Let’s make a language – Part 13a: Numerals (Intro)

After learning how to speak, counting is one of the first things children tend to figure out, for obvious reasons. And language is set up to facilitate learning how to count, simply because it’s such an important part of our existence as human beings. The familiar “one, two, three” of English has its counterparts around the world, though each language has its own way of using them.

These numerals will be our focus today. (Note that we can’t really call them numbers in a linguistic context, because we’re already using the term “number” for the singular/plural distinction.) Specifically, we’ll look at how different languages count with their numerals; in math terms, these will be the cardinal numbers. In a later post, we can add in the ordinal numbers (like “first” and “third”), fractions, quantities, measurements, and all that other good stuff. For now, let’s talk about counting.

Oh, and since numerals lie at a kind of intersection of linguistics and mathematics, it’ll help if you’re familiar with a few concepts from math. While we won’t be going into things like positional number systems—I’ll save that for a post about writing systems, far into the future—the concept of powers will be important. More information shouldn’t be that hard to find on the Internet, so I’ll leave that in your capable hands.

Count the ways

How a language counts is highly dependent on its culture. Remember that counting and numeral words predate by far the invention of writing. Now think about how you can count if you can’t write. One of the best ways is by using parts of your body. After all, it’s always with you, unlike a collection of stones or some other preliterate method. Thus, bodily terms often pop up in the context of numerals.

In fact, that’s one of the simplest methods of creating numerals: just start numbering parts of your body. A few languages from Pacific islands still use this today, and it’s entirely possible that it’s how all ancestor languages did it. Words for the fingers of one hand usually cover 1-4, with the thumb standing for 5. After that, it depends on the language. Six could be represented by the word for the palm or wrist, and larger numbers by points further up the arm. In this way, you can continue down the opposite arm, to its hand, and then on to the rest of the body.

Once you need to work with larger numbers, however, you’ll want a better way of creating them. The “pointing” method is inefficient—you need to remember each point on the body in order—and there are only so many body parts. This is fine for a hunter-gatherer society, and many of those have a very small selection of numerals (anywhere from one to five), using a word for “many” for anything higher. But we “advanced” peoples do need to refer to greater quantities. The solution, then, is to use a smaller set of numerals and construct larger ones from that. That’s how we do it in English: “twenty-five” is nothing more than “twenty” plus “five”.

For our language, the key number is 10. Every number up to this one has its own numeral, while larger ones are mostly derived. The only exceptions are words like “hundred” and “thousand” which, incidentally enough, represent higher powers of 10. Thus, we can say that English uses base-10 counting—or decimal, if you prefer fancier words.

At the base

Every language with a system of numeral words is going to have a numerical base for that system. Which number is used as the base really has a lot to do with the history of the language and how its people traditionally counted. Not every number is appropriate as a base; Douglas Adams once said that nobody makes jokes in base-13, and I can state with confidence that nobody counts in it, either. Why? Because 13 is awkward. It’s a prime number with essentially no connection to any part of the body. Since counting probably originated with body parts, there’s no reason for a culture to ever develop base-13 counting. Other numbers, though, are quite suitable.

• Decimal (base-10) counting is, far and away, the most common in the world. Look at your hands, and you’ll see why. (Unless, of course, you don’t have ten fingers.) Counting in decimal is just the finger counting most of us grew up with, and decimal systems tend to have new words for higher powers of 10. In English, we’ve got “hundred” and “thousand”, and these are pretty common in other decimal languages. For “ten thousand”, we don’t have a specific native word, but Japanese (man) and Ancient Greek (myrioi) do; the latter is where we get the word “myriad”.

• Vigesimal (base-20) is not quite as widespread as decimal, but it has plenty of supporters. A few European languages use something like base-20 up to a certain point—one hundred, in fact—where they switch to full decimal. But a “true” vigesimal system, using powers of 20 instead of 10 (and thus having separate words for 400, 8,000, etc.), can be found in Nahautl (Aztec) and Maya, as well as Dzongkha, in Bhutan. Like decimal, vigesimal most likely derives from counting, but here it would be the fingers and the toes.

• Quinary (base-5) turns up here and there, particularly in the Pacific and Australia. Again, it comes from counting, but this time with only one hand. It’s far more common for 5 to be a “sub-base” in a greater decimal system; in other words, 10 can be “two fives”, but 20 is more likely to be “two tens”. The alternative, where the core terms are for 5, 25, 125, and so on, doesn’t seem to occur, but there’s no reason why it can’t.

• Duodecimal (base-12) doesn’t appear to have an obvious body correlation, but it actually does. Using the thumb of one hand, count the finger bones on that hand. Each finger has three of them, and you’ve got four non-thumb fingers: 3 × 4 = 12. There are a few languages out there that use duodecimal numerals (including Tolkien’s Quenya), but base-12 is more common in arithmetic contexts, where its multiple factors sometimes make it easier to use than decimal. Even in English, though, we have the “dozen” (12) and “gross” (144).

• Other numbers are almost never used as the “primary” base in a language, but a few can be found as “auxiliary” bases. Base-60 (sexagesimal), like our minutes and seconds, is entirely possible, but it will likely be accompanied by decimal or duodecimal sub-bases. Some languages of Papua New Guinea and thereabouts use a quaternary (base-4) system or, far more rarely, a senary or base-6 system. Octal (base-8) can work with finger counting if you use the spaces between your fingers, and a couple of cultures do this. And, of course, it’s easy to imagine an AI using octal, hexadecimal (base-16), or plain binary (base-2).

Word problems

In general, numerals up to the primary base are all going to be different, as in English “one” through “ten”. A few powers of the base will also have their own words, but this will be dependent on how often the speakers of a language need those higher numbers. “Hundred” and “thousand” suffice for many older cultures, but the Mayans could count up to the alau, 20⁶ or 64 million, China has native terms up to 10¹⁴ (a hundred trillion), and the Vedas have lots of terms for absurdly large numerals.

No matter what the “end” of the scale, most of the numbers in between will be somehow derived. Again, the more often numbers are used, the more likely they’ll acquire specific terms, but special forms are common for multiples of the base up to its square (100 in decimal, 400 in vigesimal, and so on), like our “twenty” or “eighty”. Intermediate numbers will tend to be made from these building blocks: multiples and powers of the base. How they’re combined is up to the language, but the English phrasing, for once, is a pretty good guide.

Some languages work with a secondary base, and these may affect the way numeral words work. Twelve and twenty can almost be considered sub-bases for English with words like “dozen” and the peculiar method of constructing numbers in the teens. Twenty is a stronger force in other European languages, though. French is an example here, with 80 being quatre-vingts, literally “four twenties”. In contrast, a full vigesimal system can function just fine with the numeral for twelve derived as “ten and two”, using 10 as a sub-base, although I’m not aware of an example. Any factor can also work as a sub-base, especially in base-20, where 4 and 5 both work, or base-60, where you can use 6 and 10.

Irregularity is everywhere in natural languages, and that includes numerals. There always seem to be a few outliers that don’t fit the pattern. English has “eleven” and “twelve”, of course; it gets them from Germanic, as do many of its cousins. Spanish, among others, has veinte for 20, whereas other multiples of ten are constructed fairly regularly from their “ones” (treinte, etc.). Other examples abound.

Fitting in

How numeral words fit into a language is also a major variable. Sometimes, they’re a separate part of speech. Or they can be adjectives. Or nouns. Or some combination of all three. If they’re adjectives or nouns, then they may or may not participate in the usual grammar. Latin, for instance, requires small numerals (up to four) to be inflected, but everything larger is largely fixed in form. English lets numerals act as adjectives or nouns, as needed, and some dialects allow nouns following adjectival numerals to ignore grammatical number (“two foot of rope”, “eight head of cattle”). It’s really a mess most everywhere.

For a conlang, it’s going to come down to the necessities. Auxlangs, as always, need to be simple, logical, and reasonable, so it’s best not to get too crazy, and this extends to all aspects of numerals. You’re not going to get many followers if you make them start counting by dozens! (Confession time. I did this for a non-auxlang over ten years ago, and I still forget it uses duodecimal sometimes! Imagine how that would be for a language intended to be spoken.)

Fictional languages get a little bit of a pass. Here, it’s okay to go wild, as long as you know what you’re doing. Non-decimal bases are everywhere in conlangs, even in “professional” ones like Tolkien’s. With non-humans, you get that much more rope to hang yourself with. Four-fingered aliens (or cartoon characters) would be more likely to reckon in an octal system than a decimal one. Depending on how their digits are made, you could also make a case for base-6 or base-9, by analogy with Earthly octal and duodecimal finger counting. Advanced races will be more likely to have a sophisticated system of higher powers, like our billion, trillion, etc. And so on.

More than any other part of this series, numerals are a part of a culture. If you’re making a conlang without a culture—as in an auxlang—then think of who the speakers will be, and copy them. Otherwise, you might need to consider some of the aspects of your fictional speakers. How would they count? How would they think of numbers? Then you can start making your own.