When we compare numbers across various languages, there are perhaps two dominant counting systems we can identify. Intuitively, these rely on counting the number of digits on two hands, or perhaps including the toes as well. Many languages across the world use the former, hence having a decimal (base-10) counting system like Japanese, while some have a vigesimal (base-20) counting system like Welsh and Danish.

But there are some counting systems that deviate from this base-10 or base-20 majority. A more intuitive one is the base-5 or quinary numeral system, as it could be said that numbers are counted using only the digits on one hand. Base-8 or octal numeral systems use a slightly different approach, by counting using the spaces between digits on two hands, or on the exposed knuckles on a single closed fist.

The first example I want to look at is the senary or base-6 counting system. Here, an 11 in the senary counting system is 7 in decimal. Most of the languages that use the senary counting system are found in New Guinea, some of them belonging to a branch of languages known as the Yam languages.

But why base-6? Using digit-based tallying to reach base-6 is not really intuitive, and the closest I could come to trying to count in base-6 with my hands is to count a closed fist as “one”, counting up the digits until we get an open hand of “six”. Alternatively, on a less intuitive note, one could count the spaces between the exposed knuckles on one fist.

A more mathematical justification is, six is divisible by 2 and 3, making it easier to divide small quantities amongst very small integer denominators. After all, base-60 counting systems have been documented, as it is divisible by 2, 3, 4, 5, and 6, making it easy to divide quantities with these denominators.

After some digging about, it seems that part of the answer is in the name itself. Yam. It is a starchy tuber some people depend on as a staple. And in the Yam languages, that is essentially the case. Yam tallying is a system used in some communities in New Guinea, where 1296 yams are needed to feed a family for a year, that is, 6 raised to the power of 4.

But what about this yam tallying system? The yams eaten in these communities of New Guinea are usually teardrop-shaped, and when organised in small piles, tend to do so neatly. The issue about this is, did yam tallying necessitate a senary number system instead of a decimal number system one could naturally do with their hands, and did decimal, quinary, or vigesimal counting systems exist before these communities shifted to prefer using yam tallying? Alternatively, did a base-6 counting system already exist (using a closed fist for “one”), and the yam tallying system happen to coincide with the counting system?

Examples of Yam languages containing the senary counting system include the Yei, Ngkolmpu, and the Arammba languages. However, there are also non-Yam languages on New Guinea that use the senary counting system. One particular language of note is the Ndom language, a Kolopom language. But even within these languages that use the senary, they employ different methods of expressing larger numbers. Some have words that mean exponents of 6, while others might not have any numbers beyond 36. The former includes Ngkolmpu, which has new terms for 36 (ptae), 216 (tarumpao), 1296 (ntamnao), and 7776 (ulamaeke), while the latter includes Ndom, which has the largest number term, 36 (nif).

The other counting systems I want to explore have a distinct pattern. Instead of counting by digits on a hand, or otherwise, these numeral systems use body parts. Collectively, these counting systems are body-part counting systems. The bases used in these counting systems often extend beyond the base-10 and base-20 numerals we are more or less familiar with, and often stretching into unusual cycle numbers like 27.

In body-part tally systems, words for numerals may be synonymous with the reference body part, like “ring finger” or “wrist”. Some languages may omit certain body parts like the forearm, while others might include them. This brings about a wide variance in the cycles of body tallies, with some as low as 18, and one as high as 68. The language with the 68-cycle tally system was not identified in the source I found it in, though (Owens and Lean, 2018).

A count might typically start with the little finger, moving up the digits until the hand is complete, before moving towards the arm, shoulder, armpit, and then tallying with facial features like the eye and nose. Sometimes, upper torso features may be used in the tally, like the sternum. In some languages, this tally may proceed to the other side of the body to complete the cycle. Such words may translate as “other eye”, “other wrist”, or “other index finger”.

One might ask, what happens when all body parts are tallied, but there is still more to count? I am unable to locate resources stating what happens when a 19-cycle system has to count the number 20, but it is possible that some tally systems just reverse the tally, proceeding from the end of the first cycle, and moving towards the end point of the second cycle, which is the start point of the first cycle. It is unclear to me if there are additional words to specify the number of cycles that has been tallied, or if one word for a tally point corresponds to multiple numbers. Some languages also have separate numbers for “one” or “two”, but I cannot find any information if these words are used together with the tally to express numbers like 34 in an 18-cycle system. Examples of such languages include Kobon and Kalam, which use a total of 23 tally points, and Oksapmin with 27 tally points.

Some linguists argue that these tally systems do not inherently form a numerical base, suggesting that number systems like the ones we commonly use today use numerals as its own word class. The use of words for body parts as tally points in body-part tally systems, while occasionally having words for “one”, “two”, “few” or “many” suggest that these tally systems do not form an independent word class for numerals. But this is when some uncertainty starts to set in. If one says “I have little finger of cats”, or something that translates to that sentence, most of the languages that do not use body-part tallies would interpret this as rather illogical. However, users of body-part tallies would understand the precise number, ignoring the cycling, of cats that the speaker has. In this context, the use of body parts to denote numbers exclusively serve the expression of this numeral word class. Thus, can the n in n-cycle in these tally systems technically count as number bases, or do these languages not have numbers for the most part, at all?

But for languages that lack body-part tally systems, there are languages that lack words, or have to borrow words to express larger numbers. Most of the time, they would have words for “one” and “two”, but fewer would have words for “five”, “ten” or “twenty”. As such, to express larger numbers, speakers would have to stack these numbers together, which will become extremely impractical to express numbers like 125. To make up for these, speakers might just choose to borrow words from other languages, or use imprecise quantifiers to describe such numbers like “many” or “plenty”. These languages might also be referred to as “2-cycle languages“, with “pure 2-cycle languages” only having words for “one” and “two”.

Some anthropologists or linguists might point towards “one” as being synonymous with “few”, and “two” synonymous with “many”. And to take this to a further extreme, the Pirãha language is purported to have no precise numbers at all; the language has words to describe “few” or “many”, but nothing more. However, despite the lack of precise numbers above the single digits, speakers of these languages can still use non-verbal counting or tallying, but not report it verbally.

These languages are usually found in three particular regions: Australia, New Guinea, and the Amazon. Additionally, these societies tend to be more hunter-gatherer in nature. The main methods of forming numbers like “five” from only “one” and “two” is to compound them as “2+2+1”, translating as “two two one”, or “two and two and one”. For example, in the Madang province of Papua New Guinea, the Gende language has “mapro” for “one”, and “oroi” for “two”. “Three” is expressed as “oro gu mago“, “four” is “oroi oroi“, and “five” is “oroi oroi mago“. The Middle Watut language in the Morobe province of Papua New Guinea has “morots” for “one”, and “serok” for “two”, and makes “five” as “serok a serok a morots“, literally “two and two and one”.

	Gende	Giri	Middle Watut
1	mapro	ibabira	morots
2	oroi	ppunini	serok
3	oro gu mago	ppuni kagine	serok a morots
4	oroi oroi	ppunini ppuninin	morots a morots
5	oroi oroi mago	ppunini ppunini kagine	serok a serok a morots

One linguistic mystery is, we are not entirely sure why such extremely restricted number systems exist, despite the applications of numbers even in hunter-gatherer contexts. Trade between hunter-gatherer societies have been documented before, which one would expect to involve expression of quantities, but it has been argued that precise numbers are not entirely necessary to conduct trade. Numbers may also be used to count months of pregnancy, or number of days to a certain solar or lunar event. Yet, we still encounter languages with such restricted number systems, leaving us with a perplexing mystery.

These unconventional counting systems is something that has intrigued me, as well as some ardent language enthusiasts all around the world. In fact, the senary number system of Ndom was featured as a problem for participants in the individual round of the 2007 International Linguistics Olympiad. Such questions are a regular occurrence in the International Linguistics Olympiad, with one problem appearing almost every year. If you are interested, the Olympiad posts their problems and solutions in several languages on their website for everyone to try out. Which number system is the most interesting to you? Let us know in the comments!

Further reading

Barlow, R. (2023). Papuan-Austronesian contact and the spread of numeral systems in Melanesia. Diachronica, 40(3), pp. 287-340. https://doi.org/10.1075/dia.22005.bar.

Owens, K., Lean, G., Paraide, P. and Muke, C. (2018). History of Number. History of Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-45483-2.

Wolfers, E. P. (1971). The original counting systems of Papua and New Guinea. The Arithmetic Teacher, 18(2), pp. 77-83. http://www.jstor.org/stable/41187615.