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A vigesimal (/vɪˈdʒɛsɪməl/ vij-ESS-im-əl) 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 A_{20} , where the _{20} means base 20, to write nineteen as J_{20}, 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 I_{20} as eighteen and one, so that the number eighteen is written as J_{20}, and nineteen is written as K_{20}. The number twenty is written as 10_{20}.
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:
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.
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 | 0.1D6 | 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 |
The prime factorization of twenty is 2^{2} × 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 |
---|---|---|
√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 | 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 limiting difference between the harmonic series and the natural logarithm) | 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.^{[1]}^{[2]}^{[3]}
Vigesimal systems are common in Africa, for example in Yoruba.^{[4]} While the Yoruba number system may be regarded as a vigesimal system, it is complex.^{[further explanation needed]}
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
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").
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.^{[16]} 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 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Code digit | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | C | F | G | H | J | M | P | Q | R | V | W | X |
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 |
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 |