# Maya numerals

Maya numerals are the number symbols of the ancient Maya, written in a base-20 (vigesimal) positional system using three signs: a dot for one, a horizontal bar for five, and a shell-like glyph for zero. The system reached its fullest development in the Long Count calendar, where a modified third place value of 360 rather than 400 tied the notation to the solar year.<sup>[1](https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/)</sup><sup> • </sup><sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC8022160/)</sup>

| Key fact | Detail |
|---|---|
| Symbols | A dot for 1, a bar for 5, and a shell for 0; numbers 0 to 19 are built by stacking up to four dots above up to three bars<sup>[1](https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/)</sup><sup> • </sup><sup>[3](https://www.ancientamericas.org/sites/default/files/CKguidebook_english.compressed.pdf)</sup> |
| Base | Vigesimal (base 20), almost certainly from counting on both fingers and toes; five plays a major sub-role<sup>[1](https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/)</sup> |
| Place values | Pure base 20 gives 1, 20, 400, 8,000; the calendrical Long Count replaces the third place with 18 × 20 = 360 and continues 18 × 20², 18 × 20³, and so on<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC8022160/)</sup> |
| Earliest dated use | Stela 2 at Chiapa de Corzo carries the Long Count date 7.16.3.2.13, corresponding to 36 BCE<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC8022160/)</sup> |
| Zero | A placeholder zero appears in the earliest positional inscriptions; the Maya later developed a cardinal zero used in calculation, symbolized by the shell in the codices<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC8022160/)</sup> |
| Worked example | The Tikal plate date [8;14;3;1;12] equals 12 + 1 × 20 + 3 × 360 + 14 × 7,200 + 8 × 144,000 = 1,253,912 days, placing its carving in 320 AD<sup>[1](https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/)</sup> |

## Symbols and digit forms

Each digit from 0 to 19 has a bar-and-dot form: dots add units, bars each count five, so 19 is four dots over three bars and no single position can hold more.<sup>[3](https://www.ancientamericas.org/sites/default/files/CKguidebook_english.compressed.pdf)</sup> In the codices, four signs served as numerals: the shell for zero, the dot for one, the bar for five, and a moon sign for 20.<sup>[4](https://mayacodices.org/calendar.asp)</sup>

Numerals were arranged in columns and read from bottom upward, each level taking the power of 20 corresponding to its height.<sup>[5](https://www.gutenberg.org/files/57540/57540-h/57540-h.htm)</sup> The Smithsonian's educational materials render the same system with beans for ones, sticks for fives, and shells for zero, reflecting the likely origin of the dot in a pebble or cacao bean and the bar in a counting stick.<sup>[6](https://maya.nmai.si.edu/sites/default/files/resources/lesson-plans/Lets%20Multiply%20-%20Maya%20Style.pdf)</sup><sup> • </sup><sup>[1](https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/)</sup>

## Positional structure and the calendrical irregularity

In a true base-20 system the places would count units, twenties, four-hundreds, and so on. The Maya system begins that way, but in the version preserved in the [Dresden Codex](https://www.edgepedia.org/dresden-codex) the third place counts 360s instead of 400s, and the higher places revert to multiples of 20, giving place values 18 × 20², 18 × 20³, and onward.<sup>[1](https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/)</sup> In modern notation a Long Count number reads

\[ N = d_{k}(18 \times 20^{k-1}) + \cdots + d_{3}(18 \times 20^{2}) + d_{2}(18 \times 20) + d_{1} \cdot 20 + d_{0}. \]

This makes it a non-power positional number representation system, because the place values are not powers of a single base.<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC8022160/)</sup> The irregularity exists because 360 days approximates the solar year far better than 400: one tun is 18 winals of 20 days, one k'atun is 20 tuns (7,200 days), and one b'ak'tun is 20 k'atuns (144,000 days, about 400 years).<sup>[7](https://onlinelibrary.wiley.com/doi/full/10.1111/1467-968X.12320)</sup> A Long Count date such as 9.19.10.0.0 therefore denotes nine b'ak'tuns, 19 k'atuns, 10 tuns, zero winals, and zero k'ins.<sup>[8](https://www.livescience.com/archaeology/anthropologist-claims-hand-positions-on-1-300-year-old-maya-altar-have-a-deeper-meaning)</sup>

The Palenque building date [9;8;9;13;0] evaluates to 1,357,100 days from the creation base date, corresponding to 603 AD.<sup>[1](https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/)</sup> Because the third place is 360 rather than 400, the system lacks some properties of a true positional base; for example, dropping the final digit of a written number does not simply divide it by 20.<sup>[1](https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/)</sup>

## The Long Count and the calendars it served

The Long Count counted days linearly from a fixed creation date, most often taken as 12 August 3113 BC, giving every day a unique position in deep time; Classic-period scribes wrote of periods millions of years into the past and thousands into the future.<sup>[1](https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/)</sup><sup> • </sup><sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC8022160/)</sup> It worked alongside the 260-day ritual Tzolk'in (13 numbered days cycling against 20 named days) and the 365-day Haab (18 months of 20 days plus five unlucky wayeb days); the two together form the [Calendar Round](https://www.edgepedia.org/calendar-round), which repeats every 52 years.<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC8022160/)</sup><sup> • </sup><sup>[7](https://onlinelibrary.wiley.com/doi/full/10.1111/1467-968X.12320)</sup> The full Long Count cycle of 13 b'ak'tuns spans 5,125.37 years.<sup>[9](https://academic.oup.com/edited-volume/27948/chapter-abstract/211903925)</sup>

Monumental Long Count dates appear from the first century BC to the mid-ninth century AD, carved on hundreds of stelae in public plazas alongside records of rulers' accessions, captures, and alliances.<sup>[9](https://academic.oup.com/edited-volume/27948/chapter-abstract/211903925)</sup>

## Zero

The zero is the system's most discussed feature. MacTutor's history calls the shell-glyph zero and the positional structure the system's surprising and advanced features, while noting that the reason for the shell form cannot be explained.<sup>[1](https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/)</sup> The earliest Long Count inscriptions already require a zero sign where a place is empty, as on Stela 2 at Chiapa de Corzo in 36 BCE, strong evidence that the Maya knew zero by that date.<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC8022160/)</sup><sup> • </sup><sup>[10](https://www.mayaexploration.org/pdf/ZeroInPreColumbianAmericas_CatepillanMendez_Sept2017.pdf)</sup>

A 2021 analysis argues that positional notation and zero did not have to be invented together. The Maya were comfortable with redundant, non-unique representations, and some inscriptions, such as the date 9.13.20.0.0 at Pixoy, write both a twenty and a zero as digits side by side. On this reading the digits first ran from 1 to 20 in a bijective system, and zero emerged later as the digit 20 shifted meaning, with the explicit shell zero becoming standard by the time the surviving codices were written.<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC8022160/)</sup> A distinction is drawn between a placeholder zero, which prevents positional confusion, and a true numeral zero that can be operated on; the Maya are credited with developing the cardinal zero used in calculation.<sup>[11](https://mayan.org/civilization/mathematics/)</sup>

## Evidence, sources, and debate

All extant written evidence for large numbers relates to the calendar; some authors claim a pure vigesimal system was also used for non-calendric purposes, but no Maya document showing it survives. What is known of everyday counting comes from Bishop Diego de Landa's sixteenth-century account of cacao bean counting in Yucatán, which describes counting by fives to twenty, by twenties to four hundred, and onward.<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC8022160/)</sup> Georges Ifrah, the French historian of numbers, argued that alongside the priests' calendrical system the merchants used a true base-20 system with special symbols for 20, 400, and 8,000, though no trace of it remains.<sup>[1](https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/)</sup>

The [numeral system](https://www.edgepedia.org/numeral-system)'s origins predate the Maya: a form of vigesimal counting was present from the first Olmec civilization onward, and the earliest known Long Count inscriptions, at Chiapa de Corzo and [Tres Zapotes](https://www.edgepedia.org/tres-zapotes), were erected in the epi-Olmec period.<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC8022160/)</sup> Not all scholars accept the positional reading. Stephen Chrisomalis, author of *Numerical Notation: A Comparative History*, has argued there was no Maya positional numerical notation system because the Long Count is attested only for measuring time; the authors of the 2021 study disagree, holding that the Long Count possesses the formal structure of a positional system regardless of its restricted use.<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC8022160/)</sup>


## References

1. [Mayan mathematics, MacTutor History of Mathematics, University of St Andrews](https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/)
2. [Non-Power Positional Number Representation Systems, Bijective Numeration, and the Mesoamerican Discovery of Zero, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC8022160/)
3. [The Vigesimal (Base 20) Numbering System, Ancient Americas guidebook](https://www.ancientamericas.org/sites/default/files/CKguidebook_english.compressed.pdf)
4. [Maya Calendar: numerals in the codices, mayacodices.org](https://mayacodices.org/calendar.asp)
5. [Daniel G. Brinton, A Primer of Mayan Hieroglyphics, Project Gutenberg](https://www.gutenberg.org/files/57540/57540-h/57540-h.htm)
6. [Let's Multiply, Maya Style, Smithsonian National Museum of the American Indian](https://maya.nmai.si.edu/sites/default/files/resources/lesson-plans/Lets%20Multiply%20-%20Maya%20Style.pdf)
7. [The Ancient Maya Script of Hand Forms Embedded in Figural Art, Transactions of the Philological Society (2025)](https://onlinelibrary.wiley.com/doi/full/10.1111/1467-968X.12320)
8. [Anthropologist claims hand positions on 1,300-year-old Maya altar have a deeper meaning, Live Science](https://www.livescience.com/archaeology/anthropologist-claims-hand-positions-on-1-300-year-old-maya-altar-have-a-deeper-meaning)
9. [Mesoamerican Calendars and Archaeoastronomy, Oxford Handbook of Mesoamerican Archaeology](https://academic.oup.com/edited-volume/27948/chapter-abstract/211903925)
10. [Zero in Pre-Columbian Americas, Maya Exploration Center](https://www.mayaexploration.org/pdf/ZeroInPreColumbianAmericas_CatepillanMendez_Sept2017.pdf)
11. [Maya Mathematics: How the Ancient Maya Invented Zero, Mayan.org](https://mayan.org/civilization/mathematics/)
12. [Proposal to encode Mayan numerals, Unicode Consortium](https://www.unicode.org/wg2/docs/n4804-mayan-numerals.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Number systems › Integers and rational numbers*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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