The Ancient Maya Concept of Fractions

The word fractions comes from “fractus,” the Latin word for broken.

It represents a part of a whole or any number of equal parts of the whole. Broken or fractured from the whole.

A common, vulgar, or simple fraction consists of an integer numerator, which is displayed above a line (or before a slash), and a non-zero integer denominator, displayed below (or after) that line. The numerator represents a number of equal parts and the denominator indicates how many of those parts make up a whole. For example, in the fraction ¾ , the numerator 3 on top, tells us that the fraction represents 3 equal parts, and the denominator, 4 on the bottom, tells us that 4 parts make up a whole.

In contrast to previous claims by researchers, the Maya were familiar with a notion of fractions or “parts of a whole.” To indicate parts in general, they used the term “tzuc” which literally means “part.” The Mayan words, “tu,” “can,” “tzucil,” and “ban cah,” equals the four parts of the World (cah), or the four quarters of the whole World.

For the notion “¼,” we find the expressions, “heb” (to open) and “u” (moon).

Some examples:

  • “heb u” = moon opening or open moon.

  • “hun heb u” = 1/4 moon or moon opening of ¼.

  • “ca heb u” = 2/4 moon or moon opening of ½.

  • “ox heb u” = 3/4 moon or moon opening of ¾.

For the notion of “½,” two possible applications can be found.

First, in distance:

  • “Tan coch” = half, in the middle.

  • “lub” = “legua” (5.5 km).

  • “tan coch lub” = half a “legua.”

  • “tan coch tu cappel lub” = in the middle of the second legua (5.5 km), or 1 ½ “legua.”

Secondly, in time divisions, such as:

  • “tan coch kin tu cappel” = in the middle of the second day = 1 ½ days.

“Xel” = dividing the unit in two and subtracting one part. Xel is in fact a negative fraction:

  • “xel u ca kin bé” = – ½ + 2 days = 1 1/2 days;

  • “xel u ca cuch” = – ½ + 2 loads = 1 1/2 loads;

  • “xel u cappel lub” = – ½ + 2 leguas = 1 1/2 legua;

  • “xel u yox katun” = – ½ + 3 katun = 2 1/2 katun;

  • “xel u ca kal” = – ½ x 20 + 2×20 = -10 + 40 = 30;

  • “xel y yox bak” = – ½ x 400 (bak) + 3 x 400 = 1300.

From these astronomical and time keeping divisional uses in the Mayan language, the best inference we can gather as the most common use of fractions would be quarters or fifths of a whole that reflects the use of the Maya counting system itself.

For example: a bar being a whole is divided into five equal parts, as five dots (single units) equals one bar (5 units).
<see: Ancient Maya Arithmetic for more information>
So, with a whole bar (representing the value of 5) being broken down to single unit values, we’re able to assume the Maya also had a regular use of 5th’s ( 1/5, 2/5, 3/5, 4/5, and 5/5 = 1 whole ).

However, a better inference would be to conclude that fractional systems were in the Maya vigesimal system, no different that regular counting. Meaning the entirety of the Maya vigesimal system is already fractional.

There are two kinds of scientists;
1. Those who can extrapolate from incomplete data

So how is it possible to write and use fractions in the Maya system?

On the second tier, the single dot (value = 20) would be the whole number (or denominator).
On the first level, below it (count 0 – 19), would be the fractions (numerators) of the upper whole.

In this example we have the Maya count for 20 (201) represented as a fraction whole of 1. We then express this same whole number as a fraction using Latin numerals next to it.

Maya fractions

With this example, we could extrapolate by saying a good way to indicate a Maya fraction would be to use the shell (used for the null or zero placeholder) as the placeholder for an incomplete whole when written as a fraction. We use a shell on top to represent an incomplete whole of the dot, whereas when we use a shell below the dot on the bottom tier to represent a whole count of 20 on the second tier. To represent the fraction, we remove the dot on the second tier and replace it with a shell and the shell is now a placeholder indicating an incomplete whole number.

For example, the fraction 1/20th could be expressed in Maya as:

Using the shell (0) as a placeholder for the dot of the whole of 20. The bottom number, the numerator, is the fraction amount of the whole (201).

Addition of fractions is carried out the same manner as regular addition by simply adding the sums together.

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written by Njord Kane © 2016 Spangenhelm Publishing

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