Vigesimal

The Maya numerals are a base-20 system.

The 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).

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 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.

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. 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 (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

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.

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 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...

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[]

Inuit numerals

Asia[]

Oceania[]

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").

In Europe[]

Etymology[]

Vigesimal is derived from the Latin adjective vicesimus.

Examples[]

Related observations[]

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 Uno Nahuatl.png
20 Twenty K'áal Sempouali Cempohualli (Cempoalli) Pohualli Veinte Nahuatl.png
400 Four hundred Bak Sentsontli Centzontli Tzontli Cuatrocientos Nahuatl.png
8,000 Eight thousand Pic Senxikipili Cenxiquipilli Xiquipilli Xiquipilli.jpg
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)
Maya 1.svg Maya 2.svg Maya 3.svg Maya 4.svg Maya 5.svg Maya 6.svg Maya 7.svg Maya 8.svg Maya 9.svg Maya 10.svg
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
Maya 11.svg Maya 12.svg Maya 13.svg Maya 14.svg Maya 15.svg Maya 16.svg Maya 17.svg Maya 18.svg Maya 19.svg Maya 1.svg
Mayan00.svg
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
Maya 1.svg
Maya 1.svg
Maya 1.svg
Maya 2.svg
Maya 1.svg
Maya 3.svg
Maya 1.svg
Maya 4.svg
Maya 1.svg
Maya 5.svg
Maya 1.svg
Maya 6.svg
Maya 1.svg
Maya 7.svg
Maya 1.svg
Maya 8.svg
Maya 1.svg
Maya 9.svg
Maya 1.svg
Maya 10.svg
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
Maya 1.svg
Maya 11.svg
Maya 1.svg
Maya 12.svg
Maya 1.svg
Maya 13.svg
Maya 1.svg
Maya 14.svg
Maya 1.svg
Maya 15.svg
Maya 1.svg
Maya 16.svg
Maya 1.svg
Maya 17.svg
Maya 1.svg
Maya 18.svg
Maya 1.svg
Maya 19.svg
Maya 2.svg
Mayan00.svg
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
Maya 1.svg
Mayan00.svg
Maya 2.svg
Mayan00.svg
Maya 3.svg
Mayan00.svg
Maya 4.svg
Mayan00.svg
Maya 5.svg
Mayan00.svg
Maya 6.svg
Mayan00.svg
Maya 7.svg
Mayan00.svg
Maya 8.svg
Mayan00.svg
Maya 9.svg
Mayan00.svg
Maya 10.svg
Mayan00.svg
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
Maya 11.svg
Mayan00.svg
Maya 12.svg
Mayan00.svg
Maya 13.svg
Mayan00.svg
Maya 14.svg
Mayan00.svg
Maya 15.svg
Mayan00.svg
Maya 16.svg
Mayan00.svg
Maya 17.svg
Mayan00.svg
Maya 18.svg
Mayan00.svg
Maya 19.svg
Mayan00.svg
Maya 1.svg
Mayan00.svg
Mayan00.svg
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

Further reading[]

Notes[]

  1. ^ "google/open-location-code". GitHub. Retrieved 14 November 2018.
  2. ^ Bartley, Wm. Clark (January–February 1997). "Making the Old Way Count" (PDF). Sharing Our Pathways. 2 (1): 12–13. Retrieved February 27, 2017.
  3. ^ van Breugel, Seino. A grammar of Atong. Leiden, Boston: Brill. Chapter 11
  4. ^ Gvozdanović, Jadranka. Numeral Types and Changes Worldwide (1999), p.223.
  5. ^ Chatterjee, Suhas. 1963. On Didei nouns, pronouns, numerals, and demonstratives. Chicago: mimeo., 1963. (cf. Munda Bibliography at the University of Hawaii Department of Linguistics)
  6. ^ Comrie, Bernard. "Typology of numeral systems." Numeral types and changes worldwide. Trends in Linguistics. Studies and monographs 118 (2011).
  7. ^ Demiraj, Shaban (2006). The origin of the Albanians: linguistically investigated. Tirana: Academy of Sciences of Albania. p. 43. ISBN 978-99943-817-1-5.
  8. ^ 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
  9. ^ Artículos ..., Sabino Arana, 112–118
  10. ^ 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.
  11. ^ Fran Ramovš, Karakteristika slovenskega narečja v Reziji in: Časopis za slovenski jezik, književnost in zgodovino, no 4, 1928, pages: 107-121 [1]
  12. ^ Pavle Merku, Ljudje ob teru VI, page: 451
  13. ^ 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.)
  14. ^ Lau, S. A Practical Cantonese English Dictionary (1977) The Government Printer