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[edit]Quaternary numbers are used in the representation of 2D Hilbert curves. Here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected.
Many or all of the Chumashan languages originally used a base 4 counting system, in which the names for numbers were structured according to multiples of 4 and 16 (not 10). There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca. 1819.[1]
Parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in alphabetical order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0, 1, 2, and 3. With this encoding, the complementary digit pairs 0↔3, and 1↔2 (binary 00↔11 and 01↔10) match the complementation of the base pairs: A↔T and C↔G.
For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010 (= decimal 9156).
Quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits.
Many languages[2] use quinary number systems, including Gumatj, Nunggubuyu,[3], Kuurn Kopan Noot[4] and Saraveca. Of these, Gumatj is the only true "5-25" language known, in which 25 is the higher group of 5. The Gumatj numerals are shown below:
| Number | Numeral |
|---|---|
| 1 | wanggany |
| 2 | marrma |
| 3 | lurrkun |
| 4 | dambumiriw |
| 5 | wanggany rulu |
| 10 | marrma rulu |
| 15 | lurrkun rulu |
| 20 | dambumiriw rulu |
| 25 | dambumirri rulu |
| 50 | marrma dambumirri rulu |
| 75 | lurrkun dambumirri rulu |
| 100 | dambumiriw dambumirri rulu |
| 125 | dambumirri dambumirri rulu |
| 625 | dambumirri dambumirri dambumirri rulu |
The Ndom language of Papua New Guinea is reported to have senary numerals[5]. Mer means 6, mer an thef means 6×2 = 12, nif means 36, and nif thef means 36×2 = 72. Proto-Uralic is also suspected to have used senary numerals.[citation needed]
- The Tau of Sci-fi Table-top battle game Warhammer 40,000 use a base-7 counting system.
- In the New Series_Adventures of Doctor Who, the time lords employ a number system using base 7.
- The Halo 3 Alternate Reality Game "IRIS" used a countdown clock in base-7.
- In the online RPG Kingdom of Loathing, the Dwarven miners use a base-7 number system.
See Base 8#Usage.
The Nonary system of notation is used by the fictional civilization, The Culture, found in Iain M. Banks' books.[citation needed]
Base 11 systems appear in several science fiction stories: Carl Sagan's novel Contact references a message "hidden" inside pi that is most striking in base 11, as that permits it to be displayed in binary code. Also the fictional Psychlos (in L. Ron Hubbard's book Battlefield Earth) have a base-11 counting system.
In the show Babylon 5, the Minbari use Base-11 mathematics, according to the show's creator.
The check digit for ISBN is found as the result of taking modulo 11. Since this could give 11 possible results, the digit "X" is used in place of "10".
Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji, Gbiri-Niragu (Kahugu), the Nimbia dialect of Gwandara[6]; the Chepang language of Nepal[7] and the Mahl language of Minicoy Island in India are known to use duodecimal numerals. In fiction, J. R. R. Tolkien's Elvish languages used duodecimal.
Germanic languages have special words for 11 and 12, such as eleven and twelve in English, which are often misinterpreted as vestiges of a duodecimal system. However, they are considered to come from Proto-Germanic *ainlif and *twalif (respectively one left and two left), both of which were decimal.
Historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, and twelve European hours in a day or night. Traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches.
Being a versatile denominator in fractions may explain why we have 12 inches in an imperial foot, 12 ounces in a troy pound, 12 old British pence in a shilling, 12 items in a dozen, 12 dozens in a gross (144, square of 12), 12 gross in a great gross (1728, cube of 12), 24 (12 * 2) hours in a day, etc. The Romans used a fraction system based on 12, including the uncia which became both the English words ounce and inch. Pre-decimalisation, the United Kingdom and Republic of Ireland used a mixed duodecimal-vigesimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to the pound sterling or Irish pound), and Charlemagne established a monetary system that also had a mixed base of twelve and twenty, the remnants of which persist in many places.
In the end of The Restaurant at the End of the Universe by Douglas Adams, a possible question to get the answer "forty-two" is presented: "What do you get if you multiply six by nine?"[8] Of course, the answer is deliberately wrong, creating a humorous effect – if the calculation is carried out in base 10. People who were trying to find a deeper meaning in the passage soon noticed that in base 13, 613 × 913 is actually 4213 (as 4 × 13 + 2 = 54). When confronted with this, the author stated that it was a mere coincidence, and that "I don't write jokes in base 13." See also The Answer to Life, the Universe, and Everything.
This numeric base is infrequently used. It finds applications in mathematics as well as fields such as programming for the HP 9100A/B calculator[9], image processing applications[10] and other specialized uses.
The Huli language of Papua New Guinea is reported to have base-15 numerals[11]. Ngui means 15, ngui ki means 15×2 = 30, and ngui ngui means 15×15 = 225.
Many natural language uses: see Base 20#Use
Umbu-Ungu, also known as Kakoli, is reported to have base-24 numerals.[12][13] Tokapu means 24, tokapu talu means 24×2 = 48, and tokapu tokapu means 24×24 = 576.
It is used in two natural languages, the Telefol language and the Oksapmin language of Papua New Guinea.[citation needed]
Ngiti is reported to have a base 32 numeral system with base-4 cycles.[14] The following is a list of some Ngiti numerals.
| Number | Numeral |
|---|---|
| 1 | atdí |
| 2 | ɔyɔ |
| 3 | |
| 4 | |
| 8 | àr |
| 12 | otsi |
| 16 | ɔp |
| 20 | àbà |
| 24 | àròtsí |
| 28 | àdzòro |
| 32 | wǎdh |
| 64 | ɔyɔ wǎdh |
| 96 | |
| 128 |
See Base 60#Usage.
References
[edit]- ^ "Chumashan Numerals" by Madison S. Beeler, in Native American Mathematics, edited by Michael P. Closs (1986), ISBN 0-292-75531-7.
- ^ Harald Hammarström, Rarities in Numeral Systems: "Bases 5, 10, and 20 are omnipresent."
- ^ Harris, John (1982), Hargrave, Susanne (ed.), "Facts and fallacies of aboriginal number systems" (PDF), Work Papers of SIL-AAB Series B, 8: 153–181
- ^ Dawson, J. "Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria (1881), p. xcviii.
- ^ Owens, Kay (2001), "The Work of Glendon Lean on the Counting Systems of Papua New Guinea and Oceania", Mathematics Education Research Journal, 13 (1): 47–71
- ^ Matsushita, Shuji (1998), "Decimal vs. Duodecimal: An interaction between two systems of numeration", 2nd Meeting of the AFLANG, October 1998, Tokyo, retrieved 2008-03-17
- ^ Mazaudon, Martine (2002), "Les principes de construction du nombre dans les langues tibéto-birmanes", in François, Jacques (ed.), La Pluralité (PDF), Leuven: Peeters, pp. 91–119, ISBN 9042912952
- ^ The Restaurant at the End of the Universe, by Douglas Adams
- ^ See the HP Museum website: [1]
- ^ See one patent at Free Patents Online]
- ^ Cheetham, Brian (1978), "Counting and Number in Huli", Papua New Guinea Journal of Education, 14: 16–35
- ^ Gordon, Raymond G., Jr., ed. (2005), "Umbu-Ungu", Ethnologue: Languages of the World (15 ed.), retrieved 2008-03-16
{{citation}}: CS1 maint: multiple names: editors list (link) - ^ Bowers, Nancy; Lepi, Pundia (1975), "Kaugel Valley systems of reckoning" (PDF), Journal of the Polynesian Society, 84 (3): 309–324
- ^ Hammarström, Harald (2006), "Rarities in Numeral Systems", Proceedings of Rara & Rarissima Conference (PDF)