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 AD — were totally awesome. 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.” Plus, they created one of the most sophisticated mathematical system ever developed in the Americas.
According to Lumen Learning:
“There were two numeral systems developed by the Mayans — one for the common people and one for the priests. Not only did these two systems use different symbols, they also used different base systems. For the priests, the number system was governed by ritual. The days of the year were thought to be gods, so the formal symbols for the days were decorated heads... Since the basic calendar was based on 360 days, the priestly numeral system used a mixed base system employing multiples of 20 and 360. This makes for a confusing system, the details of which we will skip.”
Instead, we are going to focus on the simpler — more defined — math system that was relied on by both the people and the clergy. The Maya (and the Aztecs) used a vigesima or base-20 notational system for representing real numbers, as that is the total number of fingers and toes that most people have to count on; and, because there are five fingers on each hand and food, some have argued that it’s a base-5 system as well. So, if ours is a base-10 numeral system with ten possible digits for each placeholder [0 - 9], theirs was a numeral system that had nineteen digits for each placeholder [0 - 19].
The Maya represented numbers as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, and J. Their numbers, including calendar dates, were written using three simple symbols: a dot for one, a bar for five, and a shell with the plastron uppermost representing zero. "Like our system, it is positional, meaning that the position of a numeric symbol indicates its place value." But, “unlike our system, where the ones place starts on the right and then moves to the left, the Mayan systems places the ones on the bottom of a vertical orientation and moves up as the place value increases."
When writing numbers, the bars are placed horizontally and the dots are placed on top of those. Every group of five dots becomes one bar, with a maximum of four dots appearing over a single bar. A maximum of three bars may appear in a single space, as such, four bars are converted to one dot and placed in the next place up. In other words,
"numbers larger than 19 were represented by the same kind of sequence, but a dot was placed above the number for each group of 20." So, 32 "consisted of the symbols for 12, with a dot on top of the whole thing representing 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 the evidence suggesting that the Maya were working with sums up to the hundreds of millions, and with dates so large it took several lines just to represent them. It’s astounding to think that the Maya, using no tools other than sticks, were able to make calculations that were often more accurate than those being made in the West.
See? Awesome.