven notes in Western musicology — not as an actual mathematical system.

What’s the relevance of this to fiction? Well, if you’re Wendy and Richard Pini, creators of the *Elfquest* comic book series, the fact that your elves have only four fingers on each hand means that naturally their counting is done in fours and eights. (I suppose their octal, duodecimal, and vigesimal hand-counting equivalents would be senary, nonary, and hexadecimal.) It’s something to bear in mind for creatures with different body plans.

But it can also influence the setting more subtly. Sarah Monette’s Doctrine of Labyrinths series incorporates two different counting systems, one decimal, one septenary, which reflect the different layers of the culture: some things are measured in tens, others in sevens, and which gets used for what hints at some of the history of the place. Do your characters buy things by the dozen or the gross and have twelve pence to the shilling, like pre-decimal British money? Does sixty crop up as a significant number, and if so, is that six tens or five twelves? Back in Year Two we talked about the symbolic weight we ascribe to different numbers; take that and weave it into the quantities your characters deal with in their daily lives, and you’ll add a low-grade, background sense of meaning to how they interact with their world.

## 10 thoughts on “New Worlds: Count to Ten”

Phyllis Irene RadfordI once wrote an SF story where the new government, in an effort to make everything more uniform and efficient, converted military time of 24 hours a day to 20 hours a day. My characters rebelled and nearly died of exhaustion when their 8 hour work day became quite bit longer–you do the math I’m too boggled.

The story died before I finished it. Maybe now I can go back and fix it. THANK YOU!

Marie BrennanI was amused that I had ranted in this very Patreon about how monetary systems were frequently not decimal . . . and then Alyc and I, working on the Rook and Rose books, promptly gave the Liganti not only decimal money but a twenty-hour day (divided into two ten-hour blocks) and ten months in the year. ðŸ˜› But when your religion is based on the sacred implications of the numbers 1-10, it does seem the way to go!

Nobody in that society gets a nice, standard 8-hour work day, though.

drsTolkien gave his elves a preference for sixes and twelves, for no clear reason I recall. I’m not sure if he outright said they used base 12 but they counted by six a lot.

> It seems clear that the Eldar in Middle-earth, who had, as Samwise remarked, more time at their disposal, reckoned in long periods, and the Quenya word yÃ©n, often translated â€˜yearâ€™ (p. 492), really means 144 of our years. The Eldar preferred to reckon in sixes and twelves as far as possible. A â€˜dayâ€™ of the sun they called rÃ© and reckoned from sunset to sunset. The yÃ©n contained 52,596 days. For ritual rather than practical purposes the Eldar observed a week or enquiÃ« of six days; and the yÃ©n contained 8,766 of these enquier, reckoned continuously throughout the period.

— Appendix D

With ‘months’/’seasons’ of 72 or 54 days.

OTOH I think you can get through LotR, the Silmarillion, Unfinished Tales, and much related material, without a single clue of this, so so much for affecting the culture in a story-relevant way. Lots of elven things come in three though, deliberately or not (three sons of Finwe, three peoples of the elves)

Marie BrennanHah, yeah — I’ve read a fair bit of that stuff, and never noticed or possibly even encountered that! I wouldn’t be surprised if he was influenced by the archaic usages of the long hundred and the gross, and halved those for senary counting, without entirely thinking through

whythey would do it that way. (In natural languages where 6 crops up like that, it seems to largely be because 1-5 is numbers you can count on your hand and then 6+ is “a lot.”)PenceSmall correction; 12 pence to a shilling and twenty shillings to a pound (which I understand historically was the price of a sheep).

Also 21 shillings to a guinea.

Marie Brennan<facepalm> I knew that! But my fingers were not listening, apparently.

Yoon Ha LeeYou can also have bases like -1 and sqrt(2) (square root of 2) but the applications of those are more specialized and less likely to arise outside of a math/CS setting. XD

Marie BrennanYyyyeah . . . there is a(n ill-defined) line past which I decide that a subject has gotten

toospecialized for the purpose of this Patreon, and that is definitely on the far side of it. ðŸ™‚(Other things I’ve decided are too far out there are the entailment of British estates, the specific types of sail plan on sailing vessels, and the Christian doctrine of baptism by choice, i.e. if you

wantto be baptized and repent of your sins and so forth, but get killed before somebody can actually baptize you, then God considers it as good as done.)JawsBut let’s hear it for zero!

Far too many numbering systems neglect a formal assignment of the value for “there is no difference between these two quantities,” tending instead to write it out (if it’s a written numbering system) in words. Which makes the magic of Al-Jabbar very difficult indeed, as there isn’t a “divide by the absence of any difference” error to cancel out…

Mary CatelliHexadecimal is widely used in computing because of the immense ease in flipping from it to binary and back.

Take your binary. Divide into chunks of four, starting with the ones digit. Convert each one into its hex equivalent. Hence 100100110010 is 932 in hex. And vice versa