The Mayan Number System: Understanding the Mayan Calendar

December 21, 2012

We made it!

Given that you are reading this then it appears that the dire warnings about the end of the world occurring on December 21, 2012, did not come to pass. Congratulations, humanity!

As almost everyone is aware (to the point of which certain government agencies felt the need to reassure people that the end is not nigh) we have been warned that the world will end as a result of a major turning point in the ancient Mayan Long Count Calendar.

This concern was similar to people’s fears related to our modern calendar’s “turning the page” on January 1, 2000. But of course that was a fear about how our computer systems would “feel” about that particular calendar event, and fortunately many of them didn’t seem to mind too much.

Mathematics of counting days

In fact there is some interesting mathematics behind this particular calendar event. And given that you have lived through this catastrophe, perhaps you and your students might want to ponder the math behind this. It’s a good way to gain more of an appreciation of our Hindu-Arabic Base 10 positional number system (which many of us take for granted).

We are all quite used to our classic base 10 counting system, where the position of each of the 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 specifies its value in a typical number. For example, in the number 3,111 the digit 3 has a value quite different from its value in the number 1,131, representing 3 thousands in the first, and just 3 10’s in the second. It’s part of the “magic” of our modern positional system.

Things progress orderly by powers of 10: 3 then 30, then 300, all represent 10-fold increases as the digit 3 passes from the 1’s to the 10’s to the 100’s position.

Mayan number system

In the Mayan Long Count calendar, days were counted, starting with the supposed creation of the world of humans (a bit over 5,000 years ago). In typical Mayan number systems, instead of counting with base 10, the count was done essentially with a base 20 system. Instead of positions going up 10-fold, they increased 20-fold.

Thus the numbers 3, 30, and 300 represented 3, then 3 times 20, then 3 times 20 times 20, or simply 3 times 400, which is 20 squared. In Mayan counting 123 didn’t mean one 100, two 10’s, and and three 1’s. Rather, it meant one 400, two 20’s, and three 1’s (that would be 443 in our usual base 10 system). This is called a vigesimal (or base 20) system, as opposed to our decimal (base 10) system.

Question: Why did they use 20 digits? And likewise, why did we settle on 10?

Answer: Most likely it has to do with our fingers (10 of them). And probably it’s the case that some earlier cultures included their toes as well (20 fingers and toes). Perhaps this happened more often in warmer climates where toes were more likely to be exposed. Go to another planet with intelligent life forms, and they very likely will be counting using entirely different bases.

Another quick question: how many digits did the Mayans need?

Answer: They needed symbols representing 20 different possible digits instead of our 10 “Hindu-Arabic” digit symbols. These show up in Mayan stone carvings for calendar dates using pictographs (in other places they are represented with dots and bars).

The Long Count calendar

Here’s a complication. In the Mayan’s Long Count calendar, the first position counted single days, or “k’in.” The second position, “uinal,” was then equivalent to 20 k’in. But instead of 20 uinal equaling the next, third position, it made an 18-fold jump instead, so that “100”, or a “tun” in the Mayan calendar, represented 18 times 20, or 360 days. This was done so that tun more closely match the length of the solar year, which is slightly more than 365 days.

After that, the next position went back to making another 20 fold increase, so 1000, or one “k’atun” represented the value 20 x 18 x 20, or 7,200 days—a bit less than 20 years. And then the next position represented 20 x 20 x 18 x 20 days or 144,000 days, one “b’ak’tun”—a bit under 400 years.

In our base 10 system, each position goes as high as 9. In the Mayan system, each position can go up to 19, and so it’s typical to write Mayan numbers using decimal points to distinguish each digit’s position, given that otherwise writing “19” would be ambiguous: is it just 19, or does it represent one 20 plus 9, i.e. 29? So instead we write 1.17.19, for example, to represent the Mayan number that would equal 1 tun (360 days) plus 17 uinal (20 days), plus 19 k’in (days), for a total of 719 days.

Well, the Mayan Long Count calendar reached on December 20: 12 b’ak’tun, 19 k’atun, 19 tun, 17 uinal, and 19 k’in.

Just one more day ...

Here’s the big question of the day (slight pun intended): what did adding one more day to this calendar number do?

Answer: Adding one more day is the equivalent to adding 1 to a number like 99,999 for a base 10 number, going to 100,000, except that the positions roll over from 19 to 20 instead of the usual 9 to 10 (and note that the second position rolls over from 17 to 18).

Okay, so we reached on December 21. Yea! Time for a new calendar, with some new pictures for our new b’ak’tun—perhaps some nice fractal images interspersed with a few Mayan temples.

Congratulations, Earth!

So why was anyone worried? Were the Mayans triskaidekaphobic in terms of b’ak’tuns? No, to the contrary, we are actually just starting the 14th b’ak’tun after “creation.” In fact, we just lived through the 13th b’ak’tun (the same way that when you turn 20, you are actually entering your 21st year). For people who are worried about the number 13, relax, the world’s just entered a golden age: the 14th b’ak’tun!

Lastly, what about counting systems with mixed bases? Does having the second position only go up to 17 instead of 19 cause problems? We’re used to a more systematic base 10 system when we count, right?

Well, what is 2012.12.31 plus 1 day? What is 11:59:59 plus 1 second? Curiously, we’re used to using mixed bases, with positions meaning different amounts all the time (another weak pun!). Perhaps adding that wrinkle is one reason working with time on clocks and calendars can be so confusing for students.

For more on the amazing variety of number systems that humans have created, drop by a Math for Teaching Arithmetic class in our Math for Teaching program.

And our last question of the day: so who “invented” zero?