Most modern societies use a base-10, or “decimal” system:
0 1 2 3 4 5 6 7 8 9
10 …
A few languages that use(d) a decimal (base-10) system: Incan languages like Quechua and Aymara, Chinese languages, Vietnamese, Japanese, Korean, Thai (Japanese, Korean, and Thai decimal systems were adopted from the Chinese system).
The Hindu-Arabic numeral system, first developed in India, is a base-10 system, corresponding of course with languages that use a base-10 counting system.
Variations of Hindu-Arabic numerals are used in places that speak the following languages (but are not limited to them, especially in the modern age): Western languages, Arabic, Farsi, Indian languages, Burmese, Khmer, and Thai.
If we have a base-10 system in the English language, then why do “eleven” and “twelve” seem to be independently named? By the logic of the decimal system, shouldn’t eleven be called “oneteen” (one-ten) and twelve “twoteen” (two-ten)? They are, after all, followed by thirteen (three-ten), fourteen (four-ten), and fifteen (five-ten), right?
Although many European societies, such as those in the British Isles, used a duodecimal, or base-12 numeric system for measurements (one foot = 12 inches), monetary units, and time (12 months, 12 zodiac signs), the words eleven and twelve are actually thought to originate from the Proto-Germanic “ainlif” (one is left) and “twalif” (two are left). Still, in modern English this comes across as an irregularity.
French, although decimal-based, has the irregular use of base-20 for numbers 80 through 99. 80 is called quatre-vingt (4 x 20). 85 is quatre-vingt cing (4 x 20 + 5), 90 is quatre-vingt dix (4 x 20 + 10), and 95 is quatre-vingt quinze (4 x 20 + 15).
The Sumerians strategized a base-60, or sexagesimal system that was later passed on to the Babylonians. This system was used (and in many cases, is still used) to measure time (60 seconds in a minute, 60 minutes in an hour), angles, and geographic coordinates.
This base-60 system was not, however, derived from the Sumerian language itself, and the cuneiform numeric system was actually a decimal system.
The Duodecimal System (Base-12):
0 1 2 3 4 5 6 7 8 9
A B 10 11 12 13 …
Here, A=10, B=11, 10=12, etc…alphabetic letters represent numbers from 10 to the duodecimal base of 12. Numbers eleven and twelve have independent names in languages that employ the duodecimal system.
Languages that use a duodecimal (base-12) system are rare. Chepang (spoken in Nepal), Mahl (spoken on Mincoy Island, India), and Nigerian languages such as Janji, Gbiri-Niragu, and the Nimbia dialect of Gwandara all use a base-12 number system. J.R.R. Tolkein’s Elvish languages also use a duodecimal system.
The Mayans (and other Mesoamerican cultures) used a base-20 system:
0 1 2 3 4 5 6 7 8 9
A B C D E F G H I J
10 11 12 13 ….
In this system “10” represents 20. Alphabet letters are here used to represent the numbers that are between 9 and the Mayan digit-base number 20; what we call the “teen” numbers would not have been known as “sixteen” (six-ten) or “seventeen” (seven-ten) but would have had their own independent name.
Other digit-systems used in different languages include the following:
Base-4: Chumashan languages (native to southern California). Number names were structured according to multiples of 4 and 16.
Base-5 (quinary): Gumatj, Nunggubuyu, and Kuum Kopan Noot are all quinary languages indigenous to Australia. Saraveca, an extinct language spoken in what is now Bolivia, was also quinary.
Base-15: used in the Huli language of Papua New Guinea. In Huli, fifteen is called ngui, thirty is ngui ki (15x2), and 225 is ngui ngui (15x15).
Base-24: Kakoli (a.k.a. Umbu Ungu, of Papua New Guinea). Tokapu means 24. 48 is tokapu talu (24 x 2). 576 is tokapu tokapu (24 x 24)
Base-27: Telefol and Oksapmin, also in Papua New Guinea.
Base-32: Ngiti (Democratic Republic of Congo) is base-32 with base-4 cycles.
Computers use a binary, or base-2 system to store data:
0 1 10 11 100 101 110 111 1000 1001
0 1 2 3 4 5 6 7 8 9
The top row represents the computer’s base-2 digit system, and the bottom row is there so you can compare it to the base-10 system.
Computer programmers, who are after all human, use a hexadecimal (base-16) system instead of the binary system:
0 1 2 3 4 5 6 7 8 9
A B C D E F 10 …
(10=16 here)
Web page colors are based on this system. Combinations of red, green, and blue (the primary colors of light) are represented as (R,G,B) = #RRGGBB. For example, #000000 = (0,0,0) = 0 Red, 0 Green, 0 Blue = WHITE; #FFFFFF =(256 Red, 256 Green, 256 Blue) = BLACK. Likewise, #FF0000 = red, #00FF00 = green, and # 0000FF = blue.
Those of us who are not computer programmers would have a difficult time with the hexadecimal system. Likewise, I imagine it must be difficult for a language learner trying to crossover from one digit-base system to another. I am sure this has happened in history where two or more cultures have collided. Would merchants have argued in an open marketplace over basic trade units? Would colonialist pedagogues have been puzzled by their native subjects’ seeming lack of comprehension in basic math? They may have been unaware that what appears to be a logical brick wall is in fact a linguistic barrier.
I hope this post has done more than make you realize that certain languages use different digit-base systems than our familiar base-10. Depending on what digit-base is used in your language’s counting system, individual numbers can take on a different meaning or level of significance. Think of how many times you refer to the number ten. “I’ll be there in ten minutes,” or saying to an obnoxious child “I will count to ten.” Or that demonical professor who wants a ten-page essay on quantum physics. Ten is just a handy, familiar number in our minds, and this is only because we speak a language whose number system is ultimately based on the number of fingers we have. If we were Huli speakers, 225 would be more significant than 100, and your professor would want a 15-page essay instead of ten pages. And if we spoke a base-8 language, we might celebrate our 64th birthday as though it were our 100th.
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A book I came across in my research for this post:
Numbers, Language, and the Human Mind
By Heike Wiese
I found this on books.google.com, so if you are interested you may also type in the name/author of this book at that search engine.
It's interesting how a culture's number system can have an impact on its ability to advance certain fields of study. The Romans were master architects and builders, and made advances in many areas with one glaring exception. Their contribution to mathematics was negligible. I had read somewhere it might be due to their Roman numerals, which quickly become cumbersome and are not conducive to relatively simple division much less advanced mathematics.
ReplyDeleteIt's interesting how the reverse is also true (numbers affect advancement and advancement influences number-system sophistication). The Piraha speakers in the Brazilian Amazon have no monetary units and have never engaged in trade of any sort. They also lack words for numbers in their language. There is (perhaps roughly) a number "one" and a number "two," and then simply "many," though it has also been speculated that the two existing "number" words actually convey "few" and "fewer" instead of actual numbers.
ReplyDeleteKeep in mind that written numerals (1, 2, 3...) and spoken numbers ("one, two, three...") are to be considered separtely. Whereas Latin was a base-ten language like countless others, I can certainly see why Roman numerals would have been more mathematically hindering than the written numerals used by speakers of other base-ten languages.