Mayan Numerals Converter
Enter a decimal number to see its Mayan numeral representation, complete with a positional breakdown showing dots, bars, and the shell zero. Switch to Long Count calendar mode, or reverse the conversion by entering Mayan positional digits to recover the decimal value. Results update instantly as you type.
How the Mayan numeral system works
The ancient Maya developed one of the most sophisticated numeral systems in the pre-Columbian Americas. It is a positional, vigesimal (base-20) system, meaning each digit position represents a power of 20 rather than a power of 10 as in our familiar decimal system. Three symbols do all the work: a dot representing 1, a horizontal bar representing 5, and a stylised shell or oval representing 0. By stacking combinations of dots and bars, any digit from 0 to 19 can be written in a single vertical column. Larger numbers are then written in stacked tiers, with the most significant tier at the top. The existence of a dedicated zero symbol was a remarkable intellectual achievement: most ancient cultures had no concept of zero as a number in its own right, but the Maya were using it systematically for positional arithmetic long before European mathematicians adopted the concept from Indian and Arabic scholars.
Classic base-20 versus Long Count calendar mode
In the pure (classic) base-20 system, each positional tier is exactly 20 times the one below it, giving multipliers of 1, 20, 400, 8,000, 160,000, and so on. The Mayan Long Count calendar uses a slight modification: the third tier (position 2) counts 18 twenties, or 360, rather than 400. This was done so that 360 closely approximates the length of the solar year, making the calendar more practical for astronomical and agricultural purposes. The Long Count positions therefore run 1, 20, 360, 7,200, and 144,000. One full Long Count cycle of 13 baktuns (13 x 144,000 = 1,872,000 days) is roughly 5,125 years, and the completion of such a cycle on December 21, 2012, attracted widespread attention. This converter supports both systems: switch between them using the numeral system selector.
Converting decimal to Mayan: the division algorithm
To convert a decimal number to Mayan numerals, repeatedly divide by 20 (or by the appropriate Long Count place value) and record the remainders. Start with the largest place value that fits into your number and record the quotient as the top-tier digit, then continue with the remainder at the next-lower tier. For example, to convert 1,507: 1,507 divided by 400 gives 3 remainder 307; 307 divided by 20 gives 15 remainder 7; 7 divided by 1 gives 7. The three tiers are therefore 3, 15, and 7, written from top to bottom. Tier 3 (digit 3) is three dots. Tier 2 (digit 15) is three bars. Tier 1 (digit 7) is one bar plus two dots. This calculator automates that division process and displays each tier as dots and bars so you can verify every step.
Reading Mayan numerals: the reverse conversion
To convert Mayan numerals back to decimal, read each tier from top to bottom, note the digit (counting dots as 1 each and bars as 5 each), multiply by the appropriate place value, and sum the results. Using the 1,507 example: tier 2 holds 3 (three dots), multiplied by 400 gives 1,200; tier 1 holds 15 (three bars), multiplied by 20 gives 300; tier 0 holds 7 (one bar and two dots), multiplied by 1 gives 7. Total: 1,200 + 300 + 7 = 1,507. This converter lets you enter Mayan positional digits directly (as comma-separated numbers from 0 to 19) and instantly see the decimal result, the glyph representation, and the full breakdown. Switch the direction selector to "Mayan to Decimal" and type in your digits.
Mayan place-value positions (classic base-20)
| Tier | Multiplier | Long Count multiplier | Example: digit 3 |
|---|---|---|---|
| 0 | 1 | 1 | 3 |
| 1 | 20 | 20 | 60 |
| 2 | 400 | 360 | 1,080 |
| 3 | 8,000 | 7,200 | 21,600 |
| 4 | 160,000 | 144,000 | 432,000 |
Each positional tier is 20 times the one below. Long Count modifies tier 2 to 360 for calendar alignment.
Frequently asked questions
What base does the Mayan numeral system use?
The Mayan system is base-20 (vigesimal), meaning each positional tier is worth 20 times the one below it. For comparison, our decimal system is base-10 and binary is base-2. Scholars believe the Maya likely counted on both fingers and toes, which would naturally lead to a base-20 counting scheme.
How did the Maya represent zero?
The Maya used a shell-shaped or oval glyph to represent zero, which functioned as a true positional placeholder exactly as our digit 0 does today. This was one of the earliest known uses of zero as a number rather than mere absence. The Maya were using zero systematically by at least 350 CE, centuries before it became standard in Europe.
What is the difference between the classic system and the Long Count system?
Classic base-20 uses place values of 1, 20, 400, 8,000, 160,000, and so on, each exactly 20 times the previous. The Long Count calendar system modifies the third position: instead of 400 (20 x 20), it uses 360 (18 x 20) to approximate the 360-day Mayan calendar year. This makes the Long Count better suited for counting days across long time periods. All other positions in the Long Count remain strictly base-20.
How do I write 19 in Mayan numerals?
The digit 19 occupies a single tier and is written as three horizontal bars (each worth 5, totalling 15) with four dots on top (each worth 1), giving 15 + 4 = 19. It is the largest value that fits in one positional tier. The next value, 20, requires a second tier: a dot in tier 1 (worth 20) and a shell (zero) in tier 0.
What is the largest number this converter handles?
This converter handles values up to 3,199,999 in classic base-20 mode, which is the maximum representable with five positional tiers (20 to the power of 5 minus 1). In Long Count mode the practical limit is set by the same five-tier structure. Numbers used in Mayan astronomy and calendrics, such as the Long Count cycle of 1,872,000 days, are comfortably within this range.
Can the Mayan system handle fractions?
The historical record does not show widespread Mayan use of fractions in the way we use them today. The Mayan numeral system as documented is designed for whole numbers. Astronomical calculations that required precision were handled by tracking cycles and remainders rather than by introducing fractional notation. This converter therefore works with whole (integer) values only.