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A vigesimal (//) or base-20 (base-score) numeral system is based on twenty (in the same way in which the decimal numeral system is based on ten). Vigesimal is derived from the Latin adjective vicesimus, meaning 'twentieth'.
In a vigesimal place system, twenty individual numerals (or digit symbols) are used, ten more than in the decimal system. One modern method of finding the extra needed symbols is to write ten as the letter A, or A20 , where 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 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 1020.
According to this notation:
- 2020 is equivalent to forty in decimal = (2 × 201) + (0 × 200)
- D020 is equivalent to two hundred and sixty in decimal = (13 × 201) + (0 × 200)
- 10020 is equivalent to 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. Numbers in vigesimal notation use the convention that I means eighteen and J means nineteen.
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), a fraction will terminate in decimal if and only if it terminates in vigesimal.
Prime factors of the base: 2, 5
Prime factors of one below the base: 3
Prime factors of one above the base: 11
Prime factors of the base: 2, 5
Prime factors of one below the base: J
Prime factors of one above the base: 3, 7
of the denominator
|Positional representation||Positional representation||Prime factors
of the denominator
|1/3||3||0.3333... = 0.3||0.6D6D... = 0.6D||3||1/3|
|1/6||2, 3||0.16||0.36D||2, 3||1/6|
|1/10||2, 5||0.1||0.2||2, 5||1/A|
|1/12||2, 3||0.083||0.1D6||2, 3||1/C|
|1/14||2, 7||0.0714285||0.18B||2, 7||1/E|
|1/15||3, 5||0.06||0.16D||3, 5||1/F|
|1/18||2, 3||0.05||0.1248HFB||2, 3||1/I|
|1/20||2, 5||0.05||0.1||2, 5||1/10|
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.
|Algebraic irrational number||In decimal||In vigesimal|
|√ (the length of the diagonal of a unit square)||1.41421356237309...||1.85DE37JGF09H6...|
|√ (the length of the diagonal of a unit cube)||1.73205080756887...||1.ECG82BDDF5617...|
|√ (the length of the diagonal of a 1 × 2 rectangle)||2.2360679774997...||2.4E8AHAB3JHGIB...|
|φ (phi, the golden ratio = 1+√/2)||1.6180339887498...||1.C7458F5BJII95...|
|Transcendental irrational number||In decimal||In vigesimal|
|π (pi, the ratio of circumference to diameter)||3.14159265358979...||3.2GCEG9GBHJ9D2...|
|e (the base of the natural logarithm)||2.7182818284590452...||2.E7651H08B0C95...|
|γ (the pi between the harmonic series and the natural logarithm 2)||0.5772156649015328606...||0.BAHEA2B19BDIBI...|
In several European languages like French and Danish, 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).
Many cultures that use a vigesimal system count in fives to twenty, then count twenties similarly. Such a system is referred to as quinary-vigesimal by linguists. Examples include Greenlandic, Iñupiaq, Kaktovik, Maya, Nunivak Cupʼig, and Yupʼik numerals.
- Twenty is a base in the Maya and Aztec number systems. The Maya use 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 Maya numerals and Maya calendar, Nahuatl language.
- The Inuit-Yupik-Unangax languages have base-20 number systems. In 1994, Inuit students in Kaktovik, Alaska, came up with the base-20 Kaktovik numerals to better represent their language. Before this invention led to a revival, the Inuit numerals had been falling out of use.
- Dzongkha, the national language of Bhutan, has a full vigesimal system, with numerals for the powers of 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.
- In Santali, a Munda language of India, "fifty" is expressed by the phrase bār isī gäl, literally "two twenty ten." Likewise, in Didei, another Munda language spoken in India, complex numerals are decimal to 19 and decimal-vigesimal to 399.
- The Burushaski number system is base-20. For example, 20 altar, 40 alto-altar (2 times 20), 60 iski-altar (3 times 20) etc.
- 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.
- The Chukchi language has a vigesimal numeral system.
There is some evidence of base-20 usage in the Māori language of New Zealand as seen in the terms Te Hokowhitu a Tu referring to a war party (literally "the seven 20s of Tu") and Tama-hokotahi, referring to a great warrior ("the one man equal to 20").
- Twenty (otsi, ოცი) is used as a base number in Georgian for numbers 30 to 99. For example, 31 (otsdatertmeti, ოცდათერთმეტი) literally means, twenty-and-eleven. 67 (samotsdashvidi, სამოცდაშვიდი) is said as, "three-twenty-and-seven".
- Twenty (tqa) is used as a base number in the Nakh languages.
- Twenty (vingt) is used as a base number in the French 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, 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).
- Twenty (tyve) is used as a base number in the Danish names of tens from 50 to 90. For example, tres (short for 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 (ugent) is used as a base number in the Breton 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 Welsh from numbers up to 50 (deg a deugain) and from 60 to 100 (cant), although since the 1940s a decimal counting system has come to be preferred. However, the vigesimal system exclusively is used for ordinal numbers and is still required in telling the time, money, and with weights and measures. Deugain means 'two twenties' i.e. 40, trigain means 'three twenties' i.e. 60, etc. dau ar bymtheg a deugain means 57 (two on fifteen and twenty). Like with Breton, 50 is hanner cant ("half-hundred"). Prior to its withdrawal from circulation, papur chweugain (note of sixscore) was the nickname for the ten-shilling (120 pence) note; due to 120 pence = half a pound sterling. the term chweugain continues to be used to mean 50 pence in modern Welsh and phrases like pisin chweugain ('50p piece') is also not uncommon.
- Twenty (fichead) is traditionally used as a base number in Scottish Gaelic, with deich ar fhichead or fichead 's a deich being 30 (ten over twenty, or twenty and ten), dà fhichead 40 (two twenties), dà fhichead 's a deich 50 (two twenty and ten) / leth-cheud 50 (half a hundred), trì fichead 60 (three twenties) and so on up to 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 (feed) is traditionally used as a base number in Manx Gaelic, with jeih as feed being 30 (ten and twenty), daeed 40 (two twenties), jeih as daeed 50 (ten and two twenties), tree feed 60 (three twenty) and so on. A decimal system also exists, using the following tens: jeih (ten), feed (twenty), treead (thirty), daeed (forty), queigad (fifty), sheyad (sixty), shiagtad (seventy), hoghtad (eighty) and nuyad (ninety).
- Twenty (njëzet) is used as a base number in Albanian. The word for 40 (dyzet) means "two times 20". 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. Albanian is the only language in the Balkans which has retained elements of the vigesimal numeral system side by side with decimal system. The existence of the two systems in Albanian reflect the contribution of Pre-Indo-European people of the Balkans to the formation of the Paleo-Balkan Indo-European tribes and their language.
- Twenty (hogei) is used as a base number in Basque 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, and, as an alternative, a reform of the spoken language to make it decimal, but both are mostly forgotten.
- Twenty (dwisti or dwujsti) is used as a base number in the Resian dialect 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).
- In the £sd currency system (used in the United Kingdom 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, the name of the cardinal number 20 is most commonly phrased with the word 'twenty'. Counting by the score has been used historically; for example, the famous opening of the Gettysburg Address, "Four score and seven years ago...", refers to the signing of the Declaration of Independence in 1776, 87 years earlier. In the King James Bible, the term score is used over 130 times, though a single score is always expressed as "twenty". Score is still occasionally used to denote groups of 20 analogously to the use of dozen to quantify groups of 12.
- Other languages have terms similar to score, such as Danish and Norwegian snes.
- In regions where greater aspects of the Brythonic Celtic languages have not survived in modern dialect, sheep enumeration systems that are vigesimal are recalled to the present day. See Yan Tan Tethera.
Open Location Code uses a word-safe version of base 20 for its geocodes. The characters in this alphabet were chosen to avoid accidentally forming words. The developers scored all possible sets of 20 letters in 30 different languages for likelihood of forming words, and chose a set that formed as few recognizable words as possible. The alphabet is also intended to reduce typographical errors by avoiding visually similar digits, and is case-insensitive.
|Base 20 digit||0||1||2||3||4||5||6||7||8||9||10||11||12||13||14||15||16||17||18||19|
- 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). 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 to 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 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.
- 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 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), 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 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|
|160,000||One hundred sixty thousand||Calab||Sempoualxikipili||Cempohualxiquipilli||Pohualxiquipilli|
|3,200,000||Three million two hundred thousand||Kinchil||Sentsonxikipili||Centzonxiquipilli||Tzonxiquipilli|
Counting in units of twenty
|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)|
|From eleven to twenty (11 – 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)|
|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)|
|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)|
|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|
|From two hundred twenty to four hundred in steps of twenty (220 – 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|
- Nykl, Alois Richard (September 1926). "The Quinary-Vigesimal System of Counting in Europe, Asia, and America". Language. 2 (3): 165–173. doi:10.2307/408742. JSTOR 408742. OCLC 50709582 – via Google Books. p. 165:
A student of the American Indian languages is naturally led to investigate the wide-spread use of the quinary-vigesimal system of counting which he meets in the whole territory from Alaska along the Pacific Coast to the Orinoco and the Amazon.
- Eells, Walter Crosby (October 14, 2004). "Number Systems of the North American Indians". In Anderson, Marlow; Katz, Victor; Wilson, Robin (eds.). Sherlock Holmes in Babylon: And Other Tales of Mathematical History. Mathematical Association of America. p. 89. ISBN 978-0-88385-546-1 – via Google Books.
Quinary-vigesimal. This is most frequent. The Greenland Eskimo says 'other hand two' for 7, 'first foot two' for 12, 'other foot two' for 17, and similar combinations to 20, 'man ended.' The Unalit is also quinary to twenty, which is 'man completed.' ...
- Chrisomalis 2010, p. 200: "The early origin of bar-and-dot numeration alongside the Middle Formative Mesoamerican scripts, the quinary-vigesimal structure of the system, and the general increase in the frequency and complexity of numeral expressions over time all point to its indigenous development.".
- Zaslavsky, Claudia (1970). "Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria". The Two-Year College Mathematics Journal. 1 (2): 76–99. JSTOR 3027363. S2CID 163816234.
- Bartley, Wm. Clark (January–February 1997). "Making the Old Way Count" (PDF). Sharing Our Pathways. 2 (1): 12–13. Retrieved February 27, 2017.
- van Breugel, Seino. A grammar of Atong. Leiden, Boston: Brill. Chapter 11
- Gvozdanović, Jadranka. Numeral Types and Changes Worldwide (1999), p.223.
- Chatterjee, Suhas. 1963. On Didei nouns, pronouns, numerals, and demonstratives. Chicago: mimeo., 1963. (cf. Munda Bibliography at the University of Hawaii Department of Linguistics)
- Comrie, Bernard. "Typology of numeral systems." Numeral types and changes worldwide. Trends in Linguistics. Studies and monographs 118 (2011).
- Demiraj, Shaban (2006). The origin of the Albanians: linguistically investigated. Tirana: Academy of Sciences of Albania. p. 43. ISBN 978-99943-817-1-5.
- Artículos publicados en la 1.ª época de "Euzkadi" : revista de Ciencias, Bellas Artes y Letras de Bilbao por Arana-Goiri'taŕ Sabin: 1901, Artículos publicados en la 1 época de "Euskadi" : revista de Ciencias, Bellas Artes y Letras de Bilbao por Arana-Goiri'ttarr Sabin : 1901, Sabino Arana, 1908, Bilbao, Eléxpuru Hermanos. 102–112
- Artículos ..., Sabino Arana, 112–118
- Efemérides Vascas y Reforma d ela Numeración Euzkérica, Sabino Arana, Biblioteca de la Gran Enciclopedia Vasca, Bilbao, 1969. Extracted from the magazine Euskal-Erria, 1880 and 1881.
- Fran Ramovš, Karakteristika slovenskega narečja v Reziji in: Časopis za slovenski jezik, književnost in zgodovino, no 4, 1928, pages: 107-121 
- "dLib.si - LJUDJE OB TERU VI". www.dlib.si. Retrieved 2022-02-13.
- "Open Location Code: An Open Source Standard for Addresses, Independent of Building Numbers And Street Names". github.com. Retrieved 25 August 2020.
- 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.)
- Lau, S. A Practical Cantonese English Dictionary (1977) The Government Printer
- Chrisomalis, Stephen (2010-01-18). Numerical Notation: A Comparative History. Cambridge University Press. pp. 135–136. ISBN 978-0-521-87818-0.
- 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