# Vigesimal

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 ordinary 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

Vigesimal multiplication table
1 2 3 4 5 6 7 8 9 A B C D E F G H I J 10
2 4 6 8 A C E G I 10 12 14 16 18 1A 1C 1E 1G 1I 20
3 6 9 C F I 11 14 17 1A 1D 1G 1J 22 25 28 2B 2E 2H 30
4 8 C G 10 14 18 1C 1G 20 24 28 2C 2G 30 34 38 3C 3G 40
5 A F 10 15 1A 1F 20 25 2A 2F 30 35 3A 3F 40 45 4A 4F 50
6 C I 14 1A 1G 22 28 2E 30 36 3C 3I 44 4A 4G 52 58 5E 60
7 E 11 18 1F 22 29 2G 33 3A 3H 44 4B 4I 55 5C 5J 66 6D 70
8 G 14 1C 20 28 2G 34 3C 40 48 4G 54 5C 60 68 6G 74 7C 80
9 I 17 1G 25 2E 33 3C 41 4A 4J 58 5H 66 6F 74 7D 82 8B 90
A 10 1A 20 2A 30 3A 40 4A 50 5A 60 6A 70 7A 80 8A 90 9A A0
B 12 1D 24 2F 36 3H 48 4J 5A 61 6C 73 7E 85 8G 97 9I A9 B0
C 14 1G 28 30 3C 44 4G 58 60 6C 74 7G 88 90 9C A4 AG B8 C0
D 16 1J 2C 35 3I 4B 54 5H 6A 73 7G 89 92 9F A8 B1 BE C7 D0
E 18 22 2G 3A 44 4I 5C 66 70 7E 88 92 9G AA B4 BI CC D6 E0
F 1A 25 30 3F 4A 55 60 6F 7A 85 90 9F AA B5 C0 CF DA E5 F0
G 1C 28 34 40 4G 5C 68 74 80 8G 9C A8 B4 C0 CG DC E8 F4 G0
H 1E 2B 38 45 52 5J 6G 7D 8A 97 A4 B1 BI CF DC E9 F6 G3 H0
I 1G 2E 3C 4A 58 66 74 82 90 9I AG BE CC DA E8 F6 G4 H2 I0
J 1I 2H 3G 4F 5E 6D 7C 8B 9A A9 B8 C7 D6 E5 F4 G3 H2 I1 J0
10 20 30 40 50 60 70 80 90 A0 B0 C0 D0 E0 F0 G0 H0 I0 J0 100
Decimal Vigesimal
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
10 A
11 B
12 C
13 D
14 E
15 F
16 G
17 H
18 I J
19 J K

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.

 In decimal Prime factors of the base: 2, 5 Prime factors of one below the base: 3 Prime factors of one above the base: 11 In vigesimal Prime factors of the base: 2, 5 Prime factors of one below the base: J Prime factors of one above the base: 3, 7 Fraction Prime factors of the denominator Positional representation Positional representation Prime factors of the denominator Fraction 1/2 2 0.5 0.A 2 1/2 1/3 3 0.3333... = 0.3 0.6D6D... = 0.6D 3 1/3 1/4 2 0.25 0.5 2 1/4 1/5 5 0.2 0.4 5 1/5 1/6 2, 3 0.16 0.36D 2, 3 1/6 1/7 7 0.142857 0.2H 7 1/7 1/8 2 0.125 0.2A 2 1/8 1/9 3 0.1 0.248HFB 3 1/9 1/10 2, 5 0.1 0.2 2, 5 1/A 1/11 11 0.09 0.1G759 B 1/B 1/12 2, 3 0.083 0.16D 2, 3 1/C 1/13 13 0.076923 0.1AF7DGI94C63 D 1/D 1/14 2, 7 0.0714285 0.18B 2, 7 1/E 1/15 3, 5 0.06 0.16D 3, 5 1/F 1/16 2 0.0625 0.15 2 1/G 1/17 17 0.0588235294117647 0.13ABF5HCIG984E27 H 1/H 1/18 2, 3 0.05 0.1248HFB 2, 3 1/I 1/19 19 0.052631578947368421 0.1 J 1/J 1/20 2, 5 0.05 0.1 2, 5 1/10

## Real numbers

 Algebraic irrational number In decimal In vigesimal √2 (the length of the diagonal of a unit square) 1.41421356237309... 1.85DE37JGF09H6... √3 (the length of the diagonal of a unit cube) 1.73205080756887... 1.ECG82BDDF5617... √5 (the length of the diagonal of a 1×2 rectangle) 2.2360679774997... 2.4E8AHAB3JHGIB... φ (phi, the golden ratio = (1+√5)/2 1.6180339887498... 1.C7458F5BJII95... Transcendental irrational number 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 limiting difference between the harmonic series and the natural logarithm) 0.5772156649015328606... 0.BAHEA2B19BDIBI...

## 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

### 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.
• In Santali, a Munda language of India, "fifty" is expressed by the phrase bār isī gäl, literally "two twenty ten."[1] Likewise, in Didei, another Munda language spoken in India, complex numerals are decimal to 19 and decimal-vigesimal to 399.[2]
• 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. 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[citation needed] a decimal counting system has come to be preferred (particularly in the South),[citation needed] 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,[3] and, as an alternative, a reform of the spoken language to make it decimal,[4] but both are mostly forgotten.[5]
• 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.

### 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[6]). 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[7] 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 and ), but these are more seldom used. This is a historic remnant of a vigesimal system.[citation needed]
• 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
8000 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