Mayan numbers are written with just three symbols: a dot is 1, a bar is 5 and a shell is 0. Each level shows a number from 0 to 19. Bigger numbers are written as a stack of levels, because the Maya counted in twenties: the Mayan number system has base 20 and is called vigesimal.
Here you learn to write them, to convert them to decimal numbers and back, and to add them. You get charts for 1 to 100 and 100 to 1000, plus practice problems with answers.
The three symbols: dot, bar and shell
A dot is worth 1. Five dots take up a lot of room, so there is a bar for 5. For zero the Maya drew a shell. With these symbols you can write any number from 0 to 19. Since 2018 they are part of Unicode (version 11.0); here you see them as drawings.
Mayan number symbols and glyphs
The numerals on this page are the bar-and-dot numbers, the form used in everyday writing, in books and on many monuments. On carved inscriptions you also find head-variant numerals: each number from 0 to 19 has a portrait glyph of a god or a figure, so a date could be carved as a series of small faces instead of dots and bars. Both forms stand for the same numbers. They are not letters: Mayan script writes words with syllable and word signs, and numbers are one part of it, often placed next to the day and month signs of the calendar. The Mayan number glyphs are therefore best understood as a counting system with its own shapes, in which a dot is 1, a bar is 5 and the shell is zero.
Place value: level by level, from bottom to top
In the number 345 the digits are worth 1, 10 and 100. In Mayan numbers each position is a level, and each level is worth 20 times the level below it. The ones are at the bottom, the twenties above them, then the 400s (20 · 20) and the 8,000s (20 · 20 · 20): these are the powers of 20. The highest level is at the top.
The shell keeps a level empty. A dot above a shell is 1 · 20 + 0 = 20. Without the shell, the single dot would only mean 1. That is why zero matters so much in a place-value system.
Pure base 20 and the calendar count
Almost all Mayan numbers that survive are calendar dates (the Long Count). There the third level is not worth 400 but 360: 18 winals of 20 days make a tun of 360 days, close to one year. After that the system goes back to multiplying by 20.
| Level (from the bottom) | Pure base 20 | Calendar count |
|---|---|---|
| 1 | 1 | 1 (k’in) |
| 2 | 20 | 20 (winal) |
| 3 | 400 (20 · 20) | 360 (18 · 20), tun |
| 4 | 8,000 | 7,200 (k’atun) |
| 5 | 160,000 | 144,000 (b’ak’tun) |
The symbols are the same, only the values of the higher levels differ. The levels 2 | 10 | 0 mean 2 · 400 + 10 · 20 = 1,000 in the pure system, but 2 · 360 + 10 · 20 = 920 in the calendar count. For 1,000 days you need 2 | 14 | 0. On this page we use the pure system unless it says “calendar.”
How to convert Mayan numbers: in both directions
From a Mayan number to a decimal number
- Read each level on its own: bars times 5 plus dots.
- Write down the level value: from the bottom 1, 20, 400, …
- Multiply each level by its value and add.
From a decimal number to a Mayan number
Divide the number by 20 again and again and write down each remainder. The first remainder goes in the bottom level, the next one level higher. Stop when the quotient is 0. Each remainder is a number from 0 to 19 that you draw with dots and bars.
Example 365: 365 : 20 = 18 remainder 5 and 18 : 20 = 0 remainder 18. The remainders are 5 (bottom) and 18 (top): 1 bar at the bottom, 3 bars and 3 dots on top. Check: 18 · 20 + 5 = 365.
Example 1,000: 1,000 : 20 = 50 remainder 0, 50 : 20 = 2 remainder 10 and 2 : 20 = 0 remainder 2. The shell at the bottom, 2 bars in the middle, 2 dots on top. Check: 2 · 400 + 10 · 20 + 0 = 1,000.
To check your results you can use the number system converter (interface in German): it shows how the Maya wrote any number you enter.
Mayan numbers 1 to 100 and 100 to 1,000
Up to 19 one level is enough, so for 1 to 20 use the figure above (20 is a dot above a shell). From 20 on you need two levels, from 400 on three. The top level counts how many times 20 fits in. From 20 to 50, for example, 21 is 1 | 1, 25 is 1 | 5, 37 is 1 | 17 and 50 is 2 | 10.
Every multiple of 20 has the shell at the bottom, and 30, 50, 70 and 90 have two bars at the bottom. The table lists some numbers from 100 to 200 and the hundreds up to 1000, with the calculation.
| Number | Calculation | Levels (top | bottom) |
|---|---|---|
| 100 | 5 · 20 + 0 | 5 | 0 |
| 150 | 7 · 20 + 10 | 7 | 10 |
| 199 | 9 · 20 + 19 | 9 | 19 |
| 200 | 10 · 20 + 0 | 10 | 0 |
| 399 | 19 · 20 + 19 | 19 | 19 |
| 400 | 1 · 400 + 0 · 20 + 0 | 1 | 0 | 0 |
| 800 | 2 · 400 + 0 · 20 + 0 | 2 | 0 | 0 |
| 1,000 | 2 · 400 + 10 · 20 + 0 | 2 | 10 | 0 |
The next table lists numbers people look up often, including the 360 and 365 of the calendar year and the year 2026. In the pure base 20 system 360 is 18 | 0, in the calendar count it is 1 | 0 | 0.
| Number | Levels (top | bottom) | Calculation | Symbols from top to bottom |
|---|---|---|---|
| 10 | 10 | 10 | 2 bars |
| 12 | 12 | 12 | 2 bars and 2 dots |
| 15 | 15 | 15 | 3 bars |
| 19 | 19 | 19 | 3 bars and 4 dots |
| 20 | 1 | 0 | 1 · 20 + 0 | 1 dot; shell |
| 25 | 1 | 5 | 1 · 20 + 5 | 1 dot; 1 bar |
| 30 | 1 | 10 | 1 · 20 + 10 | 1 dot; 2 bars |
| 40 | 2 | 0 | 2 · 20 + 0 | 2 dots; shell |
| 50 | 2 | 10 | 2 · 20 + 10 | 2 dots; 2 bars |
| 60 | 3 | 0 | 3 · 20 + 0 | 3 dots; shell |
| 70 | 3 | 10 | 3 · 20 + 10 | 3 dots; 2 bars |
| 80 | 4 | 0 | 4 · 20 + 0 | 4 dots; shell |
| 90 | 4 | 10 | 4 · 20 + 10 | 4 dots; 2 bars |
| 360 | 18 | 0 | 18 · 20 + 0 | 3 bars and 3 dots; shell |
| 365 | 18 | 5 | 18 · 20 + 5 | 3 bars and 3 dots; 1 bar |
| 2,026 | 5 | 1 | 6 | 5 · 400 + 1 · 20 + 6 | 1 bar; 1 dot; 1 bar and 1 dot |
Two levels reach up to 399, three levels up to 7,999 and four levels up to 159,999.
How to add Mayan numbers
Mayan writing is very handy for addition. You push the dots and bars of both numbers together, level by level from the bottom up, and then regroup: 5 dots become 1 bar, and 4 bars become 1 dot in the next level up. For subtraction you remove symbols and, if needed, trade one bar for 5 dots.
Common mistakes
Practice problems with answers
Use the two practice sets as a worksheet.
A short history of the Mayan number system and zero
The Maya wrote their numbers on stone monuments, on pottery and in folding books made of bark paper. The dot and the bar, like the Long Count, already existed in Mesoamerica before the Maya cities reached their peak. The earliest known Long Count date is on Stela 2 at Chiapa de Corzo and corresponds to 36 BC. More about the calendar and zero: Mathematik der Maya (in German).
The Mayan numeral system is one of several place value systems that people invented independently. The Babylonians counted in sixties (see Babylonian numerals), the Maya in twenties, and the system we use today, with ten digits and a zero, came from India (see Hindu-Arabic numerals). Systems without place value are shown on Egyptian numerals and Roman numerals. The overview of all of them is on number systems, and the whole story is in the history of mathematics.
Frequently asked questions
How do you write Mayan numbers?
With a dot (1), a bar (5) and a shell (0). One level shows 0 to 19; for larger numbers you stack levels: ones at the bottom, twenties above, then 400s.
How do you convert Mayan numbers to regular numbers?
Multiply each level (bars times 5 plus dots) by 1, 20, 400, … from the bottom up and add: 2 | 5 is 2 · 20 + 5 = 45.
How do you write 20, 100 and 1,000 in Mayan numerals?
20 is a dot above a shell, 100 is a bar above a shell (5 · 20) and 1,000 is the levels 2 | 10 | 0: 2 dots, 2 bars and a shell.
What do the dot, the bar and the shell mean in Mayan numbers?
The dot is 1, the bar is 5 and the shell is zero. One level holds at most 4 dots and 3 bars.
Is there a Mayan numbers converter?
Yes. The number system converter on this site (interface in German) shows, for any number you enter, how the Maya wrote it with dots, bars and the shell, and converts to base 20 on request.
What is the Mayan counting system (Mayan number system)?
It is a place value system with base 20, also called vigesimal, that uses only three symbols: a dot (1), a bar (5) and a shell (0). Each level holds a number from 0 to 19, and the levels are worth 1, 20, 400 and so on from the bottom up. See place value.
How do you write the Mayan numbers from 1 to 20?
The numbers 1 to 4 are one to four dots, 5 is one bar, 6 to 9 are a bar with one to four dots, 10 is two bars, 15 is three bars and 19 is three bars with four dots. The number 20 is a dot above a shell. The drawing of 0 to 19 shows all of them.
Did the Maya have a zero?
Yes. They used a shell-shaped sign for zero, as a placeholder in the calendar count and as the sign of a completed cycle. The oldest known Long Count date (36 BC, Stela 2 at Chiapa de Corzo) contains no zero itself, but its fixed place system needs a mark for empty places; in Classic inscriptions and in the codices the zero sign is well documented. See the short history.
How did the Maya count?
In twenties. The counting words, such as k’al for 20, bak’ for 400 and pik for 8,000 in Yucatec Maya, follow base 20. A link to counting with fingers and toes is often suggested, but the reason is not recorded.
What are the main facts about the Mayan number system (for kids and KS2)?
It has base 20, only three symbols (dot 1, bar 5, shell 0), and numbers are written in levels from bottom to top. The Maya had a zero sign long before Europe. The signs and glyphs are in Mayan number symbols and glyphs.
How can children learn Mayan numbers?
With beans for dots and sticks for bars. Lay out the number on two or three levels, then practice regrouping: five beans become a stick. The practice problems with answers work as a worksheet.
Sources and further reading
- Michael D. Coe: Breaking the Maya Code, Thames & Hudson, London 1992 (several revised editions).
- MacTutor History of Mathematics: Mayan mathematics
- Wikipedia: Maya numerals
- David and Alejandra Bolles: A Grammar of the Yucatecan Mayan Language, FAMSI
- Wikipedia: Mayan Numerals (Unicode block)
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