Mayan Numbers

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.

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.

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

1. Read each level on its own: bars times 5 plus dots.
2. Write down the level value: from the bottom 1, 20, 400, …
3. 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.

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

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).

Frequently asked questions

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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