Mayan Numerals Converter (base-20 dots, bars and shell ↔ decimal)

Understanding the Mayan numerals converter — concept and principle

This converter writes a whole number in Maya numerals, base 20 with dots, bars and a shell for zero, or turns base-20 digits back into a decimal number. It shows the digits from top to bottom, how each is drawn, and the place values. Use it for history of mathematics, number-system lessons and puzzles.

How it is calculated

Decimal → base 20: divide by 20 repeatedly; the remainders, read from last to first, are the digits.

Value = Σ digit × 20^place, with places 1, 20, 400, 8,000, … from the bottom up.

• Each digit 0–19 is drawn with up to four dots (• = 1) above up to three bars (▬ = 5); zero is a shell.
• Digits are stacked vertically, the highest place at the top.

The principle

The Maya system is positional like ours, but with base 20, perhaps from counting on fingers and toes. Within one place, the digit itself is a tally of fives and ones, so only three symbols are needed. The Maya had a true zero symbol, the shell, which made the place-value system unambiguous. In calendar counts the third place was worth 360 (20 × 18) instead of 400; this converter uses the pure base-20 system.

Worked example

Default inputs: decimal → Maya numeral, 2026.

1. 2026 ÷ 20 = 101 remainder 6; 101 ÷ 20 = 5 remainder 1; 5 ÷ 20 = 0 remainder 5.
2. Digits, top place first: 5, 1, 6, using 3 places.
3. Check: 5 × 400 + 1 × 20 + 6 × 1 = 2,000 + 20 + 6 = 2,026.
4. Drawn top to bottom: 5 → ▬, 1 → •, 6 → • ▬ (one dot above one bar).

How to read the result

Read the stack from top to bottom like digits from left to right. A place holding zero is drawn with the shell, so 400 is a dot on top of two shells. Compare with our decimal system: the same idea of place value, just grouping by twenty instead of ten.

Common mistakes

• Reading the stack from bottom to top.
• Using more than four dots or three bars in one place.
• Forgetting the zero shell for an empty place.
• Applying the calendar's 360 rule to ordinary counting.

Frequently asked questions

How did the Maya write 20?

A dot in the twenties place above a shell (zero) in the ones place.

Did the Maya have zero?

Yes, they used a shell-shaped symbol for zero.

Why base 20?

It is often linked to counting on fingers and toes, though the exact origin is not known.

Sources: OpenStax, Contemporary Mathematics (2023), §4.2 Early Numeration Systems