Vigesimal

The vigesimal or base-num|20 numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten).

Places

In a vigesimal place system, twenty individual numerals (or digit symbols) are used, ten more than in the usual decimal system. One modern method of finding the extra needed symbols is to write ten as the letter A₂₀ (the ₂₀ means base 20), to write nineteen as J₂₀, and the numbers between with the corresponding letters of the alphabet. This is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters "A-F". Another method skips over the letter "I", in order to avoid confusion between I₂₀ as eighteen and 1 (one), so that the number eighteen is written as J₂₀, and nineteen is written as K₂₀. The number twenty is written as 10₂₀.

According to this notation::20₂₀ means forty in decimal {= (2 × 20¹ + (0 × 20⁰)}:DA₂₀ means [270 (number)|two hundred [and] seventy] in decimal {= (13 × 20¹) + (10 × 20⁰}:100₂₀ means four hundred in decimal {= (1 × 20²) + (0 × 20¹) + (0 × 20⁰)}.

In the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example, 10 means ten, 20 means twenty.

Vigesimal fractions

As with decimal, any number with a prime factor other than 2 or 5 will have a repeating expansion in vigesimal. However, the forms of familiar fractions are very different from those in decimal. The following table gives a list of the vigesimal expansion for some small reciprocals and for a few other denominators (listed as fractions in their "decimal" form) that yield very short vigesimal periods.

Note that J₂₀ = 18₁₀ and K₂₀ = 19₁₀.

$frac\{1\}\{3\}\; =\; .6D6D6D6D6D6D6D6D6D6D6D6D6D6D6D6D6D6D$

$frac\{1\}\{7\}\; =\; .2H2H2H2H2H2H2H2H2H2H2H2H2H2H2H2H2H2H$

$frac\{1\}\{11\}\; =\; .1G7591G7591G7591G7591G7591G7591G759$

$frac\{1\}\{13\}\; =\; .1AF7DGH94C631AF7DGH94C631AF7DGH94C63$

$frac\{1\}\{421\}\; =\; .00K00K00K00K00K00K00K00K00K00K00K00K$

$frac\{1\}\{401\}\; =\; .00KK00KK00KK00KK00KK00KK00KK00KK00KK$

$frac\{1\}\{127\}\; =\; .032KGH032KGH032KGH032KGH032KGH032KGH$

$frac\{1\}\{29\}\; =\; .0DFH4GB0DFH4GB0DFH4GB0DFH4GB0DFH4GB$

$frac\{1\}\{71\}\; =\; .05CDA8905CDA8905CDA8905CDA8905CDA89$

$frac\{1\}\{32719\}\; =\; .0004HG10004HG10004HG10004HG10004HG1$

$frac\{1\}\{160001\}\; =\; .0000KKKK0000KKKK0000KKKK0000KKKK$

The number 6D₂₀, equivalent to 133 in decimal is a cyclic number analogous to 142857 in decimal:

* 2₂₀ × 6D₂₀ = D6₂₀

1AF7DGH94C63₂₀ is also a cyclic number. It is equivalent to 315,076,919,876,923 in decimal.

160,001₁₀ is a vigesimal generalized Fermat prime. In vigesimal it is 10001₂₀ or, to describe its status as a generalized Fermat number, $20^\{2^\{\; overset\{2\}\; \{\}\; +\; 1$.

Usage

In many languages, especially in Europe, 20 is a base, at least with respect to the linguistic structure of the names of certain numbers (though a thoroughgoing consistent vigesimal system, based on the powers 20, 400, 8000 etc., is not generally used).

Asia and North America

* In Santali, a Munda language of India, "fifty" is expressed by the phrase "bār isī gäl", literally "two twenty ten." [Gvozdanović, Jadranka. "Numeral Types and Changes Worldwide" (1999), p.223.] Likewise, in Didei, another Munda language spoken in India, complex numerals are decimal to 19 and decimal-vigesimal to 399. [Chatterjee, Suhas. 1963. On Didei nouns, pronouns, numerals, and demonstratives. Chicago: mimeo., 1963. (cf. [http://www.ling.hawaii.edu/austroasiatic/AA/Munda/BIBLIO/biblio.authors Munda Bibliography] at the University of Hawaii Department of Linguistics)]
* In East Asia, the Ainu language also uses a counting system that is based around the number 20. “ _ai. hotnep” is 20, “ _ai. wanpe etu hotnep” (ten more until two twenties) is 30, “ _ai. tu hotnep” (two twenties) is 40, “ _ai. ashikne hotnep” (five twenties) is 100. Subtraction is also heavily used, e.g. “ _ai. shinepesanpe” (one more until ten) is 9.
* Twenty was a base in the Maya number systems. The Maya used the following names for the powers of twenty: " _my. kal" (20), " _my. bak" (20² = 400), " _my. pic" (20³ = 8,000), " _my. calab" (20⁴ = 160,000), " _my. kinchil" (20⁵ = 3,200,000) and " _my. alau" (20⁶ = 64,000,000). See also Maya numerals and Maya calendar, Mayan languages, Yucatec. The Aztec called them: " _na. cempoalli" (1 × 20), " _na. centzontli" (1 × 400), " _na. cenxiquipilli" (1 × 8,000), " _na. cempoalxiquipilli" (1 × 20 × 8,000 = 160,000), " _na. centzonxiquipilli" (1 × 400 × 8,000 = 3,200,000) and " _na. cempoaltzonxiquipilli" (1 × 20 × 400 × 8,000 = 64,000,000). Note that the " _na. ce(n/m)" prefix at the beginning means "one" (as in "one hundred" and "one thousand") and is replaced with the corresponding number to get the names of other multiples of the power. For example, " _na. ome" (2) × " _na. poalli" (20) = " _na. ompoalli" (40), " _na. ome" (2) × " _na. tzontli" (400) = " _na. ontzontli" (800). Note also that the " _na. -li" in " _na. poalli" (and " _na. xiquipilli") and the " _na. -tli" in " _na. tzontli" are grammatical noun suffixes that are appended only at the end of the word; thus " _na. poalli", " _na. tzontli" and " _na. xiquipilli" compound together as " _na. poaltzonxiquipilli" (instead of *" _na. poallitzontlixiquipilli"). (See also Nahuatl language.)

In Europe

According to German linguist Theo Vennemann, the vigesimal system in Europe is of Basque (Vasconic) origin and spread from the so-called Vasconic languages to other European tongues, such as many Celtic languages, French and Danish.

According to Menninger, the vigesimal system originated with the Normans and spread through them to Western Europe, the evidence being that Celtic languages often use vigesimal counting systems. Others believe that this theory is unlikely, however.

* Twenty (" _fr. vingt") is used as a base number in the French language names of numbers from 60 to 99, except in the French of Switzerland, Belgium, the Democratic Republic of the Congo, Rwanda, the Aosta Valley and the Channel Islands. For example, " _fr. quatre-vingts", the French word for 80, literally means "four twenties", "soixante-dix", the word for 70, is literally "sixty-ten", " _fr. soixante-quinze" (75) is literally "sixty-fifteen", "quatre-vingt-sept" (87) is literally "four-twenties-seven", "quatre-vingt-dix" (90) is literally "four-twenties-ten", and "quatre-vingt-seize" (96) is literally "four-twenties-sixteen". However, in the French of Belgium, the Democratic Republic of the Congo, Rwanda, the Aosta Valley, the Channel Islands and the Swiss cantons of Berne, Geneva, Jura and Neuchâtel, the numbers 70 and 90 generally have the names "septante" and "nonante" (but 80 is "quatre-vingts" except in those places, which use "octante", but Switzerland uses is "huitante"). So, the year 1996 is "mille neuf cent quatre-vingt-seize" in Parisian French, but it is "mille neuf cent nonante-six" in "e.g." Belgian French.
* Twenty (" _da. tyve") is used as a base number in the Danish language names of numbers from 50 to 99. For example, " _da. tres" (short for " _da. tresindstyve") means 3 times 20, i.e. 60. For details, see Danish numerals.
* Twenty (" _br. ugent") is used as a base number in the Breton language names of numbers from 40 to 49 and from 60 to 99. For example, " _br. daou-ugent" means 2 times 20, i.e. 40, and " _br. triwec'h ha pevar-ugent" (literally "three-six and four-twenty") means 3×6 + 4×20, i.e. 98. However, 30 is " _br. tregont" and not *" _br. dek ha ugent" ("ten and twenty"), and 50 is " _br. hanter-kant" ("half-hundred").
* Twenty (" _cy. ugain") is used as a base number in the Welsh language, although in the latter part of the twentieth century a decimal counting system has come to be preferred (particularly in the South), with the vigesimal system becoming 'traditional' and more popular in North Welsh. " _cy. Deugain" means 2 times 20 i.e. 40, " _cy. trigain" means 3 times 20 i.e. 60. Prior to the currency decimalisation in 1971, " _cy. papur chwigain" (6 times 20 paper) was the nickname for the 10 shilling (= 120 pence) note. A vigesimal system (Yan Tan Tethera) for counting sheep has also been recorded in areas of Britain that today are no longer Celtic-speaking.
* Twenty (" _ga. fiche") is used in an older counting system in Irish Gaelic, though most people nowadays use a decimal system, and this is what is taught in schools. Thirty is " _ga. fiche a deich", literally "twenty ten". Forty is " _ga. dhá fhichead", literally "two twenties" (retained in the decimal system as "daichead"). " _ga. trí fichid" is sixty ("three twenties") and " _ga. ceithre fichid" is eighty (literally "four twenties"). Similarly, Scottish Gaelic has traditionally used a vigesimal system, with (" _gd. fichead") being the word for twenty. A decimal system is now taught in schools.
* Twenty (" _sq. njëzet") is used as a base number in the Albanian language. The word for 40 (" _sq. dyzet") means two times 20.
* Twenty (" _ka. otsi") is used as a base number in the Georgian language. For example, 31 (" _ka. otsdatertmeti") literally means, "twenty-and-eleven". 67 (" _ka. samotsdashvidi") is said as, “three-twenty-and-seven”.
* Twenty (" _eu. hogei") is used as a base number in the Basque language for numbers up to 100 (" _eu. ehun"). The words for 40 (" _eu. berrogei"), 60 (" _eu. hirurogei") and 80 (" _eu. laurogei") mean "two-score", "three-score" and "four-score", respectively. The number 75 is called " _eu. hirurogeita hamabost", lit. "three-score-and ten-five". The Basque nationalist Sabino Arana proposed three vigesimal digit systems to match the spoken language but they are mostly forgottenFact|date=February 2007 .
* Twenty ("dwisti") is used as a base number in the Resian dialect of the Slovenian language in Italy's Resia valley. 60 is expressed by "trïkart dwisti" (3x20), 70 by "trïkart dwisti nu dësat" (3x20+10), 80 by "štirikrat dwisti" (4x20) and 90 by "štirikrat dwisti nu dësat" (4x20+10).
* In the old British currency system (pre-1971), there were 20 shillings to the pound. This was still the case under the decimal system introduced in 1971 for those shilling coins still in circulation (no more were minted and the shilling coin was demonetised in 1990), because the shilling - which was valued at 12 pence in the old currency - was re-valued at 5 pence in the new system. Thus, the old shilling coins still accumulate 20 to the pound, because 20 x 5 new pence = 100 new pence = 1 pound (whereas in the old system, 1 pound equalled 240 pence instead of 100 pence).
* In the imperial weight system there are twenty hundredweight in a ton.
* In English, counting by the score has been used historically, as in the famous opening of the Gettysburg Address "Four score and seven years ago…", meaning eighty-seven (87) years ago. This method has fallen into disuse, however.

Related observations

* Among multiples of 10, 20 is described in a special way in some languages. For example, the Spanish words " _es. treinta" (30) and " _es. cuarenta" (40) consist of " _es. tre(3)＋"inta" (10 times)", " _es. cuar(4)＋"enta" (10 times)", but the word " _es. veinte" (20) is not presently connected to any word meaning "two" (although historically it is [The diachronic view is like this. _es. veinte < _la. vīgintī, the IE etymology of which ( [http://starling.rinet.ru/cgi-bin/response.cgi?root=config&morpho=0&basename=%5Cdata%5Cie%5Cpiet&first=1&text_proto=&method_proto=substring&text_meaning=&method_meaning=substring&text_rusmean=&method_rusmean=substring&text_hitt=&method_hitt=substring&text_ind=&method_ind=substring&text_avest=&method_avest=substring&text_iran=&method_iran=substring&text_arm=&method_arm=substring&text_greek=&method_greek=substring&text_slav=&method_slav=substring&text_balt=&method_balt=substring&text_germ=&method_germ=substring&text_lat=v%C4%ABgint%C4%AB&method_lat=substring&text_ital=&method_ital=substring&text_celt=&method_celt=substring&text_alb=&method_alb=substring&text_tokh=&method_tokh=substring&text_refer=&method_refer=substring&text_comment=&method_comment=substring&text_any=&method_any=substring&sort=proto view] ) connects it to the roots meaning [http://starling.rinet.ru/cgi-bin/response.cgi?single=1&basename=/data/ie/pokorny&text_number=+328&root=config '2'] and [http://starling.rinet.ru/cgi-bin/response.cgi?single=1&basename=/data/ie/pokorny&text_number=+369&root=config 10'] . (The [http://starling.rinet.ru/cgi-bin/main.cgi?flags=eygtnnl etymological databases] of the [http://starling.rinet.ru/main.html Tower of Babel] project are referred here.)] ). Similarly, in Semitic languages such as Arabic and Hebrew, the numbers 30, 40 ... 90 are expressed by morphologically plural forms of the words for the numbers 3, 4 ... 9, but the number 20 is expressed by a morphologically plural form of the word for 10.
* In some languages (e.g. English, Slavic languages), the names of the two-digit numbers from 11 to 19 consist of one word, but the names of the two-digit numbers from 21 on consist of two words. So for example, the English words eleven (11), twelve (12), thirteen (13) etc., as opposed to "twenty"-one (21), "twenty"-two (22), "twenty"-three (23), etc. In a number of other languages (such as Hebrew), the names of the numbers from 11-19 contain two words, but one of these words is a special "teen" form which is different from the ordinary form of the word for the number 10, and may in fact be only found in these names of the numbers 11-19.
* The term "vicesimal" (from the Latin "vicesimus") is sometimes used

Further reading

*Karl Menninger: "Number words and number symbols: a cultural history of numbers"; translated by Paul Broneer from the revised German edition. Cambridge, Mass.: M.I.T. Press, 1969 (also available in paperback: New York: Dover, 1992 ISBN 0-486-27096-3)
*Levi Leonard Conant: "The Number Concept: Its Origin and Development"; New York, New York: MacMillon & Co, 1931. [http://www.gutenberg.org/etext/16449 Project Gutenberg EBook]

Notes