Vigesimal

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The vigesimal or base-20 numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten). Twenty is the sum of all fingers and toes on a normal human being's hands and feet, and is the product of five and four.

1 Places
2 Usage
3 Further reading
4 Notes

[edit] 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 an extension of the common computer-science practice of writing hexadecimal numerals over 9 with the letters "A-F".^{[citation needed]}) 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 two hundred 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.

[edit] 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).

[edit] 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 (vingt) is used as a base number in the French language names of numbers from 60 to 99. So for example, quatre-vingts, the French word for 80 literally means "four twenties", and soixante quinze, the word for 75 is literally "sixty-fifteen.").
Twenty (tyve) is used as a base number in the Danish language names of numbers from 50 to 99. For example, Tres (short for tresindstyve) means 3 times 20, i.e. 60. For details, see Danish numerals.
Twenty (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, daou-ugent means 2 times 20, i.e. 40, and triwec’h ha pevar-ugent (literally "three-six and four-twenty") means 3×6 + 4×20, i.e. 98. However, 30 is tregont and not *dek ha ugent ("ten and twenty"), and 50 is hanter-kant ("half-hundred").
Twenty (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. Deugain means 2 times 20 i.e. 40, trigain means 3 times 20 i.e. 60. Prior to the currency decimalisation in 1971, papur chwigain (6 times 20 paper) was the nickname for the 10 shilling (= 120 pence) note.
Twenty (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 fiche is deag, literally twenty and ten. Forty is dhá fhichid, literally two twenties. Trichid is sixty (three twenties) and ceithre fichid is eighty (literally four twenties).
Twenty (njëzet) is used as a base number in the Albanian language. The word for 40 (dyzet) means two times 20.
Twenty (hogei) is used as a base number in the Basque language for numbers up to 100 (ehun). The words for 40 (berrogei), 60 (hirurogei) and 80 (laurogei) mean "two-score", "three-score" and "four-score", respectively. The number 75 is called 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 forgotten.
In the old British currency system, there were 20 shillings in a pound.
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.
Twenty (otsi) is used as a base number in the Georgian language. For example, 31 (otsdatertmeti) literally means, twenty-and-eleven. 67 (samotsdashvidi) is said as, “three-twenty-and-seven”.

[edit] Outside Europe

In East Asia, the Ainu language also uses a counting system that is based around the number 20. “hotnep” is 20, “wanpe etu hotnep” (ten more until two twenties) is 30, “tu hotnep” (two twenties) is 40, “ashikne hotnep” (five twenties) is 100. Subtraction is also heavily used, e.g. “shinepesanpe” (one more until ten) is 9.
Twenty was a base in the Maya and Aztec number systems. The Maya used the following names for the powers of twenty: kal (20), bak (20² = 400), pic (20³ = 8,000), calab (20⁴ = 160,000), kinchil (20⁵ = 3,200,000) and alau (20⁶ = 64,000,000). See also Maya numerals and Maya calendar, Mayan languages, Yucatec. The Aztec called them: cempoalli (1 × 20), centzontli (1 × 400), cenxiquipilli (1 × 8,000), cempoalxiquipilli (1 × 20 × 8,000 = 160,000), centzonxiquipilli (1 × 400 × 8,000 = 3,200,000) and cempoaltzonxiquipilli (1 × 20 × 400 × 8,000 = 64,000,000). Note that the 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, ome (2) × poalli (20) = ompoalli (40), ome (2) × tzontli (400) = ontzontli (800). Note also that the -li in poalli (and xiquipilli) and the -tli in tzontli are grammatical noun suffixes that are appended only at the end of the word; thus poalli, tzontli and xiquipilli compound together as poaltzonxiquipilli (instead of *poallitzontlixiquipilli). (See also Nahuatl language.)

[edit] Related observations

Among multiples of 10, 20 is described in a special way in some languages. For example, the Spanish words treinta (30) and cuarenta (40) consist of "tre(3)＋inta (10 times)", "cuar(4)＋enta (10 times)", but the word veinte (20) is not presently connected to any word meaning "two" (although historically it is^[1]). 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, 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.

[edit] 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. Project Gutenberg EBook

[edit] Notes

^ The diachronic view is like this. Spanish: veinte < Latin: vīgintī, the IE etymology of which (view) connects it to the roots meaning '2' and 10'. (The etymological databases of the Tower of Babel project are referred here.)