The Maya Created the Most Sophisticated Math System in the World, This Is How It Worked

by Liz Publika

The Maya — who occupied and thrived in a nearly continuous stretch of territory in southern Mexico, Guatemala, and northern Belize from about 1500 BCE to 900 CE — developed one of the most sophisticated civilizations in the ancient Americas. They formed one of the most densely populated and culturally dynamic societies in the world, excelled at agriculture, built great stone buildings and pyramid temples, worked gold and copper, and used a form of hieroglyphic writing that has now largely been deciphered. They also created one of the most advanced mathematical systems ever developed in pre-Columbian America.

Maya Hieroglyphs Plate 32 of the Postclassic Dresden Codex

The Maya developed two distinct numeral systems — one used by common people and another employed by the priestly class. The priestly system was governed by ritual and incorporated the 360-day calendar, using a mixed base system with multiples of 20 and 360. According to Lumen Learning, the days themselves were considered divine, with formal symbols appearing as decorated heads. Because of its complexity and ritual-specific nature, we'll focus instead on the more consistent system used in everyday life.

The Maya (and later the Aztecs) used a vigesimal or base-20 notational system for representing real numbers, derived from the total number of fingers and toes that people possess. Some scholars argue it also functions as a base-5 system due to the five digits on each hand and foot. In contrast to our base-10 system with ten possible digits for each placeholder [0-9], the Mayan system had 20 positions for each placeholder [0-19].

The Maya represented these 20 values as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, and J using just three simple symbols: a dot for one, a horizontal bar for five, and a shell symbol with the plastron uppermost representing zero. Like our system, it was positional, meaning that a numeric symbol's position indicated its place value. However, unlike our system, where the ones place starts on the right and moves left, the Mayan system placed the ones at the bottom of a vertical orientation and moved upward as place value increased.

When writing numbers, the bars were placed horizontally with dots positioned above them. Every group of five dots became one bar, with a maximum of four dots appearing over a single bar. A maximum of three bars could appear in a single space; four bars converted to one dot in the next place up.

Numbers larger than 19 were represented by adding additional places. For example, 32 consisted of the symbols for 12, with a dot placed above the entire grouping to represent an additional group of 20. The system could thus be extended infinitely.

The fact that the concept of zero as a placeholder was already in use by the Maya as early as 36 BCE is absolutely remarkable, considering that most of the world's civilizations had not yet developed it. Even more impressive is evidence suggesting that the Maya were working with sums reaching into the hundreds of millions, with dates so large they required several lines just to represent them. The Maya achieved calculations of exceptional accuracy using no tools beyond sticks, often surpassing the precision of contemporary calculations in the West.

Note* Images are available via Fair Use or are in the Public Domain.