Editing Vigesimal

{{Short description|Base-20 numeral system}} 
[[File:Maya.svg|thumb|The [[Maya numerals]] are an example of a base-20 numeral system.]]
{{Table Numeral Systems}}
A '''vigesimal''' ({{IPAc-en|v|ɪ|ˈ|dʒ|ɛ|s|ɪ|m|əl}} {{respell|vij|ESS|im|əl}}) or '''base-20''' ('''base-score''') numeral system is based on [[20 (number)|twenty]] (in the same way in which the [[decimal|decimal numeral system]] is based on [[10 (number)|ten]]). ''[[wikt:vigesimal#English|Vigesimal]]'' is derived from the Latin adjective {{wikt-lang|la|vicesimus}}, meaning 'twentieth'.

== Places ==
In a vigesimal [[Numerical digit|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 [[10 (number)|ten]] as the letter A, or {{vigesimal|10}} , where the <sub>20</sub> means [[radix|base]] {{num|20}}, to write [[19 (number)|nineteen]] as {{vigesimal|19}}, and the numbers between with the corresponding letters of the alphabet. This is similar to the common [[Computer science|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<sub>20</sub> as [[18 (number)|eighteen]] and [[1 (number)|one]], so that the number eighteen is written as J<sub>20</sub>, and nineteen is written as K<sub>20</sub>. The number twenty is written as {{vigesimal|20}}.

{|
|valign=top|
{|class="wikitable" style="text-align:center;"
|+Comparison
!Decimal!!colspan="2"|Vigesimal||Name spelled out<br>(in English)
|-
|0||colspan="2"|0||zero
|-
|1||colspan="2"|1||one
|-
|2||colspan="2"|2||two
|-
|3||colspan="2"|3||three
|-
|4||colspan="2"|4||four
|-
|5||colspan="2"|5||five
|-
|6||colspan="2"|6||six
|-
|7||colspan="2"|7||seven
|-
|8||colspan="2"|8||eight
|-
|9||colspan="2"|9||nine
|-
|10||colspan="2"|A||ten
|-
|11||colspan="2"|B||eleven
|-
|12||colspan="2"|C||twelve
|-
|13||colspan="2"|D||thirteen
|-
|14||colspan="2"|E||fourteen
|-
|15||colspan="2"|F||fifteen
|-
|16||colspan="2"|G||sixteen
|-
|17||colspan="2"|H||seventeen
|-
|18||I||J||eighteen
|-
|19||J||K||nineteen
|-
|20||colspan="2"|10||twenty
|-
|400||colspan="2"|100||four hundred
|-
|8000||colspan="2"|1000||eight thousand
|-
|160000||colspan="2"|10000||one hundred and<br>sixty thousand
|}
|width=35|
|valign=top|

{| class="wikitable" style="text-align:center;"
|+ 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:
:{{vigesimal|40}} is equivalent to [[40 (number)|forty]] in decimal = {{nowrap|(2 × 20<sup>1</sup>) + (0 × 20<sup>0</sup>)}}
:{{vigesimal|260}} is equivalent to [[260 (number)|two hundred and sixty]] in decimal = {{nowrap|(13 × 20<sup>1</sup>) + (0 × 20<sup>0</sup>)}}
:{{vigesimal|400}} is equivalent to [[400 (number)|four hundred]] in decimal = {{nowrap|(1 × 20<sup>2</sup>) + (0 × 20<sup>1</sup>) + (0 × 20<sup>0</sup>)}}.

In the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example, '''10''' means [[10 (number)|ten]], '''20''' means [[20 (number)|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 ({{sfrac|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.

{|class="wikitable"
! colspan="3" align="center" | In decimal<br/><small>Prime factors of the base: {{color|green|'''2'''}}, {{color|green|'''5'''}}</small><br/><small>Prime factors of one below the base: {{color|blue|'''3'''}}</small><br/><small>Prime factors of one above the base: {{color|Magenta|'''11'''}}</small>
! colspan="3" align="center" |In vigesimal<br/><small>Prime factors of the base: {{color|green|'''2'''}}, {{color|green|'''5'''}}</small><br/><small>Prime factors of one below the base: {{color|blue|'''J'''}}</small><br/><small>Prime factors of one above the base: {{color|Magenta|'''3'''}}, {{color|Magenta|'''7'''}}</small>
|-
| align="center" | Fraction
| align="center" | <small>Prime factors<br/>of the denominator</small>
| align="center" | Positional representation
| align="center" | Positional representation
| align="center" | <small>Prime factors<br/>of the denominator</small>
| align="center" | Fraction
|-
| align="center" | {{sfrac|1|2}}
| align="center" | {{color|green|'''2'''}}
| '''0.5'''
| '''0.A'''
| align="center" | {{color|green|'''2'''}}
| align="center" | {{sfrac|1|2}}
|-
| align="center" | {{sfrac|1|3}}
| align="center" | {{color|blue|'''3'''}}
| bgcolor=#c0c0c0 | '''0.'''3333... = '''0.'''{{overline|3}}
| bgcolor=#c0c0c0 | '''0.'''6D6D... = '''0.'''{{overline|6D}}
| align="center" | {{color|Magenta|'''3'''}}
| align="center" | {{sfrac|1|3}}
|-
| align="center" | {{sfrac|1|4}}
| align="center" | {{color|green|'''2'''}}
| '''0.25'''
| '''0.5'''
| align="center" | {{color|green|'''2'''}}
| align="center" | {{sfrac|1|4}}
|-
| align="center" | {{sfrac|1|5}}
| align="center" | {{color|green|'''5'''}}
| '''0.2'''
| '''0.4'''
| align="center" | {{color|green|'''5'''}}
| align="center" | {{sfrac|1|5}}
|-
| align="center" | {{sfrac|1|6}}
| align="center" | {{color|green|'''2'''}}, {{color|blue|'''3'''}}
| bgcolor=#c0c0c0 | '''0.1'''{{overline|6}}
| bgcolor=#c0c0c0 | '''0.3'''{{overline|6D}}
| align="center" | {{color|green|'''2'''}}, {{color|Magenta|'''3'''}}
| align="center" | {{sfrac|1|6}}
|-
| align="center" | {{sfrac|1|7}}
| align="center" | {{color|red|'''7'''}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|142857}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|2H}}
| align="center" | {{color|Magenta|'''7'''}}
| align="center" | {{sfrac|1|7}}
|-
| align="center" | {{sfrac|1|8}}
| align="center" | {{color|green|'''2'''}}
| '''0.125'''
| '''0.2A'''
| align="center" | {{color|green|'''2'''}}
| align="center" | {{sfrac|1|8}}
|-
| align="center" | {{sfrac|1|9}}
| align="center" | {{color|blue|'''3'''}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|1}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|248HFB}}
| align="center" | {{color|Magenta|'''3'''}}
| align="center" | {{sfrac|1|9}}
|-
| align="center" | {{sfrac|1|10}}
| align="center" | {{color|green|'''2'''}}, {{color|green|'''5'''}}
| '''0.1'''
| '''0.2'''
| align="center" | {{color|green|'''2'''}}, {{color|green|'''5'''}}
| align="center" | {{sfrac|1|A}}
|-
| align="center" | {{sfrac|1|11}}
| align="center" | {{color|Magenta|'''11'''}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|09}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|1G759}}
| align="center" | {{color|red|'''B'''}}
| align="center" | {{sfrac|1|B}}
|-
| align="center" | {{sfrac|1|12}}
| align="center" | {{color|green|'''2'''}}, {{color|blue|'''3'''}}
| bgcolor=#c0c0c0 | '''0.08'''{{overline|3}}
| bgcolor=#c0c0c0 | '''0.1'''{{overline|D6}}
| align="center" | {{color|green|'''2'''}}, {{color|Magenta|'''3'''}}
| align="center" | {{sfrac|1|C}}
|-
| align="center" | {{sfrac|1|13}}
| align="center" | {{color|red|'''13'''}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|076923}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|1AF7DGI94C63}}
| align="center" | {{color|red|'''D'''}}
| align="center" | {{sfrac|1|D}}
|-
| align="center" | {{sfrac|1|14}}
| align="center" | {{color|green|'''2'''}}, {{color|red|'''7'''}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|714285}}
| bgcolor=#c0c0c0 | '''0.1'''{{overline|8B}}
| align="center" | {{color|green|'''2'''}}, {{color|Magenta|'''7'''}}
| align="center" | {{sfrac|1|E}}
|-
| align="center" | {{sfrac|1|15}}
| align="center" | {{color|blue|'''3'''}}, {{color|green|'''5'''}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|6}}
| bgcolor=#c0c0c0 | '''0.1'''{{overline|6D}}
| align="center" | {{color|Magenta|'''3'''}}, {{color|green|'''5'''}}
| align="center" | {{sfrac|1|F}}
|-
| align="center" | {{sfrac|1|16}}
| align="center" | {{color|green|'''2'''}}
| '''0.0625'''
| '''0.15'''
| align="center" | {{color|green|'''2'''}}
| align="center" | {{sfrac|1|G}}
|-
| align="center" | {{sfrac|1|17}}
| align="center" | {{color|red|'''17'''}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|0588235294117647}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|13ABF5HCIG984E27}}
| align="center" | {{color|red|'''H'''}}
| align="center" | {{sfrac|1|H}}
|-
| align="center" | {{sfrac|1|18}}
| align="center" | {{color|green|'''2'''}}, {{color|blue|'''3'''}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|5}}
| bgcolor=#c0c0c0 | '''0.1'''{{overline|248HFB}}
| align="center" | {{color|green|'''2'''}}, {{color|Magenta|'''3'''}}
| align="center" | {{sfrac|1|I}}
|-
| align="center" | {{sfrac|1|19}}
| align="center" | {{color|red|'''19'''}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|052631578947368421}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|1}}
| align="center" | {{color|blue|'''J'''}}
| align="center" | {{sfrac|1|J}}
|-
| align="center" | {{sfrac|1|20}}
| align="center" | {{color|green|'''2'''}}, {{color|green|'''5'''}}
| '''0.05'''
| '''0.1'''
| align="center" | {{color|green|'''2'''}}, {{color|green|'''5'''}}
| align="center" | {{sfrac|1|10}}
|}

== Cyclic numbers ==

The prime factorization of twenty is 2<sup>2</sup>&nbsp;×&nbsp;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 number|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==
{|class="wikitable"
! Algebraic irrational numbers
! In decimal
! In vigesimal
|-
| align="center" | [[Square root of 2|{{radical|2}}]] <small>(the length of the [[diagonal]] of a unit [[Square (geometry)|square]])</small>
| 1.41421356237309...
| 1.85DE37JGF09H6...
|-
| align="center" | [[Square root of 3|{{radical|3}}]] <small>(the length of the diagonal of a unit [[cube]])</small>
| 1.73205080756887...
| 1.ECG82BDDF5617...
|-
| align="center" | [[Square root of 5|{{radical|5}}]] <small>(the length of the [[diagonal]] of a 1&nbsp;×&nbsp;2 [[rectangle]])</small>
| 2.2360679774997...
| 2.4E8AHAB3JHGIB...
|-
| align="center" | {{mvar|[[Golden ratio|φ]]}} <small>(phi, the [[golden ratio]] = {{sfrac|1+{{radical|5}}|2}})</small>
| 1.6180339887498...
| 1.C7458F5BJII95...
|-
! Transcendental irrational numbers
! In decimal
! In vigesimal
|-
| align="center" | ''[[Pi|{{pi}}]]'' <small>(pi, the ratio of [[circumference]] to [[diameter]])</small>
| 3.14159265358979...
| 3.2GCEG9GBHJ9D2...
|-
| align="center" | {{mvar|[[E (mathematical constant)|e]]}} <small>(the base of the [[natural logarithm]])</small>
| 2.7182818284590452...
| 2.E7651H08B0C95...
|-
| align="center" | {{mvar|[[Euler-Mascheroni constant|γ]]}} <small>(the [[limit (mathematics)|limiting difference]] between the [[harmonic series (mathematics)|harmonic series]] and the natural logarithm)</small>
| 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 [[Greenlandic language|Greenlandic]], [[Iñupiaq language#Numerals|Iñupiaq]], [[Kaktovik numerals|Kaktovik]], [[Maya numerals|Maya]], [[Nunivak Cupʼig language|Nunivak Cupʼig]], and [[Central Alaskan Yupʼik|Yupʼik]] numerals.<ref name="Nykl">{{cite journal |first=Alois Richard |last=Nykl |date=September 1926 |title=The Quinary-Vigesimal System of Counting in Europe, Asia, and America |pages=165–173 |journal=[[Language (journal)|Language]] |volume=2 |issue=3 |url=https://books.google.com/books?id=1GwUAAAAIAAJ&q=Nykl&pg=RA1-PA165 |quote-page=165|quote=A student of the American Indian languages is naturally led to investigate the wide-spread use of the quinary-vigesimal system of counting which he meets in the whole territory from Alaska along the Pacific Coast to the Orinoco and the Amazon.|doi=10.2307/408742 <!--|issn=0097-8507 |eissn=1535-0665 |didn't work for .Raven--> |oclc=50709582 |jstor=408742 |via=[[Google Books]]|url-access=subscription }}</ref><ref>{{cite book |first=Walter Crosby |last=Eells |chapter=Number Systems of the North American Indians |editor-first1=Marlow |editor-last1=Anderson |editor-first2=Victor |editor-last2=Katz |editor-first3=Robin |editor-last3=Wilson |date=October 14, 2004 |title=Sherlock Holmes in Babylon: And Other Tales of Mathematical History |page=89 |publisher=[[Mathematical Association of America]] |isbn=978-0-88385-546-1 |quote=''Quinary-vigesimal''. This is most frequent. The Greenland Eskimo says 'other hand two' for 7, 'first foot two' for 12, 'other foot two' for 17, and similar combinations to 20, 'man ended.' The Unalit is also quinary to twenty, which is 'man completed.' ... |chapter-url=https://books.google.com/books?id=BKRE5AjRM3AC&pg=PA89 |via=[[Google Books]]}}</ref>{{sfn|Chrisomalis|2010|loc=[https://books.google.com/books?id=ux--OWgWvBQC&pg=PA200 p. 200:] "The early origin of bar-and-dot numeration alongside the Middle Formative Mesoamerican scripts, the quinary-vigesimal structure of the system, and the general increase in the frequency and complexity of numeral expressions over time all point to its indigenous development."}}<!-- pentavigesimal{{cn|date=May 2023}}-->

===Africa===
Vigesimal systems are common in Africa, for example in [[Yoruba numerals|Yoruba]].<ref>{{cite journal | last=Zaslavsky |first=Claudia |author-link=Claudia Zaslavsky | title=Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria| journal= The Two-Year College Mathematics Journal| volume= 1 | issue=2 | pages=76–99 | year=1970|doi=10.2307/3027363 | jstor=3027363| s2cid= 163816234}}</ref> While the Yoruba number system may be regarded as a vigesimal system, it is complex.{{Explain|date=March 2021}}

===Americas===
*Probably the best-known instance of the use of the vigesimal system in the Americas is in Lincoln's Gettysburg address: "Four score and seven years ago ..." (see below).
* Twenty is a base in the [[Maya civilization|Maya]] and [[Aztec]] number systems. The Maya use the following names for the powers of twenty: {{lang|myn|kal}} (20), {{lang|myn|bak}} (20<sup>2</sup> = 400), {{lang|myn|pic}} (20<sup>3</sup> = 8,000), {{lang|myn|calab}} (20<sup>4</sup> = 160,000), {{lang|myn|kinchil}} (20<sup>5</sup> = 3,200,000) and {{lang|myn|alau}} (20<sup>6</sup> = 64,000,000). See [[Maya numerals]] and [[Maya calendar]], [[Nahuatl language]].
* The [[Eskimo–Aleut languages|Inuit-Yupik-Unangan languages]] have base-20 number systems. 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.<ref name="kakt">{{cite journal |last=Bartley |first=Wm. Clark |date=January–February 1997 |title=Making the Old Way Count |url=http://www.ankn.uaf.edu/sop/SOPv2i1.pdf |journal=Sharing Our Pathways |volume=2 |issue=1 |pages=12–13 |access-date=February 27, 2017}}</ref>  The Kaktovik numerals are:
{| class=wikitable style="text-align: center;"
|- style="vertical-align: bottom;"
|{{Kaktovik digit|0|x32px}}
|{{Kaktovik digit|1|x32px}}
|{{Kaktovik digit|2|x32px}}
|{{Kaktovik digit|3|x32px}}
|{{Kaktovik digit|4|x32px}}
|{{Kaktovik digit|5|x32px}}
|{{Kaktovik digit|6|x32px}}
|{{Kaktovik digit|7|x32px}}
|{{Kaktovik digit|8|x32px}}
|{{Kaktovik digit|9|x32px}}
|{{Kaktovik digit|10|x32px}}
|{{Kaktovik digit|11|x32px}}
|{{Kaktovik digit|12|x32px}}
|{{Kaktovik digit|13|x32px}}
|{{Kaktovik digit|14|x32px}}
|{{Kaktovik digit|15|x32px}}
|{{Kaktovik digit|16|x32px}}
|{{Kaktovik digit|17|x32px}}
|{{Kaktovik digit|18|x32px}}
|{{Kaktovik digit|19|x32px}}
|-
|0||1||2||3||4||5||6||7||8||9||10||11||12||13||14||15||16||17||18||19
|}

===Asia===
* [[Dzongkha numerals|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 language (Sino-Tibetan)|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.<ref>{{cite book |last=van Breugel |first=Seino |title=A grammar of Atong |publisher=Brill |chapter=11}}</ref>
* In [[Santali language|Santali]], a [[Munda languages|Munda language]] of [[India]], "fifty" is expressed by the phrase ''bār isī gäl'', literally "two twenty ten."<ref>{{cite book |last=Gvozdanović |first=Jadranka |title=Numeral Types and Changes Worldwide |year=1999 |page=223}}</ref> Likewise, in [[Gataq language|Didei]], another Munda language spoken in India, complex numerals are decimal to 19 and decimal-vigesimal to 399.<ref>Chatterjee, Suhas. 1963. On Didei nouns, pronouns, numerals, and demonstratives. Chicago: mimeo., 1963. (cf. [http://www.ling.hawaii.edu/austroasiatic/AA/Munda/BIBLIO/biblio.authors Munda Bibliography] at the University of Hawaii Department of Linguistics)</ref>
* 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. "{{lang|ain|hotnep}}" is 20, "{{lang|ain|wanpe etu hotnep}}" (ten more until two twenties) is [[30 (number)|30]], "{{lang|ain|tu hotnep}}" (two twenties) is 40, "{{lang|ain|ashikne hotnep}}" (five twenties) is [[100 (number)|100]]. Subtraction is also heavily used, e.g. "{{lang|ain|shinepesanpe}}" (one more until ten) is 9.{{citation needed|date=January 2019}}
* The [[Chukchi language]] has a vigesimal numeral system.<ref>{{cite journal |last=Comrie |first=Bernard |url=https://mpi-lingweb.shh.mpg.de/numeral/TypNumCuhk_11ho.pdf |title=Typology of numeral systems. Numeral types and changes worldwide. |journal=Trends in Linguistics |series=Studies and monographs |volume=118 |year=2011 |archive-url=https://web.archive.org/web/20210622052221/https://mpi-lingweb.shh.mpg.de/numeral/TypNumCuhk_11ho.pdf |archive-date=2021-06-22}}</ref>

===Oceania===
There is some evidence of base-20 usage in the [[Māori language]] of New Zealand with the suffix ''hoko-'' (i.e. ''hokowhitu''<!--see war party Te Hokowhitu a Tu-->, ''hokotahi''<!--vide Tama-hokotahi-->).{{citation needed|date=May 2025}}

===Caucasus===
* Twenty ({{lang|ka-Latn|otsi}}, {{lang|ka|ოცი}}) is used as a base number in [[Georgian language|Georgian]] for numbers 30 to 99. For example, [[40 (number)|40]] ({{lang|ka-Latn|ormotsi}}, {{lang|ka|ორმოცი}}) literally means two-times-twenty, whereas [[80 (number)|80]] ({{lang|ka-Latn|otkhmotsi}}, {{lang|ka|ოთხმოცი}}), means four-times-twenty. On the other hand, [[31 (number)|31]] ({{lang|ka-Latn|otsdatertmeti}}, {{lang|ka|ოცდათერთმეტი}}) literally means, ''twenty-and-eleven''. [[67 (number)|67]] ({{lang|ka-Latn|samotsdashvidi}}, {{lang|ka|სამოცდაშვიდი}}) is said as, "three-twenty-and-seven".
* Twenty ({{lang|ce|tq’a, ткъа, ტყა}}) is used as a base number in the [[Nakh languages]] ([[Chechen language|Chechen]], [[Ingush language|Ingush]], and [[Bats language|Batsbi]]).

===Europe===
In several European languages like [[French language|French]] and [[Danish language|Danish]], [[20 (number)|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).

* Twenty ({{lang|fr|vingt}}) is used as a base number in the [[French language|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, {{lang|fr|quatre-vingts}}, the French word for "[[80 (number)|80]]", literally means "four-twenties"; ''soixante-dix'', the word for "[[70 (number)|70]]", is literally "sixty-ten"; {{lang|fr|soixante-quinze}} ("[[75 (number)|75]]") is literally "sixty-fifteen"; {{lang|fr|quatre-vingt-sept}} ("[[87 (number)|87]]") is literally "four-twenties-seven"; {{lang|fr|quatre-vingt-dix}} ("[[90 (number)|90]]") is literally "four-twenties-ten"; and {{lang|fr|quatre-vingt-seize}} ("[[96 (number)|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 {{lang|fr|septante}} and {{lang|fr|nonante}}. Therefore, the year 1996 is {{lang|fr|mille neuf cent quatre-vingt-seize}} in Parisian French, but it is {{lang|fr|mille neuf cent nonante-six}} in Belgian French. In Switzerland, "80" can be {{lang|fr|quatre-vingts}} (Geneva, Neuchâtel, Jura) or {{lang|fr|huitante}} (Vaud, Valais, Fribourg).
* Twenty ({{lang|da|tyve}}) is used as a base number in the [[Danish language|Danish]] names of tens from 50 to 90. For example, {{lang|da|tres}} (short for {{lang|da|tresindstyve}}) means 3 times 20, i.e. [[60 (number)|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 {{lang|fr|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 language#Numerals|Danish numerals]].
* Twenty ({{lang|br|ugent}}) is used as a base number in the [[Breton language|Breton]] names of numbers from 40 to 49 and from 60 to 99. For example, {{lang|br|daou-ugent}} means 2 times 20, i.e. [[40 (number)|40]], and {{lang|br|triwec'h ha pevar-ugent}} (literally "three-six and four-twenty") means 3×6&nbsp;+&nbsp;4×20, i.e. 98. However, 30 is {{lang|br|tregont}} and not *{{lang|br|dek ha ugent}} ("ten and twenty"), and 50 is {{lang|br|hanter-kant}} ("half-hundred").
* Twenty ({{lang|cy|ugain}}) is used as a base number in [[Welsh language|Welsh]] for numbers from 20 to 99 (e.g. 50 is {{lang|cy|deg a deugain}}, "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. {{lang|cy|Deugain}} means "two twenties" i.e. 40, {{lang|cy|trigain}} means 'three twenties' i.e. 60, etc. {{lang|cy|dau ar bymtheg a deugain}} means 57 (two on fifteen and forty). As with Breton, 50 can also be expressed as {{lang|cy|hanner cant}} ("half hundred"). Prior to its withdrawal from circulation, {{lang|cy|papur chweugain}} (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 {{lang|cy|chweugain}} continues to be used to mean 50 pence in modern Welsh, and phrases like {{lang|cy|pisin chweugain}} ('50p piece') are also not uncommon.
* Twenty ({{lang|gd|fichead}}) is traditionally used as a base number in [[Scottish Gaelic]], with {{lang|gd|deich ar fhichead}} or {{lang|gd|fichead 's a deich}} being 30 (ten over twenty, or twenty and ten), {{lang|gd|dà fhichead}} 40 (two twenties), {{lang|gd|dà fhichead 's a deich}} 50 (two twenty and ten) / {{lang|gd|leth-cheud}} 50 (half a hundred), {{lang|gd|trì fichead}} 60 (three twenties) and so on up to {{lang|gd|naoidh fichead}} 180 (nine twenties). Nowadays a decimal system is taught in schools, but the vigesimal system is still used by many, particularly older speakers.
* Twenty ({{lang|gv|feed}}) is traditionally used as a base number in [[Manx Gaelic]], with {{lang|gv|jeih as feed}} being 30 (ten and twenty), {{lang|gv|daeed }} 40 (two twenties), {{lang|gv|jeih as daeed}} 50 (ten and two twenties), {{lang|gv|tree feed}} 60 (three twenty) and so on. A decimal system also exists, using the following tens: {{lang|gv|jeih}} (ten), {{lang|gv|feed}} (twenty), {{lang|gv|treead}} (thirty), {{lang|gv|daeed}} (forty), {{lang|gv|queigad}} (fifty), {{lang|gv|sheyad}} (sixty), {{lang|gv|shiagtad}} (seventy), {{lang|gv|hoghtad}} (eighty) and {{lang|gv|nuyad}} (ninety).
* Twenty ({{lang|sq|njëzet}}) is used as a base number in [[Albanian language|Albanian]]. The word for 40 ({{lang|sq|dyzet}}) means "two times 20". The [[Arbëreshë people|Arbëreshë]] in Italy may use {{lang|sq|trizetë}} for 60. Formerly, {{lang|sq|katërzetë}} was also used for 80. Today [[Cham Albanians]] in Greece use all {{lang|sq|zet}} numbers. Basically, 20 means 1 {{lang|sq|zet}}, 40 means 2 {{lang|sq|zet}}, 60 means 3 {{lang|sq|zet}} and 80 means 4 {{lang|sq|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.<ref name="Demiraj">{{cite book |last=Demiraj|first=Shaban|title=The origin of the Albanians: linguistically investigated |url=https://books.google.com/books?id=aXIbAQAAIAAJ |year=2006 |location=Tirana |publisher=Academy of Sciences of Albania |isbn=978-99943-817-1-5 |page=43 }}</ref>
* Twenty ({{lang|eu|hogei}}) is used as a base number in [[Basque language|Basque]] for numbers up to 100 ({{lang|eu|ehun}}). The words for 40 ({{lang|eu|berrogei}}), 60 ({{lang|eu|hirurogei}}) and 80 ({{lang|eu|laurogei}}) mean "two-score", "three-score" and "four-score", respectively. For example, the number 75 is called {{lang|eu|hirurogeita hamabost}}, lit. "three-score-and ten-five". The Basque nationalist [[Sabino Arana]] proposed a vigesimal digit system to match the spoken language,<ref name="AranaVigesimal">''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.
[http://www.kultura.ejgv.euskadi.net/r46-19239/es/q56War/q56ControladorServlet?mapping=detalleMonografia.do&accion=4&idObjeto=2422376&idLibro=09600015620 102–112]</ref> and, as an alternative, a reform of the spoken language to make it decimal,<ref name="AranaDecimal">''Artículos ...'', Sabino Arana, [http://www.kultura.ejgv.euskadi.net/r46-19239/es/q56War/q56ControladorServlet?mapping=detalleMonografia.do&accion=4&idObjeto=2422386&idLibro=09600015620  112&ndash;118]</ref> but both are mostly forgotten.<ref name="Arana">''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.</ref><!-- Probably also in: Análisis y reforma de la numeración euzkérica, Arana eta Goiri'tar Sabin, Euzkadi : revista trimestral de ciencias, bellas artes, letras. Año 1, n. 2&ndash;3 (jun.&ndash;sept. 1901), p. 189&ndash;221, [1] h. pleg., 299&ndash;334. The title sounds like it but I have not checked. -->
* Twenty ({{lang|sl-rozaj|dwisti}} or {{lang|sl-rozaj|dwujsti}}) is used as a base number in the [[Resian dialect]] {{lang|sl-rozaj|trïkrat dwisti}} (3×20), 70 by {{lang|sl-rozaj|trïkrat dwisti nu dësat}} (3×20&nbsp;+&nbsp;10), 80 by {{lang|sl-rozaj|štirikrat dwisti}} (4×20) and 90 by {{lang|sl-rozaj|štirikrat dwisti nu dësat}} (4×20&nbsp;+&nbsp;10).<ref name="Romavsh">Fran Ramovš, Karakteristika slovenskega narečja v Reziji in: Časopis za slovenski jezik, književnost in zgodovino, no 4, 1928, pages: 107-121 [http://abaoaqu.maldura.unipd.it:8081/resianica/slv/ramkarak.do]</ref><ref name="Merku">{{Cite web |title=dLib.si - LJUDJE OB TERU VI |url=http://www.dlib.si/details/URN:NBN:SI:doc-ZYCM5U86 |access-date=2022-02-13 |website=www.dlib.si }}</ref>
* In the [[£sd|£sd currency system]] (used in the [[United Kingdom]] pre-1971), there were 20 [[shilling]]s (worth 12 pence each) to the [[pound Sterling|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 language|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 (United States)|Declaration of Independence]] in 1776, 87 years earlier. In the [[Authorized King James Version|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 (language)|Danish]] and [[Norwegian (language)|Norwegian]] {{wikt-lang|no|snes}}.
* Even in regions where greater aspects of the [[Brittonic languages|Brythonic Celtic]] languages may be less apparent in modern dialect, sheep enumeration systems that are vigesimal are recalled to the present day. See {{lang|xcb|[[Yan Tan Tethera]]}}.

=== Software applications ===

[[Open Location Code]] uses a word-safe version of base 20 for its [[geocoding|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.<ref>{{cite web |title=Open Location Code: An Open Source Standard for Addresses, Independent of Building Numbers And Street Names |url=https://github.com/google/open-location-code/blob/master/docs/olc_definition.adoc#open-location-code |website=github.com |access-date=25 August 2020}}</ref> The alphabet is also intended to reduce typographical errors by avoiding visually similar digits, and is case-insensitive.

{| class="wikitable"
|+Word-safe base 20
|-
! style="text-align: left" | Base 20 digit
| 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18 || 19
|-
! style="text-align: left" | Code digit
| 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || C || F || G || H || J || M || P || Q || R || V || W || X
|}

===Related observations===
* Among [[Multiple (mathematics)|multiple]]s of [[10 (number)|10]], 20 is described in a special way in some languages. For example, the [[Spanish language|Spanish]] words {{lang|es|treinta}} ([[30 (number)|30]]) and {{lang|es|cuarenta}} (40) consist of "{{lang|es|tre(3)+inta}} (10 times)", "{{lang|es|cuar(4)+enta}} (10 times)", but the word {{lang|es|veinte}} ([[20 (number)|20]]) is not presently connected to any word meaning "two" (although historically it is<ref>The [[Historical linguistics|diachronic]] view is like this. {{langx|es|veinte}} < {{langx|la|vīgintī}}, the [[Indo-European languages|IE]] [[etymology]] of which ([http://starling.rinet.ru/cgi-bin/response.cgi?root=config&morpho=0&basename=%5Cdata%5Cie%5Cpiet&first=1&text_proto=&method_proto=substring&text_meaning=&method_meaning=substring&text_rusmean=&method_rusmean=substring&text_hitt=&method_hitt=substring&text_ind=&method_ind=substring&text_avest=&method_avest=substring&text_iran=&method_iran=substring&text_arm=&method_arm=substring&text_greek=&method_greek=substring&text_slav=&method_slav=substring&text_balt=&method_balt=substring&text_germ=&method_germ=substring&text_lat=v%C4%ABgint%C4%AB&method_lat=substring&text_ital=&method_ital=substring&text_celt=&method_celt=substring&text_alb=&method_alb=substring&text_tokh=&method_tokh=substring&text_refer=&method_refer=substring&text_comment=&method_comment=substring&text_any=&method_any=substring&sort=proto view]) connects it to the roots meaning [http://starling.rinet.ru/cgi-bin/response.cgi?single=1&basename=/data/ie/pokorny&text_number=+328&root=config '2'] and [http://starling.rinet.ru/cgi-bin/response.cgi?single=1&basename=/data/ie/pokorny&text_number=+369&root=config  10']. (The [http://starling.rinet.ru/cgi-bin/main.cgi?flags=eygtnnl etymological databases] of the [http://starling.rinet.ru/main.html Tower of Babel] project are referred here.)</ref>). 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 (number)|11]] to [[19 (number)|19]] consist of one word, but the names of the two-digit numbers from [[21 (number)|21]] on consist of two words. So for example, the English words eleven ([[11 (number)|11]]), twelve ([[12 (number)|12]]), thirteen ([[13 (number)|13]]) etc., as opposed to ''twenty''-one ([[21 (number)|21]]), ''twenty''-two ([[22 (number)|22]]), ''twenty''-three ([[23 (number)|23]]), etc. In French, this is true up to 16. In a number of other languages (such as [[Hebrew language|Hebrew]]), the names of the numbers from 11 to 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]]<ref>Lau, S. ''A Practical Cantonese English Dictionary'' (1977) The Government Printer</ref> and [[Wu Chinese]] frequently use the single unit {{lang|zh-Hans|廿}} (Cantonese ''yàh'', [[Shanghainese]] ''nyae'' or ''ne'', Mandarin ''niàn'') for twenty, in addition to the fully decimal {{lang|zh-Hans|二十}} (Cantonese ''yìh sàhp'', Shanghainese ''el sah'', Mandarin ''èr shí'') which literally means "two ten". Equivalents exist for 30 and 40 ({{lang|zh-Hans|卅}} and {{lang|zh-Hans|卌}} respectively: Mandarin ''sà'' and ''xì''), but these are more seldom used. This is a historic remnant of a vigesimal system.{{citation needed|date=August 2015}}
* Although Khmer numerals have represented a [[decimal]] [[positional notation]] system since at least the 7th century, [[Old Khmer]], or Angkorian Khmer, also possessed separate symbols for the numbers 10, 20, and 100. Each multiple of 20 or 100 would require an additional stroke over the character, so the number 47 was constructed using the 20 symbol with an additional upper stroke, followed by the symbol for number 7. This suggests that spoken Angkorian Khmer used a vigesimal system.
* [[Thai language|Thai]] uses the term {{lang|th|ยี่สิบ}} (''yi sip'') for 20. Other multiples of ten consist of the base number, followed by the word for ten, e.g. {{lang|th|สามสิบ}} (''sam sip''), lit. 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 language|Lao]] similarly forms multiples of ten by putting the base number in front of the word ten, so {{lang|la|ສາມສິບ}} (''sam sip''), litt. three ten, for thirty. The exception is twenty, for which the word {{lang|la|ຊາວ}} (''xao'') is used. ({{lang|la|ซาว}} ''sao'' is also used in the North-Eastern and Northern dialects of Thai, but not in standard Thai.)
* The [[Kharosthi#Numerals|Kharosthi numeral]] system behaves like a partial vigesimal system.

==Examples in Mesoamerican languages==

=== Powers of twenty in Yucatec Maya and Nahuatl ===
{|
! colspan="10" style="color:white; background-color:#970a0a;" | Powers of twenty in Yucatec Maya and Nahuatl
|-
!  Number !! English !! [[Yucatec Maya|Maya]] !! [[Nahuatl]] (modern orthography) !! [[Classical Nahuatl]] !! Nahuatl root !! Aztec pictogram
|-
| 1 || [[One]] || Hun || Se || Ce || Ce || [[File:Maya 1.svg|20px]]
|-
| 20 || [[20 (number)|Twenty]] || K'áal || Sempouali || Cempohualli (Cempoalli) || Pohualli || [[File:Veinte Nahuatl.png|20px]]
|-
| 400 || Four hundred || Bak || Sentsontli || Centzontli || Tzontli || [[File:Cuatrocientos Nahuatl.png|20px]]
|-
| 8,000 || Eight thousand || Pic || Senxikipili || Cenxiquipilli || Xiquipilli || [[File:Xiquipilli.jpg|20px]]
|-
| 160,000 || One hundred sixty thousand || Calab || Sempoualxikipili || Cempohualxiquipilli || Pohualxiquipilli || &nbsp;
|-
| 3,200,000 || Three million two hundred thousand || Kinchil || Sentsonxikipili || Centzonxiquipilli || Tzonxiquipilli || &nbsp;
|-
| 64,000,000 || Sixty-four million || Alau || Sempoualtzonxikipili || Cempohualtzonxiquipilli || Pohualtzonxiquipilli || &nbsp;
|}

=== Counting in units of twenty ===
This table shows the [[Maya numerals]] and the [[Numeral (linguistics)|number names]] in [[Yucatec Maya]], [[Nahuatl]] in modern orthography and in [[Classical Nahuatl]].
{|
! colspan="10" style="color:white; background-color:#970a0a;" | From one to ten (1&nbsp;–&nbsp;10)
|-
!  [[1]] &nbsp;(one)!! [[2]]&nbsp;(two) !! [[3]]&nbsp;(three) !! [[4]]&nbsp;(four) !! [[5]]&nbsp;(five) !! [[6]]&nbsp;(six) !! [[7]]&nbsp;(seven) !! [[8]]&nbsp;(eight) !! [[9]]&nbsp;(nine) !! [[10]]&nbsp;(ten)
|-
| [[File:Maya 1.svg|40px]] || [[File:Maya 2.svg|40px]] || [[File:Maya 3.svg|40px]] || [[File:Maya 4.svg|40px]] || [[File:Maya 5.svg|40px]] || [[File:Maya 6.svg|40px]] || [[File:Maya 7.svg|40px]] || [[File:Maya 8.svg|40px]] || [[File:Maya 9.svg|40px]] || [[File:Maya 10.svg|40px]]
|-
| 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
|-
! colspan="10" style="color:white; background-color:#970a0a;" |  From eleven to twenty (11&nbsp;–&nbsp;20)
|-
!  11 !! 12 !! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 19 !! 20 
|-
| [[File:Maya 11.svg|40px]] || [[File:Maya 12.svg|40px]] || [[File:Maya 13.svg|40px]] || [[File:Maya 14.svg|40px]] || [[File:Maya 15.svg|40px]] || [[File:Maya 16.svg|40px]] || [[File:Maya 17.svg|40px]] || [[File:Maya 18.svg|40px]] || [[File:Maya 19.svg|40px]] || [[File:Maya 1.svg|40px]]<br/>[[File:Mayan00.svg|40px]]
|-
| 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
|-
! colspan="10" style="color:white; background-color:#970a0a;" | From twenty-one to thirty (21&nbsp;–&nbsp;30)
|-
!  21 !! 22 !! 23 !! 24 !! 25 !! 26 !! 27 !! 28 !! 29 !! 30
|-
| [[File:Maya 1.svg|40px]]<br />[[File:Maya 1.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 2.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 3.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 4.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 5.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 6.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 7.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 8.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 9.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 10.svg|40px]]
|-
| 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
|-
! colspan="10" style="color:white; background-color:#970a0a;" |  From thirty-one to forty (31&nbsp;–&nbsp;40)
|-
!  31 !! 32 !! 33 !! 34 !! 35 !! 36 !! 37 !! 38 !! 39 !! 40
|-
| [[File:Maya 1.svg|40px]]<br />[[File:Maya 11.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 12.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 13.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 14.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 15.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 16.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 17.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 18.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Maya 19.svg|40px]] || [[File:Maya 2.svg|40px]]<br />[[File:Mayan00.svg|40px]]
|-
| 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
|-
! colspan="10" style="color:white; background-color:#970a0a;" | From twenty to two hundred in steps of twenty (20&nbsp;–&nbsp;200)
|-
! 20 !! 40 !! 60 !! 80 !! 100 !! 120 !! 140 !! 160 !! 180 !! 200
|-
| [[File:Maya 1.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 2.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 3.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 4.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 5.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 6.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 7.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 8.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 9.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 10.svg|40px]]<br />[[File:Mayan00.svg|40px]]
|-
| 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
|-
! colspan="10" style="color:white; background-color:#970a0a;" | From two hundred twenty to four hundred in steps of twenty (220&nbsp;–&nbsp;400)
|-
! 220 !! 240 !! 260 !! 280 !! 300 !! 320 !! 340 !! 360 !! 380 !! 400
|-
| [[File:Maya 11.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 12.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 13.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 14.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 15.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 16.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 17.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 18.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 19.svg|40px]]<br />[[File:Mayan00.svg|40px]] || [[File:Maya 1.svg|40px]]<br />[[File:Mayan00.svg|40px]]<br />[[File:Mayan00.svg|40px]]
|-
| 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
|}

==Notes==
{{Reflist}}

==Sources==
* {{Cite book | url=https://books.google.com/books?id=ux--OWgWvBQC&pg=PA135 |title = Numerical Notation: A Comparative History |language=en |publisher=[[Cambridge University Press]] |isbn=978-0-521-87818-0|last1 = Chrisomalis| first1 = Stephen |date = 2010-01-18 |at=pp. [https://books.google.com/books?id=ux--OWgWvBQC&pg=PA135 135]–[https://books.google.com/books?id=ux--OWgWvBQC&pg=PA136 136]}}

==Further reading==
*[[Karl Menninger (mathematics)|Karl Menninger]]: ''Number words and number symbols: a cultural history of numbers''; translated by Paul Broneer from the revised German edition. Cambridge, Mass.: M.I.T. Press, 1969 (also available in paperback: New York: Dover, 1992  {{isbn|0-486-27096-3}})
*Levi Leonard Conant: ''The Number Concept: Its Origin and Development''; New York, New York: Macmillan & Co, 1931. [https://www.gutenberg.org/ebooks/16449 Project Gutenberg EBook]

{{Wiktionary}}
{{Authority control}}

[[Category:Positional numeral systems]]