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 , meaning 'twentieth'.
Places 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 , where the 20 means base , to write nineteen as , 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 .
+ComparisonDecimalVigesimalName spelled out(in English)00zero11one22two33three44four55five66six77seven88eight99nine10Aten11Beleven12Ctwelve13Dthirteen14Efourteen15Ffifteen16Gsixteen17Hseventeen18IJeighteen19JKnineteen2010twenty400100four hundred80001000eight thousand16000010000one hundred andsixty thousand+ 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 According to this notation: is equivalent to forty in decimal = is equivalent to two hundred and sixty in decimal = is equivalent to four hundred in decimal = .
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 ( = 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.
In decimalPrime factors of the base: , Prime factors of one below the base: Prime factors of one above the base: In vigesimalPrime factors of the base: , Prime factors of one below the base: Prime factors of one above the base: , Fraction Prime factorsof the denominator Positional representation Positional representation Prime factorsof the denominator Fraction 0.5 0.A 0.3333... = 0. 0.6D6D... = 0. 0.25 0.5 0.2 0.4 , 0.1 0.3 , 0. 0. 0.125 0.2A 0. 0. , 0.1 0.2 , 0. 0. , 0.08 0.1 , 0. 0. , 0.0 0.1 , , 0.0 0.1 , 0.0625 0.15 0. 0. , 0.0 0.1 , 0. 0. , 0.05 0.1 ,
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.
Irrational numbers Algebraic irrational numbers 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... (phi, the golden ratio = ) 1.6180339887498... 1.C7458F5BJII95... Transcendental irrational numbers In decimal In vigesimal (pi, the ratio of circumference to diameter) 3.14159265358979... 3.2GCEG9GBHJ9D2... (the base of the natural logarithm) 2.7182818284590452... 2.E7651H08B0C95... (the limiting difference between the harmonic series and the natural logarithm) 0.5772156649015328606... 0.BAHEA2B19BDIBI...
Use
Quinary-vigesimal 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 Maya and Inuit-Yupik-Unangan languages.
Africa Vigesimal systems are common in Africa, for example in Yoruba. While the Yoruba number system may be regarded as a vigesimal system, it is complex. Igbo, Khana, and Krongo also have full vigesimal systems, while Jola-Fonyi, Gola, and Maasina Fulfulde use a mixed or hybrid vigesimal-decimal system (i.e. vigesimal up to 100, decimal after). Supyire is unique in using a base 80 system with subbases of 20, 10, and 5, but does also have a word for 400.
Americas In English, vigesimal is occasionally used in phrases like "Four score and seven years ago ..." in Lincoln's Gettysburg address (see below). Twenty is a base in the Maya and Aztec number systems. The Maya use the following names for the powers of twenty: (20), (202 = 400), (203 = 8,000), (204 = 160,000), (205 = 3,200,000) and (206 = 64,000,000). Vigesimal systems predated the Maya in the area, possibly going back to the Olmec, and were commonly used in other Central American languages as well. See Maya numerals and Maya calendar, Nahuatl language. The Caddoan language family in central North America traditionally used quinary-vigesimal numbers, as did some tribes along the Pacific coast of North America such as the Haida. The Inuit-Yupik-Unangan languages, including Greenlandic, Iñupiaq, Nunivak Cupʼig, and Yupʼik, among others, all have base-20 number systems with sub-base 5. 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. The Kaktovik numerals are: 012345678910111213141516171819
Asia 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.Chatterjee, Suhas. 1963. On Didei nouns, pronouns, numerals, and demonstratives. Chicago: mimeo., 1963. (cf. Munda Bibliography at the University of Hawaii Department of Linguistics) 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. "" is 20, "" (ten more until two twenties) is 30, "" (two twenties) is 40, "" (five twenties) is 100. Subtraction is also heavily used, e.g. "" (one more until ten) is 9. The Chukchi language has a vigesimal numeral system.
Oceania There is some evidence of base-20 usage in the Māori language of New Zealand with the suffix hoko- (i.e. hoko-whitu, hoko-tahi).
Caucasus Twenty (, ) is used as a base number in Georgian for numbers 30 to 99. For example, 40 (, ) literally means two-times-twenty, whereas 80 (, ), means four-times-twenty. On the other hand, 31 (, ) literally means, twenty-and-eleven. 67 (, ) is said as, "three-twenty-and-seven". Twenty () is used as a base number in the Nakh languages (Chechen, Ingush, and Batsbi).
Europe 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). The Celtic languages use a base 20 counting system, and French having extensive influence from Gaulish also uses a base 20 system. Most variations of French in the modern day uses a hybrid system, with more numbers in Quebec and Wallonia being base 10. In other regions like French Switzerland base 10 is exclusively used.
Twenty () 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, , the French word for "80", literally means "four-twenties"; soixante-dix, the word for "70", is literally "sixty-ten"; ("75") is literally "sixty-fifteen"; ("87") is literally "four-twenties-seven"; ("90") is literally "four-twenties-ten"; and ("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 and . Therefore, the year 1996 is in Parisian French, but it is in Belgian French. In Switzerland, "80" can be (Geneva, Neuchâtel, Jura) or (Vaud, Valais, Fribourg). Twenty () is used as a base number in the Danish names of tens from 50 to 90. For example, (short for ) 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 , the units only go from zero to nine between each ten which is a defining trait of a decimal system. Twenty () is used as a base number in the Breton names of numbers from 40 to 49 and from 60 to 99. For example, means 2 times 20, i.e. 40, and (literally "three-six and four-twenty") means 3×6 + 4×20, i.e. 98. However, 30 is and not * ("ten and twenty"), and 50 is ("half-hundred"). Twenty () is used as a base number in Welsh for numbers from 20 to 99 (e.g. 50 is , "ten and twoscore"), although since the 1940s a decimal counting system is often used for cardinal numbers. However, the vigesimal system exclusively is used for ordinal numbers, and is still required in telling the time, money, and with weights and measures. means "two twenties" i.e. 40, means 'three twenties' i.e. 60, etc. means 57 (two on fifteen and forty). As with Breton, 50 can also be expressed as ("half hundred"). Prior to its withdrawal from circulation, (note of sixscore) was the nickname for the ten-shilling (120 pence) note, as 120 (old) pence was equal to half a pound sterling. the term continues to be used to mean 50 pence in modern Welsh, and phrases like ('50p piece') are also not uncommon.