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All About That Numerical Base

Congratulations, Purple Parrots! You didn’t die!

Congratulations, Purple Parrots! You didn’t die!

Happy anniversary, everyone!

What, you're not sure what I'm talking about? How could you all forget? I'm so insulted. Yesterday was the second anniversary of the December 21st, 2012, the day the world was supposed to end according to a number of nutjobs, crackpots, and professional film-flam artists. Congratulations on surviving another year past fake doomsday!

These whackadoos got the idea based on an incredibly incorrect and, honestly, kind of culturally insulting interpretation of some ancient Mayan ideas. Primarily they had this idea that because the Mayan calendar ended that day, the whole world would end. Leaving aside the weird idea that somehow the Maya, alone among all cultures, knew the true End Times, we still have to contend with the fact that nothing was even ending at all! Their calendar was cyclical, just like ours, with a nested hierarchy of units of time. We have days, months, years, decades, centuries, and millennia. and when one ends it just rolls over into the next.

The Mayan Long Count Calendar had k'in (1 day), winal (20 days), tun (~1 year), k'atun (~20 years), b'ak'tun (~394 years), and piktun, or greater b'ak'tun (~7,885 years). There were actually even higher order periods than that, up to the alautun, which counted over 63 million years. All that happened two years ago yesterday was that a greater b'ak'tun rolled over into the next one. It's hard to argue that the Maya thought the world would end with the current (when they were at their height) piktun when they had higher order periods in their calendar and, in fact, prophesied events that would happen far past the Gregorian year 2012.

What's really interesting here, though, isn't a non-event that didn't happen 2 tuns ago. What's really interesting is the system of math that the Maya used to generate that calendar. With a single exception, every step is twenty times the one before it. A winal is 20 k'in. A greater b'ak'tun is 20 b'ak'tun. The only exception is the tun, which is actually 18 winal.

The Mayan Long Count Calendar was built (mostly) in multiples of 20 because the Mayan culture actually counted using a base-20 number system.

These days most of us just tend to think totally in base-10. This is perfectly natural, because our system of counting, the Hindu-Arabic numeral system, is base-10.The numbers roll over at every ten. Tens, twenties, thirties, up to hundreds, thousands, millions, billions, all of them multiples of ten. But base-10 is only one of the ways to construct a counting system. In principle, I suppose, you could use any base you want, but historically, for whatever reason, this isn't the case. Cultures tended to gravitate towards certain bases over others. I'm sure smarter people than I have speculated as to why this is but I, personally, have not a freakin' clue.

The way a base-20 number system works is just like base-10 except the rollover happens at 20. The image to the left are the numerals the Maya used to represent the numbers 0 through 19. You can see that to simplify things, a bar represents each group of five, kind of like counting by drawing hashmarks or using an abacus. Once they hit twenty they would add another "place," just like we add another "place" (the "tens place") when we hit 10, and put a dot in it to represent a single new 20. But instead of adding it to the left of the previous numeral, as we do, they added it above. So 23 would be a dot (the 20) on top of 3 dots (for the 3). 60 would be three dots above the shell, the numeral for 0, and so on. The next place was added at 400 (20x20, like we add a place at 100, which is 10x10). Their math was no less sophisticated than that of their contemporaries in the Old World, they just used a base-20 system to get there.

In fact, in one way it far outstripped the supposed geniuses traipsing about Ancient Greece. When Pythagoras and his buddies were having deep philosophical discussions about the concept of zero (I mean, how can nothing be a thing, man? It doesn't make sense!), the Maya simply recognized its utility and adopted it into their system.

As I implied earlier, the Maya weren't the only culture to deviate from base-10. You can look at modern languages to figure out how the early speakers of the progenitors of those languages probably counted. There's evidence in French for both a history of base-16 and base-20. Their "teens" don't truly begin until 17, "dix-sept," or "ten-seven." The words before that (onze, douze, treize, quatorze, quinze, and seize) cannot be split into a ten and a single digit; this implies that they were generated by people counting in base-16, or hexadecimal. Get up to 70 and you see the history of base-20. Everything is cool and base-10 until sixty, soixante. Then 70, soixante-dix, or sixty-ten, comes along, and everything gets weird. Suddenly you're counting in twenties just like the Maya: sixty-eleven, sixty-12, and so on, all the way up to eighty, which is quatre-vignts: four-twenties!

In fact, we can find evidence of a different base in our very on linguistic history. When we count, 9 becomes 10 and from there we hit the teens. Thirteen, fourteen, and so on. But what about eleven and twelve? They're the two numbers that don't fit the pattern. This implies that early Germanic language speakers might have used a base-12 system.

There are 10 types of people: those who understand binary and those who do not.

There are 10 types of people: those who understand binary and those who do not.

There were even cultures, like the Babylonians, who used a base-60 counting system. This may seem weird until you think about how, in the present day, we measure time, the degrees of a circle, and, by extension, all angles. Seconds and hours are in base-60 and a circle is 360 (60x6) degrees around, and each degree can be subdivided into 60 arcminutes, with each arcminute breaking down into 60 arcseconds. We're constantly surrounded by base-60 and we usually don't even think about it.

Heck, the computer, tablet, or smartphone you're reading this on uses binary code for everything, which is just another way of saying base-2. We are not just surrounded, we are suffocating in different numerical bases!

There are so many different ways to think of numbers, and so many ways that cultures have thought of numbers. We may think that we count in base-10, and usually we do, but our language, our computers, our very notion of time itself, all show that we're far more mathematically versatile than we think.

Copyright 2017 by Aaron Berenbach and Don Riefler

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