Number system
From KneeQuickie
A number system is the system used by a language for constructing number-names. All languages form number-names systematically (if they have a developed number system at all); all known natlangs use a basal system, although some non-basal systems have been developed for conlangs. The most common base, in both natlangs and conlangs, is 10 (decimal).
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Base-ten number systems
In a base-ten system, higher numbers are formed by counting the number of tens they contain. For instance, the English word "forty-five" is clearly derived from "four tens and five". Numbers above 100 are named by dividing them into hundreds, tens and units; the special role of 100 derives from the fact that one hundred is ten tens.
There are many possible ways to derive number-names. For example, suppose that the words for "four", "five", and "ten" in some language are for, fiv, and ten. The following are only a few of the possible ways to form the number "forty-five":
- for-ten-fiv (i.e. four tens and five)
- for-ten-na-fiv (again, four tens and five; "na" might mean "and")
- forti-fiv (again, four tens and five, but now multiples of ten are contracted)
- for-fiv (a verbal place-value system: the "for" is interpreted as 40 and not 4 because of its position)
- fiv-forti (placing the units before the tens)
- fiv-na-forti (the same, but using the word for "and")
- fiv-la-fiv-ten (i.e. five less than five tens)
- fiv-de-fifti (ie. five of the fifth ten)
Note that many languages have exceptions to perfect systematicity. For instance, in English, the words for 13–19 derive from 3 + 10, 4 + 10 etc., but eleven and twelve have their own unique names. Breton has a largely base-20 number system, but derives the word for eighteen from 3 × 6.
The language is likely to have names for some higher powers of ten, but not all. For instance, in English we have separate names for 100, 1000, and 1000000, but 10000 and 100000 have to be expressed by multiplying lower powers: "ten thousand" and "one hundred thousand". Other languages have separate words for these but not for 1000 and 1000000.
Other bases
In principle, any number above 1 could be used as the base of a number system. The powers of this number would play the same role that powers of ten play in a base-10 system. For instance, in base-6, forty-five would be expressed as 36 + 6 + 3.
Twenty is the most common number base for natlangs, after base ten (decimal). Base twenty has also been used by many conlangs, including Archeía and Vardeu.
The reason proposed to explain the prevalence of base 10 is simply that we have ten fingers (and as we might expect, after decimal, the next most common bases, 5 and 20, correspond to the number of fingers on one hand and to the total of fingers and toes). Ironically, using 10 as a base makes it is nearly impossible to make use of the concept of base in counting, because all digits are used up simply in counting to 10. Languages with smaller bases can take advantage of the fact that humans have 10 fingers; for instance, in base 6, each hand can represent one digit (0-5) and allow users to count up to 35; in binary each finger can be used as a digit, allowing a person to count on their fingers up to 1023. Some languages are known that use base 2, 4, 6, 8, or 12 (of these, base 8 may be based on counting using the spaces between the fingers); or even 60 or 100.
When choosing a base for a conlang, bear in mind that the more small-number factors a base has, the easier it will be to work with; this makes 6 and 12 more attractive from a purely mathematical standpoint than 10, as these two bases are easily divisible by 2, 3, and (in the case of base 12) 4, as compared with decimal which facilitates division by 2 and 5 but renders division by 4 difficult and 3 virtually impossible. This principle of selecting a base which includes many small factors in accounts for the popularity of bases such as 12 and 60.
Of course, the convenience of multiplying a dividing by common numbers is only one point to consider in selecting a base. If you were designing a conlang for an alien race, they might well have a different number of fingers, in which case they would be likely to count using that number as a base. Also, there have been studies which find that less developed cultures sometimes do not have words for numbers above a low limit, such as four; this suggests that an extremely intelligent and mathematically inclined race might find it a simple matter to use a very high base such as 360 to speed communication; in such a language numbers in the millions could be expressed as a mere three digits.
A related consideration is that if you work with a base smaller than 10, you will not need as many names for individual digits, but you will need more names for higher powers of the base to express high numbers; the reverse will hold true if you use a base above 10.
Because of the way computers use simple on-off switches in logic and counting, the binary (base-2), octal (base-8) and hexadecimal (base-16) number systems are commonly seen in association with computer programming.
Mixed-base systems
Many languages have a mixed-base system to some extent. An example is the system of Roman numerals, which mixes the bases 5 and 10: like a base-10 system there are separate symbols for each power of ten, but numbers less than ten are grouped by fives.
An alternative is to have a secondary base higher than the main base: for instance, French uses a secondary base of 20. Thus the words for seventy and ninety are derived from "sixty and ten" and "eighty and ten". Danish goes even further, forming its words for fifty, seventy, and ninety from 2½ × 20, 3½ × 20, and 4½ × 20. Yet in both languages the system overall is base-10, not base-20, because there are separate words for 100, 1000 etc. and not for 400 and 8000 (the powers of twenty).
In all known mixed-base systems, one base is a multiple of the other. The most common mixed-base systems on Earth are 5–10 and 10–20 systems; 5–20 and 20–100 systems are also known.
Thus, in a 5–10 mixed system, the name for 77 might derive from 70 + 5 + 2 or even 50 + 20 + 5 + 2; in a 10–20 mixed system, it could instead derive from (3 × 20) + 10 + 7.
Non-basal systems
One type of non-basal system is that based on a multiple sequence – a sequence of numbers in which each term is a multiple of the previous term. Examples are the Asimov sequence 1, 2, 6, 12, 60, 420, 840, 2520... (in which each term is the lowest common multiple of the first n numbers) and the factorial sequence 1, 2, 6, 24, 120, 720, 5040... (in which each term is the first n numbers multiplied together).
To express numbers in such a system, first note which terms of the sequence the number falls between. Then take the lower number from it as many times as it will go, and repeat with each lower term of the sequence. For example, 1000 = 840 + (2 × 60) + (3 × 12) + (2 × 2) using the Asimov sequence, or 720 + (2 × 120) + 24 + (2 × 6) + (2 × 2) using the factorial sequence. Then write down the multipliers in order, remembering to write a zero for each term of the sequence that was not used: so 1000 = 1023020 in Asimov or 121220 in factorial.
(These are of course numeral systems rather than verbal number systems. I leave to the interested conlanger the task of constructing a verbal number system based on either one.)
