Previous post on number systems
Counting by tens is obvious but boring. Which is why the Mesopotamians and the Mayans went the haute culture option and found other bases.
We’ve got ten fingers, so almost every society in history has used a base-ten number system. That’s the one we currently use, with ten different symbols starting at 0 and going to 9. To make a bigger number, say 243, we put 2 in the 100s spot, 4 in the 10s spot, and then 3 in the ones spot. We’ve discussed the standard base-ten systems, so now let’s focus on the rebels.
Have you ever wondered why there are 60 minutes in an hour? You can thank (or blame) the ancient Mesopotamians for that. This is the region of modern-day Iraq, Syria, Iran, and Jordan, and a number of civilizations rose and fell here. Almost all of them wrote in cuneiform and used the sexigesimal system. It sounds sexy, and base sixty is nothing if not sexy – and annoying to those of us who didn’t grow up using it.
The nice thing about base sixty is how many numbers divide into it. Ten is only divisible by two and five, which means if we have the fraction 1/3 and turn it into a decimal, it looks horrible and the numbers go on forever. But three goes into sixty, which means it’s suddenly a nice fraction and a nice decimal. If you have a third of an hour, you have exactly twenty minutes.
The Mesopotamians didn’t end their weirdness with just base sixty. Oh no, they were way too cool for that. They used base sixty with only two symbols: a symbol for one which looks like a T, and a symbol for ten which looks like a less than sign, <. (They don’t quite look like that, but that’s the easier keyboard symbols I’ve got.)
So, to write 43, it looked like TTT<<< or <<<<TTT because writing stuff in a consistent order is for weaklings (see: the Egyptians and their art). It gets complicated the higher we go. To write 72, first we have to express that as 72 = 1×60 + 12×1 – we’ll have a 1 in the 60s spot and then 12 in the 1s spot. So we write T <TT. To write this is annoying for us, because we have to convert to base sixty, but we can figure it out. To write 2023, we need to pull it apart as 3×360 (which is 60×60, the equivalent of the 100s spot) + 15×60 + 1×43. That looks like TTT <TTTTT <<<<TTT.
A pain for modern people, but it makes sense. The real problem comes with zero. We don’t appreciate today just how big of a deal zero is, but boy is it. Mathematicians spent literally thousands of years coming to terms with zero. For us, we can write 2023 and understand that the zero means we have no number in the 100s spot. For the other civilizations we talked about earlier, they just didn’t write any of that symbol. IE, when the Egyptians wrote 2023, they didn’t write any of the symbol for hundred (which if you remember is a rope. If you don’t remember, that’s why I’m saying it again. Teaching is all about repetition.)
The Mesopotamians didn’t have a zero, and this is where archaeologists and mathematicians start banging their heads when looking at Mesopotamian numbers. 123 = 2×60 + 1×3, so we write that as TT TTT. But 723 = 2×360 + 1×3, and there’s nothing in the 60s spot. In other words, we still write it as TT TTT – it looks exactly the same.
If there’s context with the number, we can maybe figure out whether it’s 123 or 723. No context at all, and just a number hanging out? Good luck with that.
Much later, the Mesopotamians did come up with what we call a place-holder zero. That’s a symbol to insert to indicate there is no number in that place. So for 723, they would write TT 0 TTT, and now you can tell it isn’t 123. We say this is a place-holder zero, because this is only used here — they don’t just have zero as a number. I’ll talk more about zero a little later.
We have tons of cuneiform tablets covered in sexigesimal numbers. The clay tablets used in Mesopotamia preserve better than paper, papyrus, or many other things written on by ancients. That means we have lots and lots of examples. Some of them are very important. Some of them are tax records. (Taxation: the reason we have math. Yay?) Some of them are clearly student work, which is really cool to see. Then there’s the one I love to show my students.
There are lots of ways a person can go down in history. Some are good ways, like inventing something or helping people or being a war hero. Some are bad ways — I’m looking at you, dictators and murderers. Some are kinda pathetic.
Archaeologists found a clay tablet with taxation information, signed by the scribe who made it and also signed by the supervisor. There are two parts. One part is the person tallying up numbers (probably counting wheat or sheep or something), so we see TTTT then TTTTTT then <<TTT and so on. The other part is the summary of the tally, where the numbers are written in a way that’s easier to read. The only problem is that two different sides add up to different values.
In other words, this poor scribe has now gone down in history as the person who made the oldest recorded math mistake. So that’s one thing you don’t have to worry about!
Let’s leave that man in peace and hop over the Atlantic to the Americas. Specifically, where we can find modern-day southern Mexico, Belize, Guatemala, and Honduras. This is the cradle of the Mayan civilization.
Does anyone remember how the world was supposed to end in 2012? If you don’t, there was a huge craze. Lots of people took it seriously. There was a movie about it (though the movie had very little to do with the Mayans). The reason for this is based on the Mayan calendar, because the Mayans had an awesome calendar. And they had an awesome calendar because they were great at math.
The reason I call them number system rebels, though, comes back to that calendar. See, the Mayans loved the calendar. They wanted everything to match up nicely. There are 360 days in a year (not quite, but we’ll talk about that later). The Mayans knew that, they had their awesome calendar, and so they wanted their numbers to work well with their calendar.
So the Mayans used a base-twenty system, but with a caveat. (Everyone loves a caveat, right? My students are so thrilled when I tell them the exceptions that they go right back to staring at their cell phones.) See, 20 x 20 = 400, which is too much, but 20 x 18 = 360. Cue lightbulbs over heads.
The Mayans used 18 as their ones place, then 360 (20 x 18) for the tens place, then 7200 (20 x 20 x 18) for the hundreds place, and so on. Since most of the places use twenty, we say it’s a base twenty number system, but it’s only 18 for the ones place.
The way the Mayans wrote numbers is pretty simple: all lines and dots. One is one dot, two is two dots, three is three dots, and four is four dots. But five is a line. To write six, you have one got and one line, because 1 + 5 = 6. Seven is two dots and one line, eight is three dots and one line. Skipping ahead, fourteen is four dots and two lines, because 4 + 5 + 5 = 14. Nineteen, the highest digit, is four dots and three lines.
This resembles a lot of number systems, because you count dots and lines. But, like the Romans, the Mayans never have to count more than four of anything, so that makes reading numbers a little faster.
So, to write the number 256, the Mayans wrote line-line-dot-dot (for 12 in the 20s place), then line-line-line-dot (for 16 in the ones place). It’s less complicated than some of the systems we’ve mentioned, with the 18 for the ones place being the only weird part.
There’s one other special thing about the Mayans, though, and that’s zero.
Zero is fantastic. My students don’t appreciate it when I give them problems with lots of zeros, but that’s because they use calculators for everything instead of computing by hand. How to use zero in math is actually very complicated and very recent. You might remember that zero times any number equals zero — most ancient civilizations didn’t know that, even though they could multiply like champions. They had the concept of nothing, but that’s different from using zero in any mathematical sense. Even using some sort of place-holder zero is pretty recent — that’s why we have trouble with Mesopotamian numbers until they came up with the place-holder zero.
The Mayans had zero, though. They had a symbol for it, which archaeologists tell me looks like a conch shell, and I just shrug and agree with them, because they tell me what the numbers are and I just play around with numbers. Archaeologists have found tons of Mayan stuff with both words and numbers, and they’ve found lots of examples of that conch shell being used as a placeholder.
For old-world mathematics (by which I mean Africa, Europe, and Asia), the first place we see zero is in India. Remember, that’s where our modern number system comes from. Since we use only ten digits and then arrange them in a very specific order, we have to have zero as a placeholder. It doesn’t work otherwise, because 2023 and 223 would look the same. The Indians didn’t stop at just making an awesome placeholder, though. They actually started doing math with zero, but I’ll talk more about that in the section on zero.
