Vigesimal

A vigesimal or base-20 numeral system is based on twenty. Vigesimal is derived from the Latin adjective, meaning 'twentieth'.

Places

In a vigesimal place system, twenty individual numerals 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.
According to this notation:
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. In decimal, dividing by three twice only gives one digit periods 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, 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. 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

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.

Americas

In English, vigesimal is occasionally used in phrases like "Four score and seven years ago..." in Lincoln's Gettysburg address.
Twenty is a base in the Maya and Aztec number systems. The Maya use the following names for the powers of twenty: kal, bak, pic, calab, kinchil and alau. 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:


0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19

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.
The Burushaski number system is base-20. For example, 20 altar, 40 alto-altar, 60 iski-altar 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" is 30, "tu hotnep" is 40, "ashikne hotnep" is 100. Subtraction is also heavily used, e.g. "shinepesanpe" 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-.

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.
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. 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 20. In other regions like French Switzerland base 20 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, quatre-vingts, the French word for "80", literally means "four-twenties"; soixante-dix, the word for "70", is literally "sixty-ten"; soixante-quinze is literally "sixty-fifteen"; quatre-vingt-sept is literally "four-twenties-seven"; quatre-vingt-dix is literally "four-twenties-ten"; and quatre-vingt-seize 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 or huitante.
Twenty is used as a base number in the Danish names of tens from 50 to 90. For example, tres 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 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 means 3×6 + 4×20, i.e. 98. However, 30 is tregont and not *dek ha ugent, and 50 is hanter-kant.
Twenty is used as a base number in Welsh for numbers from 20 to 99, 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. Deugain means "two twenties" i.e. 40, trigain means 'three twenties' i.e. 60, etc. dau ar bymtheg a deugain means 57. As with Breton, 50 can also be expressed as hanner cant. Prior to its withdrawal from circulation, papur chweugain was the nickname for the ten-shilling note, as 120 pence was equal to half a pound sterling. the term chweugain continues to be used to mean 50 pence in modern Welsh, and phrases like pisin chweugain are also not uncommon.
Twenty is traditionally used as a base number in Scottish Gaelic, with deich ar fhichead or fichead 's a deich being 30, dà fhichead 40, dà fhichead 's a deich 50 / leth-cheud 50, trì fichead 60 and so on up to naoidh fichead 180. Nowadays a decimal system is taught in schools, but the vigesimal system is still used by many, particularly older speakers.
Twenty is traditionally used as a base number in Manx Gaelic, with jeih as feed being 30, daeed 40, jeih as daeed 50, tree feed 60 and so on. A decimal system also exists, using the following tens: jeih, feed, treead, daeed, queigad, sheyad, shiagtad, hoghtad and nuyad.
Twenty is used as a base number in Albanian. The word for 40 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 is used as a base number in Basque for numbers up to 100. The words for 40, 60 and 80 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 is used as a base number in the Resian dialect trïkrat dwisti, 70 by trïkrat dwisti nu dësat, 80 by štirikrat dwisti and 90 by štirikrat dwisti nu dësat.
In the £sd currency system, there were 20 shillings to the pound. Under the decimal system introduced in 1971, the shilling coins still in circulation were re-valued at 5 pence.
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.
Even in regions where greater aspects of the Brythonic Celtic languages may be less apparent in modern dialect, sheep enumeration systems that are vigesimal are recalled to the present day. See Yan Tan Tethera.