View source for Maya numerals

{{Short description|System used by the ancient Mayan civilization to represent numbers and dates}}
{{stack begin|float=right}}
[[Image:maya.svg|thumb|right|Maya numerals]]
{| class="wikitable" style="text-align:center; margin-left:1em; float: right" [[Mayan Numbers/Numerals: Explained in a table]]
|-
|400s
| 
| {{Horizontal Maya|1}}
| {{Horizontal Maya|12}}
|- 
| 20s
| {{Horizontal Maya|1}}
| {{Horizontal Maya|1}}
| {{Horizontal Maya|16}}
|-
| 1s
| {{Horizontal Maya|13}}
| {{Horizontal Maya|9}}
| {{Horizontal Maya|5}}
|-
| Total(s)
| 33
| 429
| 5125
|}
{{Table Numeral Systems}}
{{stack end}}

The '''Mayan numeral system''' was the system to represent [[number]]s and [[calendar date]]s in the [[Maya civilization]]. It was a [[vigesimal]] (base-20) [[positional notation|positional]] [[numeral system]]. The numerals are made up of three symbols: [[Zero number#The Americas|zero]] (a shell),<ref>{{Cite journal |last=Batz |first=J. Mucía |date=March 29, 2021 |title="Nik" — The Zero in Vigesimal Maya Mathematics |url=https://baas.aas.org/pub/2021n1i336p03/release/2 |url-status=live |archive-url=https://archive.today/20240910192515/https://baas.aas.org/pub/2021n1i336p03/release/2 |archive-date=September 10, 2024 |access-date=October 30, 2024 |website=Bulletin of the AAS|volume=53 |issue=1 }}</ref> [[1 (number)|one]] (a dot) and [[5 (number)|five]] (a bar). For example, thirteen is written as three dots in a horizontal row above two horizontal bars; sometimes it is also written as three vertical dots to the left of two vertical bars. With these three symbols, each of the twenty vigesimal digits could be written.

Numbers after 19 were written vertically in powers of twenty. The Mayan used powers of twenty, just as the [[Hindu–Arabic numeral system]] uses powers of ten.<ref>{{cite web |author=Saxakali |date=1997 |title=Mayan Numerals |url=http://saxakali.com/historymam2.htm |archive-url=https://web.archive.org/web/20060714025120/http://www.saxakali.com/historymam2.htm |archive-date=July 14, 2006 |access-date=July 29, 2006 |website=Saxakali}}</ref>

For example, thirty-three would be written as one dot, above three dots atop two bars. The first dot represents "one twenty" or "1×20", which is added to three dots and two bars, or thirteen. Therefore, (1×20) + 13 = 33.

:{| class="mw-collapsible mw-collapsed" style="text-align:center;"
|+Addition (single)
|- style="font-size: 150%;"
| (1×20)
| +
| 13
| =
| 33
|-
| {{Horizontal Maya|1}}
|
| {{Horizontal Maya|13}}
|
| {{Horizontal Maya|1}}{{Horizontal Maya|13}}
|}

Upon reaching 20<sup>2</sup> or 400, another row is started (20<sup>3</sup> or 8000, then 20<sup>4</sup> or 160,000, and so on). The number 429 would be written as one dot above one dot above four dots and a bar, or (1×20<sup>2</sup>) + (1×20<sup>1</sup>) + 9 = 429.


:{| class="mw-collapsible mw-collapsed" style="text-align:center;"
|+Addition (multiple)
|- style="font-size: 150%;"
| (1×20<sup>2</sup>)
| +
| (1×20<sup>1</sup>)
| +
| 9
| =
| 429
|-
| {{Horizontal Maya|1}}
|
| {{Horizontal Maya|1}}
|
| {{Horizontal Maya|9}}
|
| {{Horizontal Maya|1}}{{Horizontal Maya|1}}{{Horizontal Maya|9}}
|}

Other than the bar and dot notation, Maya numerals were sometimes illustrated by face type glyphs or pictures. The face glyph for a number represents the deity associated with the number. These face number glyphs were rarely used, and are mostly seen on some of the most elaborate monumental carvings.

[[File:Dresden_Codex_f8461796.png|thumb|266px|Section of page 43b of the [[Dresden Codex]] showing the different representations of zero.]]

There are different representations of zero in the [[Dresden Codex]], as can be seen at page 43b (which is concerned with the synodic cycle of Mars).<ref>{{cite web
| url = http://digital.slub-dresden.de/id280742827
| title = Codex Dresdensis - Mscr.Dresd.R.310
| publisher = Saxon State and University Library (SLUB) Dresden
}}</ref> It has been suggested that these pointed, oblong "bread" representations are calligraphic variants of the PET logogram, approximately meaning "circular" or "rounded", and perhaps the basis of a derived noun meaning "totality" or "grouping", such that the representations may be an appropriate marker for a number position which has reached its totality.<ref>{{cite web
| url = https://mayadecipherment.com/2012/06/15/the-calligraphic-zero
| title = The Calligraphic Zero
| author = David Stuart
| date = June 15, 2012
| website = Maya Decipherment: Ideas on Maya Writing and Iconography -- Boundary End Archaeological Research Center
| accessdate = Mar 11, 2024
}}</ref>

== Addition and subtraction ==
Adding and subtracting numbers below 20 using Mayan numerals is very simple.
[[Addition]] is performed by combining the numeric symbols at each level:<br>
[[Image:Maya add.png|210px]]

If five or more dots result from the combination, five dots are removed and replaced by a bar. If four or more bars result, four bars are removed and a dot is added to the next higher row. This also means that the value of 1 bar is 5.

Similarly with [[subtraction]], remove the elements of the [[subtrahend]] Symbol from the [[minuend]] symbol:<br>
[[Image:Mayan subtract.png|210px]]

If there are not enough dots in a minuend position, a bar is replaced by five dots. If there are not enough bars, a dot is removed from the next higher minuend symbol in the column and four bars are added to the minuend symbol which is being worked on.

== Modified vigesimal system in the Maya calendar ==
[[File:La Mojarra Estela 1 (Escritura superior).jpg|thumb|266x266px|Detail showing in the right columns glyphs from [[La Mojarra Stela 1]].  The left column uses Maya numerals to show a [[Mesoamerican Long Count calendar|Long Count date]] of 8.5.16.9.7 or 156 CE.]] The "Long Count" portion of the [[Maya calendar]] uses a variation on the strictly vigesimal numerals to show a [[Mesoamerican Long Count calendar|Long Count date]]. In the second position, only the digits up to 17 are used, and the [[positional notation|place value]] of the third position is not 20×20 = 400, as would otherwise be expected, but 18×20 = 360 so that one dot over two zeros signifies 360.  Presumably, this is because 360 is roughly the number of days in a [[year]]. (The Maya had however a quite accurate estimation of 365.2422 days for the [[Solar Year|solar year]] at least since the early [[Maya civilization#Classic period|Classic era]].)<ref>{{cite book | title=The Mayans | publisher=Lucent Books, Inc. | author=Kallen, Stuart A. | year=1955 | location=San Diego, CA | pages=[https://archive.org/details/mayans00kall/page/56 56] | isbn=1-56006-757-8 | url-access=registration | url=https://archive.org/details/mayans00kall/page/56 }}</ref> Subsequent positions use all twenty digits and the place values continue as 18×20×20 = 7,200 and 18×20×20×20 = 144,000, etc.

Every known example of large numbers in the Maya system uses this 'modified vigesimal' system, with the third position representing multiples of 18×20. It is reasonable to assume, but not proven by any evidence, that the normal system in use was a pure base-20 system.<ref>Anderson, W. French. “Arithmetic in Maya Numerals.” American Antiquity, vol. 36, no. 1, 1971, pp. 54–63</ref>

== Origins ==
Several Mesoamerican cultures used similar numerals and base-twenty systems and the [[Mesoamerican Long Count calendar]] requiring the use of zero as a place-holder. The earliest long count date (on [[Chiapa de Corzo Stela 2#Notable finds|Stela 2]] at Chiappa de Corzo, [[Chiapas]]) is from 36 BC.{{refn|group=lower-alpha|No long count date actually using the number 0 has been found before the 3rd century, but since the long count system would make no sense without some placeholder, and since Mesoamerican glyphs do not typically leave empty spaces, these earlier dates are taken as indirect evidence that the concept of 0 already existed at the time.}}

Since the eight earliest Long Count dates appear outside the Maya homeland,<ref>{{cite book|title=The Olmecs: America's First Civilization|last=Diehl|first=Richard|publisher=Thames & Hudson|year=2004|isbn=0-500-02119-8|location=London|page=[https://archive.org/details/olmecsamericasfi0000dieh/page/186 186]|oclc=56746987|author-link=Richard Diehl|url=https://archive.org/details/olmecsamericasfi0000dieh/page/186}}</ref> it is assumed that the use of zero and the Long Count calendar predated the Maya, and was possibly the invention of the [[Olmec]]. Indeed, many of the earliest Long Count dates were found within the Olmec heartland. However, the Olmec civilization had come to an end by the 4th century BC, several centuries before the earliest known Long Count dates—which suggests that zero was ''not'' an Olmec discovery.

== Unicode ==
{{main|Mayan Numerals (Unicode block)}}
Mayan numerals codes in Unicode comprise the block 1D2E0 to 1D2F3
{{Unicode chart Mayan Numerals}}

==See also==
*[[Kaktovik numerals]], a similar system from another culture, created in the late 20th century.

== Notes ==
{{reflist|group=lower-alpha}}

== References ==
{{reflist}}

== Further reading ==
{{refbegin|indent=yes}}<!-- BEGIN biblio style.  -->
* {{cite book |last=Coe |first=Michael D. |author-link=Michael D. Coe |year=1987 |title=The Maya |publisher=Thames & Hudson |location=London; New York |edition=4th edition (revised) |isbn=0-500-27455-X |oclc=15895415}}
* {{cite journal |last=Díaz Díaz |first=Ruy |date=December 2006 |title=Apuntes sobre la aritmética Maya |url=http://www.scielo.org.ve/scielo.php?script=sci_arttext&pid=S1316-49102006000400007&lng=en&nrm=iso&tlng=es |format=online reproduction |journal=Educere |volume=10 |issue=35 |pages=621–627 |location=Táchira, Venezuela |publisher=[[University of the Andes, Venezuela|Universidad de los Andes]] |issn=1316-4910 |oclc=66480251|language=es}}
*Davidson, Luis J. “The Maya Numerals.” Mathematics in School, vol. 3, no. 4, 1974, pp. 7–7
*{{cite book |last=Thompson |first=J. Eric S. |author-link=J. Eric S. Thompson |year=1971 |title=Maya Hieroglyphic ting; An Introduction |series=Civilization of the American Indian Series, No. 56 |edition=3rd |location=Norman |publisher=University of Oklahoma Press |isbn=0-8061-0447-3 |oclc=275252}}

{{refend}}<!-- END biblio style -->

== External links ==
{{Commons category|Mayan numerals}}
* [https://lovasoa.github.io/maya_numerals_converter/ Maya numerals converter] - online converter from decimal numeration to Maya numeral notation.
* [http://www.archimedes-lab.org/numeral2.html Anthropomorphic Maya numbers] - online story of number representations.
* [http://www.babelstone.co.uk/Fonts/Mayan.html BabelStone Mayan Numerals] - free font for Unicode Mayan numeral characters.

{{Maya}}

{{DEFAULTSORT:Maya Numerals}}
[[Category:Maya science and technology|Numerals]]
[[Category:Mathematics of ancient history]]
[[Category:Numerals]]
[[Category:Numeral systems]]
[[Category:Maya script]]
[[Category:Vigesimal numeral systems]]