Vigesimal

Numeral systems
Hindu–Arabic numeral system
Western Arabic Eastern Arabic Bengali Gurmukhi Indian Sinhala Tamil Balinese Burmese Dzongkha Javanese Khmer Lao Mongolian Thai
East Asian
Chinese Suzhou Hokkien Japanese Korean Vietnamese Counting rods
Alphabetic
Abjad Armenian Āryabhaṭa Cyrillic Ge'ez Georgian Greek Hebrew Roman
Former
Aegean Attic Babylonian Brahmi Egyptian Etruscan Inuit Kharosthi Mayan Muisca Quipu Prehistoric
Positional systems by base
2 3 4 5 6 8 10 12 16 20 60
Non-standard positional numeral systems
Bijective numeration (1) Signed-digit representation (Balanced ternary) factorial negative Complex-base system (2i) Non-integer representation (φ) mixed
List of numeral systems

The Maya numerals are a base-20 system.

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 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 one, so that the number eighteen is written as J₂₀, and nineteen is written as K₂₀. The number twenty is written as 10₂₀.

Converting table

According to this notation:

20₂₀ means forty in decimal = (2 × 20¹) + (0 × 20⁰)

D0₂₀ means two hundred and sixty in decimal = (13 × 20¹) + (0 × 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.

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 (1/9 = 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), any fraction that terminates in decimal will terminate in vigesimal, and any fraction that does not terminate in decimal will not terminate in vigesimal either: the converses of these statements are also true.

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).

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: 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.)
• The Tlingit people use base 20.

Inuit numerals

• 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. ^[1]
• In Santali, a Munda language of India, "fifty" is expressed by the phrase bār isī gäl, literally "two twenty ten."^[2] Likewise, in Didei, another Munda language spoken in India, complex numerals are decimal to 19 and decimal-vigesimal to 399.^[3]
• 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.

In Europe

Origins

"Vigesimal" has its roots in the Latin adjective vicesimus (in its first or second declension).

Examples

• Twenty (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, quatre-vingts, the French word for "80", literally means "four-twenties"; soixante-dix, the word for "70", is literally "sixty-ten"; 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, 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); in the past octante was also in use.
• 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. However, Danish numerals are not truly vigesimal since it is only the names of some of the tens that are formed in an etymologically 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 (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. A vigesimal system (Yan Tan Tethera) for counting sheep has also been recorded in areas of Britain that today are no longer Celtic-speaking.
• Scottish Gaelic traditionally uses a vigesimal system similar to that of traditional Irish, with (fichead) being the word for twenty, deich ar fhichead being 30 (ten over twenty), dà fhichead 40 (two twenties), dà fhichead 's a deich 50 (two twenty and ten), trì fichead 60 (three twenties) and so on up to naoidh fichead 180 (nine twenties). A decimal system is now taught in schools.
• Twenty (njëzet) is used as a base number in the Albanian language. The word for 40 (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 (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 (tqa) is used as a base number in the Nakh languages.
• 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. For example, the number 75 is called hirurogeita hamabost, lit. "three-score-and ten-five". The Basque nationalist Sabino Arana proposed a vigesimal digit system to match the spoken language,^[4] and, as an alternative, a reform of the spoken language to make it decimal,^[5] but both are mostly forgotten.^[6]
• 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 (3×20), 70 by trïkart dwisti nu dësat (3×20 + 10), 80 by štirikrat dwisti (4×20) and 90 by štirikrat dwisti nu dësat (4×20 + 10).
• 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.
• 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 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^[7]). 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^[8] and Wu Chinese frequently use the single unit 廿 (Cantonese yàh, Shanghainese nyae or ne, Mandarin niàn) for twenty, in addition to the fully decimal 二十 (Cantonese yìh sàhp, Shanghainese el sah, Mandarin èr shí) which literally means "two ten". Equivalents exist for 30 and 40 (卅 and 卌 respectively: Mandarin sà and xì), but these are more seldom used. This is a historic remnant of a vigesimal system.
• Thai uses the term ยี่สิบ (yi sip) for 20. Other multiples of ten consist of the base number, followed by the word for ten, e.g. สามสิบ (sam sip), litt. 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 ສາມສິບ (sam sip), litt. three ten, for thirty. The exception is twenty, for which the word ຊາວ (xao) is used. (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

Powers of twenty in Yucatec Maya and Nahuatl
Number	English	Maya	Nahuatl (modern orthography)	Classical Nahuatl	Nahuatl root	Aztec pictogram
1	One	Hun	Se	Ce	Ce
20	Twenty	K'áal	Sempouali	Cempohualli (Cempoalli)	Pohualli
400	Four hundred	Bak	Sentsontli	Centzontli	Tzontli
8,000	Eight thousand	Pic	Senxikipili	Cenxiquipilli	Xiquipilli
160,000	One hundred sixty thousand	Calab	Sempoualxikipili	Cempohualxiquipilli	Pohualxiquipilli
3,200,000	Three million two hundred thousand	Kinchil	Sentsonxikipili	Centzonxiquipilli	Tzonxiquipilli
64,000,000	Sixty-four million	Alau	Sempoualtzonxikipili	Cempohualtzonxiquipilli	Pohualtzonxiquipilli

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.

From one to ten (1 – 10)
1 (one)	2 (two)	3 (three)	4 (four)	5 (five)	6 (six)	7 (seven)	8 (eight)	9 (nine)	10 (ten)
Hun	Ka'ah	Óox	Kan	Ho'	Wak	Uk	Waxak	Bolon	Lahun
Se	Ome	Yeyi	Naui	Makuili	Chikuasen	Chikome	Chikueyi	Chiknaui	Majtlaktli
Ce	Ome	Yei	Nahui	Macuilli	Chicuace	Chicome	Chicuei	Chicnahui	Matlactli
From eleven to twenty (11 – 20)
11	12	13	14	15	16	17	18	19	20
Buluk	Lahka'a	Óox lahun	Kan lahun	Ho' lahun	Wak lahun	Uk lahun	Waxak lahun	Bolon lahun	Hun k'áal
Majtlaktli onse	Majtlaktli omome	Majtlaktli omeyi	Majtlaktli onnaui	Kaxtoli	Kaxtoli onse	Kaxtoli omome	Kaxtoli omeyi	Kaxtoli onnaui	Sempouali
Matlactli huan ce	Matlactli huan ome	Matlactli huan yei	Matlactli huan nahui	Caxtolli	Caxtolli huan ce	Caxtolli huan ome	Caxtolli huan yei	Caxtolli huan nahui	Cempohualli
From twenty-one to thirty (21 – 30)
21	22	23	24	25	26	27	28	29	30
Hump'éel katak hun k'áal	Ka'ah katak hun k'áal	Óox katak hun k'áal	Kan katak hun k'áal	Ho' katak hun k'áal	Wak katak hun k'áal	Uk katak hun k'áal	Waxak katak hun k'áal	Bolon katak hun k'áal	Lahun katak hun k'áal
Sempouali onse	Sempouali omome	Sempouali omeyi	Sempouali onnaui	Sempouali ommakuili	Sempouali onchikuasen	Sempouali onchikome	Sempouali onchikueyi	Sempouali onchiknaui	Sempouali ommajtlaktli
Cempohualli huan ce	Cempohualli huan ome	Cempohualli huan yei	Cempohualli huan nahui	Cempohualli huan macuilli	Cempohualli huan chicuace	Cempohualli huan chicome	Cempohualli huan chicuei	Cempohualli huan chicnahui	Cempohualli huan matlactli
From thirty-one to forty (31 – 40)
31	32	33	34	35	36	37	38	39	40
Buluk katak hun k'áal	Lahka'a katak hun k'áal	Óox lahun katak hun k'áal	Kan lahun katak hun k'áal	Ho' lahun katak hun k'áal	Wak lahun katak hun k'áal	Uk lahun katak hun k'áal	Waxak lahun katak hun k'áal	Bolon lahun katak hun k'áal	Ka' k'áal
Sempouali ommajtlaktli onse	Sempouali ommajtlaktli omome	Sempouali ommajtlaktli omeyi	Sempouali ommajtlaktli onnaui	Sempouali onkaxtoli	Sempouali onkaxtoli onse	Sempouali onkaxtoli omome	Sempouali onkaxtoli omeyi	Sempouali onkaxtoli onnaui	Ompouali
Cempohualli huan matlactli huan ce	Cempohualli huan matlactli huan ome	Cempohualli huan matlactli huan yei	Cempohualli huan matlactli huan nahui	Cempohualli huan caxtolli	Cempohualli huan caxtolli huan ce	Cempohualli huan caxtolli huan ome	Cempohualli huan caxtolli huan yei	Cempohualli huan caxtolli huan nahui	Ompohualli
From twenty to two hundred in steps of twenty (20 – 200)
20	40	60	80	100	120	140	160	180	200
Hun k'áal	Ka' k'áal	Óox k'áal	Kan k'áal	Ho' k'áal	Wak k'áal	Uk k'áal	Waxak k'áal	Bolon k'áal	Lahun k'áal
Sempouali	Ompouali	Yepouali	Naupouali	Makuilpouali	Chikuasempouali	Chikompouali	Chikuepouali	Chiknaupouali	Majtlakpouali
Cempohualli	Ompohualli	Yeipohualli	Nauhpohualli	Macuilpohualli	Chicuacepohualli	Chicomepohualli	Chicueipohualli	Chicnahuipohualli	Matlacpohualli
From two hundred twenty to four hundred in steps of twenty (220 – 400)
220	240	260	280	300	320	340	360	380	400
Buluk k'áal	Lahka'a k'áal	Óox lahun k'áal	Kan lahun k'áal	Ho' lahun k'áal	Wak lahun k'áal	Uk lahun k'áal	Waxak lahun k'áal	Bolon lahun k'áal	Hun bak
Majtlaktli onse pouali	Majtlaktli omome pouali	Majtlaktli omeyi pouali	Majtlaktli onnaui pouali	Kaxtolpouali	Kaxtolli onse pouali	Kaxtolli omome pouali	Kaxtolli omeyi pouali	Kaxtolli onnaui pouali	Sentsontli
Matlactli huan ce pohualli	Matlactli huan ome pohualli	Matlactli huan yei pohualli	Matlactli huan nahui pohualli	Caxtolpohualli	Caxtolli huan ce pohualli	Caxtolli huan ome pohualli	Caxtolli huan yei pohualli	Caxtolli huan nahui pohualli	Centzontli

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

Notes