1. Places
     Converting table  

2. Fractions

3. Cyclic numbers

4. Real numbers

5. Use
    Africa  Americas  Asia  In Europe  Etymology  Examples  Related observations 

6. Examples in Mesoamerican languages
     Powers of twenty in Yucatec Maya and Nahuatl    Counting in units of twenty  

7. Further reading

8. Notes

The vigesimal or base 20 numeral system is based on twenty (in the same way in which the 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 {{vigesimal|10}} (the ₂₀ means base {{num|20}}), to write nineteen as {{vigesimal|19}}, 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 less common method skips over the letter "I", in order to avoid confusion between I₂₀ as eighteen and one, so that the number eighteen is written as J₂₀, and nineteen is written as K₂₀. The number twenty is written as {{vigesimal|20}}.

Converting table

According to this notation:

{{vigesimal|40}} means forty in decimal = {{nowrap|(2 × 20¹) + (0 × 20⁰)}}

{{vigesimal|260}} means two hundred and sixty in decimal = {{nowrap|(13 × 20¹) + (0 × 20⁰)}}

{{vigesimal|400}} means four hundred in decimal = {{nowrap|(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. Numbers in vigesimal notation use the convention that I means eighteen and J means nineteen.

Fractions

As 20 is divisible by two and five and is adjacent to 21, the product of three and seven, thus covering the first four prime numbers, many vigesimal fractions have simple representations, whether terminating or recurring (although thirds are more complicated than in decimal, repeating two digits instead of one). In decimal, dividing by three twice (ninths) only gives one digit periods ({{sfrac|9}} = 0.1111.... for instance) because 9 is the number below ten. 21, however, the number adjacent to 20 that is divisible by 3, is not divisible by 9. Ninths in vigesimal have six-digit periods. As 20 has the same prime factors as 10 (two and five), a fraction will terminate in decimal if and only if it terminates in vigesimal.

Cyclic numbers

The prime factorization of twenty is 2² × 5, so it is not a perfect power. However, its squarefree part, 5, is congruent to 1 (mod 4). Thus, according to Artin's conjecture on primitive roots, vigesimal has infinitely many cyclic primes, but the fraction of primes that are cyclic is not necessarily ~37.395%. An UnrealScript program that computes the lengths of recurring periods of various fractions in a given set of bases found that, of the first 15,456 primes, ~39.344% are cyclic in vigesimal.

Real numbers

Use

In many European languages, 20 is used as 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).

• The Open Location Code, used for encoding geographic areas uses a base 20 encoding of coordinates.^[1]

Africa

Vigesimal systems are common in Africa, for example in Yoruba.

Ogún, 20, is the basic numeric block.

Ogójì, 40, (Ogún-meji) = 20 multiplied by 2 (èjì).

Ogota, 60, (Ogún-mẹ̀ta) = 20 multiplied by 3 (ẹ̀ta).

Ogorin, 80, (Ogún-mẹ̀rin) = 20 multiplied by 4 (ẹ̀rin).

Ogorun, 100, (Ogún-màrún) = 20 multiplied by 5 (àrún).

16 (Ẹẹ́rìndílógún) = 4 less than 20.

17 (Etadinlogun) = 3 less than 20.

18 (Eejidinlogun) = 2 less than 20.

19 (Okandinlogun) = 1 less than 20.

21 (Okanlelogun) = 1 increment on 20.

22 (Eejilelogun) = 2 increment on 20.

23 (Etalelogun) = 3 increment on 20.

24 (Erinlelogun) = 4 increment on 20.

25 (Aarunlelogun) = 5 increment on 20.

Americas

• Twenty was a base in the Maya and Aztec number systems. The Maya used the following names for the powers of twenty: {{lang|myn|kal}} (20), {{lang|myn|bak}} (20² = 400), {{lang|myn|pic}} (20³ = 8,000), {{lang|myn|calab}} (20⁴ = 160,000), {{lang|myn|kinchil}} (20⁵ = 3,200,000) and {{lang|myn|alau}} (20⁶ = 64,000,000). See also Maya numerals and Maya calendar, Mayan languages, Yucatec. The Aztec called them: {{lang|nah|cempoalli}} (1 × 20), {{lang|nah|centzontli}} (1 × 400), {{lang|nah|cenxiquipilli}} (1 × 8,000), {{lang|nah|cempoalxiquipilli}} (1 × 20 × 8,000 = 160,000), {{lang|nah|centzonxiquipilli}} (1 × 400 × 8,000 = 3,200,000) and {{lang|nah|cempoaltzonxiquipilli}} (1 × 20 × 400 × 8,000 = 64,000,000). Note that the {{lang|nah|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, {{lang|nah|ome}} (2) × {{lang|nah|poalli}} (20) = {{lang|nah|ompoalli}} (40), {{lang|nah|ome}} (2) × {{lang|nah|tzontli}} (400) = {{lang|nah|ontzontli}} (800). Note also that the {{lang|nah|-li}} in {{lang|nah|poalli}} (and {{lang|nah|xiquipilli}}) and the {{lang|nah|-tli}} in {{lang|nah|tzontli}} are grammatical noun suffixes that are appended only at the end of the word; thus {{lang|nah|poalli}}, {{lang|nah|tzontli}} and {{lang|nah|xiquipilli}} compound together as {{lang|nah|poaltzonxiquipilli}} (instead of {{lang|nah|poallitzontlixiquipilli}}). (See also Nahuatl language.)
• The Tlingit people use base 20.

• The Inuit numbering system is base 20.

Asia

• Dzongkha, the national language of Bhutan, has a full vigesimal system, with numerals for the powers of twenty 20, 400, 8,000, and 160,000.
• Atong, a language spoken in the South Garo Hills of Meghalaya state, Northeast India, and adjacent areas in Bangladesh, has a full vigesimal system that is nowadays considered archaic. ^[2]
• In Santali, a Munda language of India, "fifty" is expressed by the phrase bār isī gäl, literally "two twenty ten."^[3] Likewise, in Didei, another Munda language spoken in India, complex numerals are decimal to 19 and decimal-vigesimal to 399.^[4]
• In East Asia, the Ainu language also uses a counting system that is based around the number 20. “{{lang|ain|hotnep}}” is 20, “{{lang|ain|wanpe etu hotnep}}” (ten more until two twenties) is 30, “{{lang|ain|tu hotnep}}” (two twenties) is 40, “{{lang|ain|ashikne hotnep}}” (five twenties) is 100. Subtraction is also heavily used, e.g. “{{lang|ain|shinepesanpe}}” (one more until ten) is 9.{{citation needed|date=January 2019}}
• The Chukchi language has a vigesimal numeral system.^[5]

In Europe

Etymology

Vigesimal is derived from the Latin adjective vicesimus.

Examples

• The Kaktovik Inupiaq numerals uses a base 20 system. In 1994, Students from Kaktovik, Alaska, came up with the Kaktovik Inupiaq numerals in 1994. Before the numerals had been developed the Inuit names had been falling out of favor. ^[6]
• Twenty ({{lang|fr|vingt}}) is used as a base number in the French language names of numbers from 70 to 99, except in the French of Belgium, Switzerland, the Democratic Republic of the Congo, Rwanda, the Aosta Valley and the Channel Islands. For example, {{lang|fr|quatre-vingts}}, the French word for "80", literally means "four-twenties"; soixante-dix, the word for "70", is literally "sixty-ten"; {{lang|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, Switzerland, the Democratic Republic of the Congo, Rwanda, the Aosta Valley, and the Channel Islands, the numbers 70 and 90 generally have the names septante and nonante. Therefore, the year 1996 is "mille neuf cent quatre-vingt-seize" in Parisian French, but it is "mille neuf cent nonante-six" in Belgian French. In Switzerland, "80" can be quatre-vingts (Geneva, Neuchâtel, Jura) or huitante (Vaud, Valais, Fribourg); octante is also in use in rural parts of Southern France.{{cn|date=November 2017}}
• Twenty ({{lang|da|tyve}}) is used as a base number in the Danish language names of numbers from 50 to 99. For example, {{lang|da|tres}} (short for {{lang|da|tresindstyve}}) means 3 times 20, i.e. 60. However, Danish numerals are not vigesimal since it is only the names of some of the tens that are etymologically formed in a vigesimal way. In contrast with e.g. French quatre-vingt-seize, the units only go from zero to nine between each ten which is a defining trait of a decimal system. For details, see Danish numerals.
• Twenty ({{lang|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, {{lang|br|daou-ugent}} means 2 times 20, i.e. 40, and {{lang|br|triwec'h ha pevar-ugent}} (literally "three-six and four-twenty") means 3×6 + 4×20, i.e. 98. However, 30 is {{lang|br|tregont}} and not {{lang|br|dek ha ugent}} ("ten and twenty"), and 50 is {{lang|br|hanter-kant}} ("half-hundred").
• Twenty ({{lang|cy|ugain}}) is used as a base number in the Welsh language, although in the latter part of the twentieth century{{Citation needed|date=January 2012}} a decimal counting system has come to be preferred (particularly in the South),{{Citation needed|date=January 2012}} with the vigesimal system becoming 'traditional' and more popular in North Welsh. {{lang|cy|Deugain}} means 2 times 20 i.e. 40, {{lang|cy|trigain}} means 3 times 20 i.e. 60. Prior to the currency decimalisation in 1971, {{lang|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 ({{lang|gd|fichead}}) is traditionally used as a base number in Scottish Gaelic, with {{lang|gd|deich ar fhichead}} or {{lang|gd|fichead 's a deich}} being 30 (ten over twenty, or twenty and ten), {{lang|gd|dà fhichead}} 40 (two twenties), {{lang|gd|dà fhichead 's a deich}} 50 (two twenty and ten), {{lang|gd|trì fichead}} 60 (three twenties) and so on up to {{lang|gd|naoidh fichead}} 180 (nine twenties). Nowadays a decimal system is taught in schools, but the vigesimal system is still used by many, particularly older speakers.
• Twenty ({{lang|sq|njëzet}}) is used as a base number in the Albanian language. The word for 40 ({{lang|sq|dyzet}}) means two times 20 (some Gheg subdialects, however, use 'katërdhetë'). The Arbëreshë in Italy may use 'trizetë' for 60. Formerly, 'katërzetë' was also used for 80. Today Cham Albanians in Greece use all zet numbers. Basically 20 means 1 zet, 40 means 2 zet, 60 means 3 zet and 80 means 4 zet.
• Twenty ({{lang|ka|otsi}}) is used as a base number in the Georgian language. For example, 31 ({{lang|ka|otsdatertmeti}}) literally means, twenty-and-eleven. 67 ({{lang|ka|samotsdashvidi}}) is said as, “three-twenty-and-seven”.
• Twenty ({{lang|ce|tqa}}) is used as a base number in the Nakh languages.
• Twenty ({{lang|eu|hogei}}) is used as a base number in the Basque language for numbers up to 100 ({{lang|eu|ehun}}). The words for 40 ({{lang|eu|berrogei}}), 60 ({{lang|eu|hirurogei}}) and 80 ({{lang|eu|laurogei}}) mean "two-score", "three-score" and "four-score", respectively. For example, the number 75 is called {{lang|eu|hirurogeita hamabost}}, lit. "three-score-and ten-five". The Basque nationalist Sabino Arana proposed a vigesimal digit system to match the spoken language,^[7] and, as an alternative, a reform of the spoken language to make it decimal,^[8] but both are mostly forgotten.^[9]
• Twenty (dwisti or dwujsti) is used as a base number in the Resian dialect of the Slovenian language in Italy's Resia Valley. 60 is expressed by trïkrat dwisti (3×20), 70 by trïkrat dwisti nu dësat (3×20 + 10), 80 by štirikrat dwisti (4×20) and 90 by štirikrat dwisti nu dësat (4×20 + 10)^[10][11].
• In the old British currency system (pre-1971), there were 20 shillings (worth 12 pence each) to the pound. Under the decimal system introduced in 1971 (1 pound equals 100 new pence instead of 240 pence in the old system), the shilling coins still in circulation were re-valued at 5 pence (no more were minted and the shilling coin was demonetised in 1990).
• 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. In the Authorised Version of the Bible the term score is used over 130 times although only when prefixed by a number greater than one while a single "score" is always expressed as twenty. The use of the term score to signify multiples of twenty has fallen into disuse in modern English.
• Other languages have terms similar to the old English score, for example Danish and Norwegian snes.
• In regions where traces of the Brythonic Celtic languages have survived in dialect, sheep enumeration systems that are vigesimal are recalled to the present day. See Yan Tan Tethera.

Related observations

• Among multiples of 10, 20 is described in a special way in some languages. For example, the Spanish words {{lang|es|treinta}} (30) and {{lang|es|cuarenta}} (40) consist of "{{lang|es|tre(3)+inta}} (10 times)", "{{lang|es|cuar(4)+enta}} (10 times)", but the word {{lang|es|veinte}} (20) is not presently connected to any word meaning "two" (although historically it is^[12]). 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. The Japanese language has a special word (hatachi) for 20 years (of age), and for the 20th day of the month (hatsuka).
• In some languages (e.g. English, Slavic languages and German), 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 French, this is true up to 16. 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 it may in fact be only found in these names of the numbers 11-19.
• Cantonese^[13] and Wu Chinese frequently use the single unit {{lang|zh-Hans|廿}} (Cantonese yàh, Shanghainese nyae or ne, Mandarin niàn) for twenty, in addition to the fully decimal {{lang|zh-Hans|二十}} (Cantonese yìh sàhp, Shanghainese el sah, Mandarin èr shí) which literally means "two ten". Equivalents exist for 30 and 40 ({{lang|zh-Hans|卅}} and {{lang|zh-Hans|卌}} respectively: Mandarin sà and xì), but these are more seldom used. This is a historic remnant of a vigesimal system.{{citation needed|date=August 2015}}
• Although Khmer numerals have represented a decimal positional notation system since at least the 7th century, Old Khmer, or Angkorian Khmer, also possessed separate symbols for the numbers 10, 20, and 100. Each multiple of 20 or 100 would require an additional stroke over the character, so the number 47 was constructed using the 20 symbol with an additional upper stroke, followed by the symbol for number 7. This suggests that spoken Angkorian Khmer used a vigesimal system.
• Thai uses the term {{lang|th|ยี่สิบ}} (yi sip) for 20. Other multiples of ten consist of the base number, followed by the word for ten, e.g. {{lang|th|สามสิบ}} (sam sip), lit. three ten, for thirty. The yi of yi sip is different from the number two in other positions, which is สอง (song). Nevertheless, yi sip is a loan word from Chinese.
• Lao similarly forms multiples of ten by putting the base number in front of the word ten, so {{lang|la|ສາມສິບ}} (sam sip), litt. three ten, for thirty. The exception is twenty, for which the word {{lang|la|ຊາວ}} (xao) is used. ({{lang|la|ซาว}} sao is also used in the North-Eastern and Northern dialects of Thai, but not in standard Thai.)
• The Kharosthi numeral system behaves like a partial vigesimal system.

Examples in Mesoamerican languages

Powers of twenty in Yucatec Maya and Nahuatl

Counting in units of twenty

This table shows the Maya numerals and the number names in Yucatec Maya, Nahuatl in modern orthography and in Classical Nahuatl.

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: Macmillan & Co, 1931. Project Gutenberg EBook

Notes

• 21st Division (German Empire)
• 21st Dynasty
• 21st dynasty of egypt
• 21st Dáil
• 21st (Empress of India's) Lancers
• 21st Figher-Bomber Wing
• 21st Fighter-Bomber Wing
• 21st Foot
• 21st G7 summit
• 21st G8 summit
• 21st Gemini Awards
• 21st General Assembly of Prince Edward Island
• 21st Genie Awards
• 21st Golden Globe Awards
• 21st Hong Kong Film Awards
• 21st Illinois Infantry Regiment
• 21st Independent Spirit Awards
• 21st Infantry Division
• 21st Infantry Division (France)
• 21st Infantry Division (Poland)
• 21st Infantry Division (Wehrmacht)
• 21st Iowa Volunteer Infantry Regiment
• 21st Lancers (Empress of India's)
• 21st Los Angeles Film Critics Association Awards
• 21st Marine Regiment