Vigesimal

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The vigesimal or base-twenty 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 human hands and feet, and is the product of five and four.

Contents 1 Places 2 Usage 3 Origin in Europe 4 Further reading 5 See also 6 Notes

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 A, to write nineteen as J, and the numbers between with the corresponding letters of the alphabet. (This is an extension of the common 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 I as eighteen and 1 (one), so that the number eighteen is written as J, and nineteen is written as K. The number twenty is written as "10".

According to this notation, 20 means forty (= two times twenty), DA means two hundred seventy (= thirteen and ten-twentieth times twenty), 100 means four hundred (= twenty times twenty). In the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example, 10 means ten, 20 means twenty.

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):

• Twenty (vingt) is used as a base number in the French language names of numbers from 70-99. So for example, quatre-vingts means 4 times 20, i.e. is the French word for 80, and soixante quinze (literally "sixty-fifteen") means 75. Controversial Japanese politician Shintaro Ishihara used this to accuse the French of being unable to count.

• Twenty (tyve) is used as a base number in the Danish language: Tres (short for tresindstyve) means 3 times 20, i.e. 60; firs (short for firsindstyve) means 4 times 20 i.e. 80. halvtreds means (3 － 1/2) times 20, i.e. 50; halvfjerds means (4 － 1/2) times 20, i.e. 70; and halvfems means (5 － 1/2) times 20, i.e. 90.

• 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 came to be preferred, with the vigesimal system becoming 'traditional'. 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 (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”.

• 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".

• Twenty was the base of 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.)

• In the old British currency system, there were 20 shillings in a pound.

• 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.

• 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 synchronically connected to any word meaning "two" (although, diachronically, 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 many 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.

• 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.

Origin in Europe

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

However according to Menninger the vigesimal system originated with the Normans and spread through them to Western Europe.

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)

See also

• quinary
• decimal
• duodecimal
• sexagesimal

Notes

1. ^  The diachronic view is like this. Template:Lang-es < Template:Lang-la, 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.)de:Vigesimalsystem

es:Sistema vigesimal fr:Nombres en français ja:二十進記数法 pt:Sistema vigesimal