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Tuesday, June 16, 2015

Counting Systems: Maya

The great pyramid at the Mayan city Chichen Itza
(it was featured in one of our favorite blog posts)
The Region: The Mayan civilization can be split up into several distinct periods: the preclassic, classic, and postclassic. During the final period, explorers from Western Europe made their first contact with the Mayan people (around 1500 CE). 

Over the 3500 years that the Mayan culture was prominent, the lands under their control remained expanded and shrank several times. They were limited to the Yucatan Penninsula (present day, a part of Mexico) but had strong ties to the Olmec and Toltec cities in surrounding Mesoamerican lands.

The Numerals: In many ways, the Mayan numeral system is more intuitive than our own. They used a Base-20, or vigesimal, system.

Click on any image in the post to expand
These 20 symbols comprise all the digits needed to make any number. And, they seem relatively simple in design. The symbols for 1 through 4 are horizontal rows of dots, with the number of dots depicting the digits. Then, the symbol for 5 is a solid line, like a tally mark. Then, on top of the line for 5, more dots are added to make each progressing number. This continues until their symbol for 19, which has three lines of value 5 and 4 dots.

The symbol for 0 does not follow the rules. In fact, it couldn't possibly follow the same system, because the system is to represent a number with countable "tick marks." The Mayans had a symbol that we associate with equaling 0, and that symbol is in the shape of a shell. But the Mayans did not use their 0 as anything more than a placeholder. In fact, the use of 0 as a placeholder in Central America might predate the Mayans and go back to the Olmec culture.

What about when numbers larger than 19 had to be represented? The Base-20 system works very similarly to our decimal system. A number such as 56,091 is easy for us to understand, but we don't realize that we intuitively see this as 5 groups of 10,000 plus 6 (1,000) plus 0 (100) plus 9 (10) plus 1. The system that the Mayans used for large numbers would have been just as intuitive to them. The exact number I just used as an example is represented in Mayan numerals in the image directly above.

We write our numbers horizontally, with the largest place value at the far left. The Mayans wrote vertically, with the largest value at the top. Above, we have 4 symbols (an 7, 0, 4, and 11) written top to bottom. We know that the bottom value is in ones. Then the second value is multiplied by 20 and added to the ones. The third is multiplied by 20x20. And the top value is the 8,000s place, and 7 is multiplied by 20x20x20.

The Math: Due to the tally-mark nature of Mayan numbers, it is plain to see how they were used in simple equations. The numbers were easy to add and subtract: simply combine the numerals. If you have a group of five dots, exchange them for a line. If you have a group of four lines, exchange them for a dot in the next place value. See some exaples right below:

Mayans didn't just use math for simple transactions; they used complex numbers for projects relating to astrology and astronomy. The Mayan religious beliefs were tied closely to their calender, which they derived to be a 18 month (with 20 days each month) year, with 5 extra days- totalling to 365 days per year.

The Problems:  One significant problem is that the Mayans didn't just use these numerals, though they are the most simple to interpret. They also used a series of pictograms, where one image was equal to a significant number or value. There were several different images used to represent zero. 

The Mayans also had no concept of fractions, and no way to write them. We also have little knowledge of how they multiplied or came up with their fantastic approximations for astrological time spans. For example, they estimated 29.5308 days for the lunar month, compared with the modern value of 29.54059 days.

Thank you for reading! In case you missed them, I covered Ancient Chinese numerals and Babylonian numerals already. I plan to post about a more modern "suggestion" within the next week.


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