Monday, August 9, 2010

An extensible number system

Earlier this year, I pointed to John Nystrom's 1862 proposal for a tonal (hexadecimal) number and measurement system to replace that of English in lieu of the metric system, and speculated that a system similar to Nystrom's could be used to generate a flexible number system that can be extended to describe any base. I've been noodling around with the idea since then, and here's what I've come up with.

Each of the numbers from 1 to 20 has a simple word that represents its basic identity. These words are simple consonant-vowel syllables; these should be generally accessible to speakers of numerous widespread languages, and contrast in place and manner of articulation as much as possible.

A table of the numbers and their names is below. The numbers are named as follows: "Ye" is 1; "bi" is 2; "sa" is 3; "te" is 4; "fu" is 5; "go" is 6; "mi" is 7; "pa" is 8; "ze" is 9; "du" is 10; "vo" is 11; "ki" is 12; "hu" is 13; "be" is 14; "su" is 15; "to" is 16; "fi" is 17; "ga" is 18; "me" is 19; and "pu" is 20.

The consonants are pronounced roughly as in English, and the vowels are pronounced as in Spanish or Japanese. A different orthography might be appropriate for English speakers: the word for 2 rhymes with "bumblebee", and the words for both 14 and 19 rhyme with "meh", but the words for 2 and 14 have different vowels. It'd probably be almost impossible to get English speakers to pronounce everything properly, though ;)

Each of these numbers can be used as the basis for a number system. To indicate that a number is the base, append "-n" to that number's name. For example, the word "du" represents the idea of 10. Thus, "dun" represents 10 within the context of the decimal (base-10) system. Effectively, appending "-n" to "du" means "10 raised to the 1st power".

This may not seem terribly useful. But the system can regularly extrapolate a word for any position in the number system as an exponent of the base. The decimal word for 100, for example, means literally "10 raised to the 2nd power". To do this, append "-l" to the word for the base, and follow it with the word for 2 ("bi") appended by "-n". So the word for a hundred is "dulbin". The word for 1,000 is "dulsan".

Overall, the system is strict place-value notation, similar to East Asian numbers. Numbers larger than the base are said as they are written in positional notation, multiplying and adding as necessary. Within the context of decimal math, "dun ye" (literally, "ten and one") is 11. "Bi dulbin dun bi" (literally, "two hundreds, ten, and two") is 212. "Sa dulsan ye dulbin bi dun sa" (literally, "three thousands, one hundred, two tens, and three") is 3,123. Since "dun" is the word for 10 in a base-10 context, the decimal system is internally known as the "dunal system".

This system's main feature is that it can predictably extend any base up to 20. The octal, or base-8, system, is the "panal system", since "pan" is the word for 8 in a base-8 context. "Te pan fu" (literally, "four 8s and five") is decimal 37. "Mi palbin fu pan sa" (literally, "seven 64s, five 8s, and three") is decimal 491. In hexadecimal, "te tolbin" (literally, "four 256s") is decimal 1,024.

This is more limited than scientific notation, of course. But without resorting to scientific notation, in vigesimal, the system can count to more than decimal 2 octillion. In decimal, the highest it can count is one short of one sextillion.

For convenience and clarity, binary can be handled in a somewhat different manner, the short binal system. In short binal, 1 is "ye", 2 is "bin", and 4 is "tel". Subsequent numbers are created by appending "-b" to the name of the power of 2. For example, "sab" is 8 (rather than "bilsan"), and "teb" is 16 (rather than "bilten"). Thus, "teb sab tel ye" is decimal 29.

One clear drawback is the high level of rhyming. These words are more similar than the same words in English; the example of decimal 212 above adequately demonstrates how repeating the same sounds could be confusing. Of course, explicitly repeating the base might make it easier to break the number up. And the difference in scale between "dulsan", "dulgon", and "dulzen" might be more readily apparent than between "thousand", "million", and "billion", given that people often fail to easily conceptualize the difference in degree. Certainly, this is a limited system deficient for the purposes that John Nystrom envisioned (replacing the number system of a natural language); but it may be useful within the scope of its intent.

NumberNameDecimalHexadecimalVigesimalShort Binal
1 Ye ye ye ye ye
2 Bi bi bi bi bin
3 Sa sasasabin ye
4 Te tetetetel
5 Fu fufufutel ye
6 Go gogogotel bin
7 Mi mi mi mi tel bin ye
8 Pa pa pa pa sab
9 Ze ze ze ze sab ye
10 Du ye dun du du sab bin
11 Vo ye dun ye vo vo sab bin ye
12 Ki ye dun bi ki ki sab tel
13 Hu ye dun sa hu hu sab tel ye
14 Be ye dun te be be sab tel bin
15 Su ye dun fu su su sab tel bin ye
16 To ye dun go ye ton to teb
17 Fi ye dun mi ye ton ye fi teb ye
18 Ga ye dun pa ye ton bi ga teb bin
19 Me ye dun ze ye ton sa me teb bin ye
20 Pu bi dun ye ton te ye pun teb tel
21 bi dun ye ye ton fu ye pun ye teb tel ye
22 bi dun bi ye ton go ye pun bi teb tel bin
23 bi dun sa ye ton mi ye pun sa teb tel bin ye
24 bi dun te ye ton pa ye pun te teb sab
25 bi dun fu ye ton ze ye pun fu teb sab ye
100 ye dulbin go ton te fu pun gob fub tel
256 bi dulbin fu dun go ye tolbin ki pun to pab
400 te dulbin ye tolbin ze ton ye pulbin pab mib teb

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