I like numbers — not just number properties (although I definitely like those, hello number theory!) but the actual numbers. In particular, I’ve been into number systems recently.
Number system means the symbols and values we use for numbers. We’ve got a base-ten number system, because we have ten different symbols for numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. When we hit ten, we then have to use two symbols combined, same for all other two-digit numbers. Similarly, for a three-digit number, we combine three digits and call it the 100s place, and so on.
Almost every civilization in history has used a base-ten number system. Not all, and I’ll talk about those crazy cats later. Base-ten is obvious for us, since we’ve got ten fingers and ten toes. I think that if humans had twelve fingers instead of ten, we’d probably use a base-twelve number system and think that ten is for weirdos.
There’s nothing in the numbers themselves that make ten better or worse than another number for the base. In fact, ten isn’t as good as some options, because it’s only divisible by two and five. That’s why if we write the fraction 1/3, we get a horrible decimal that goes on forever. If we used base-twelve instead, 1/3 would be a nice fraction. If I have a twelve-day vacation and I use a third of it to laze around the beach, that means I’ve spent four days on the beach — three times four is twelve, it works out nicely.
But humans have got ten fingers, and so most of us decided to use ten for our numbers. But even within that, you can get pretty creative. There are, in general terms, two ways you can make a number system.
Option 1: our system, where you pick ten different symbols. To make larger numbers, put those symbols together.
Option 2: pick a symbol for each place value.
You’ve probably seen Roman Numerals, which is a variation on option 2. The Romans got a little weird, so I’ll instead talk about the ancient Egyptians.
The Egyptians have a symbol for the one’s place, which looks like a line. Then they had a symbol for the tens place, which looks like a horseshoe, a symbol for the hundreds place, which looks like a coiled rope, a symbol for the thousands place, which looks like a flower, and so on.
That means, to write the number 4578, Egyptians wrote four flowers, five ropes, seven horseshoes, and eight lines. To write 3201, Egyptians wrote three flowers, two ropes, and one line — no horseshoes, since there’s a zero in the tens place.
It’s a pretty obvious system, and lots of other ancient number systems use a similar idea. It’s also really easy to add and subtract, since you just put down more of a symbol, or cross some of them out. Since the Egyptians were super into art, they also didn’t feel the need to write numbers in the same directions. Sometimes they wrote numbers left to right, sometimes right to left. Often, especially in temples, they wrote the numbers symmetrically, because that was prettier. (Here’s the equation to summarize: beauty > consistency.)
The downside of a system like this is it takes longer to read a number. If I’ve got eight flowers, I have to count them, and that takes time and I might lose count. Even if I group symbols into fours or fives, it still takes longer to read that than a single symbol.
But it’s a good, obvious system that makes basic arithmetic simple and which can be very beautiful. You’ll have a finite number of numbers (I love saying number of numbers), because we can’t draw infinitely many symbols, but as far as applied mathematics is concerned, that’s fine, we don’t need infinity. Well, we’ll need infinity if we want to do calculus, but we can make a symbol for that, which is exactly what modern mathematicians do — it looks like a sideways eight.
When I talk about number systems, I always start with the Egyptians. They’ve got such a nice system which is easy to explain and understand. While plenty of other civilizations had similar ideas, many of them were not as beautiful with their simple logic as the Egyptians. They were all, dude, that’s fine, what they’ve got in Egypt, but we want to be different (by which they of course meant better).
Which means, let’s talk about the Romans.
The Romans also used base ten and needed to write symbols many times. But the Romans thought, I don’t want to have to count that many symbols if I need eight of something! That takes forever!
So the Romans broke up their numbers halfway: meaning at five, fifty, five hundred, and so on. I’m sure some of you are already having flashbacks to school when you have to decode V, five, and L, fifty.
But the Romans weren’t completely crazy — in fact, they were logical, efficient people, which is why they didn’t want to waste so much time counting numbers. If the Romans needed to write 4578, they wrote MMMMCLXXVIII — M is thousand and I’ve got three of them, C is five hundred so I only need one, L is fifty and X is ten so I add one fifty and two tens, V is five and I is one so I add one five and three ones. I’ve got to memorize more symbols, but once I do, I just add them up and I can read the number.
But, I can hear you say, what about IV for four? Well, you can’t actually blame the ancient Romans for that one. Instead, blame Medieval monks in Europe. (I mean, technically you can blame whoever you want. They’re all dead, so they can’t complain or send you nasty messages on social media.)
The Medieval monks had lots of time on their hands, but they didn’t feel like counting four symbols, because that is deeply arduous and takes away time they could use to starve to death. (Because lack of food security during the Middle Ages was always a thing.) Those monks were the ones who came up with the idea of using IV for four, IX for ten, XL for forty, and so on. You look at actual ancient Roman ruins, you’ll see IIII for four.
You can see the logic of this modified system. If I take one away from five, I get four, so I can write IV and see where it comes from. Here’s a secret: half the reason us evil math teachers make you learn Roman Numerals in school is because we want you to practice subtraction in a different form than what you’re used to. We’re mean like that.
There weren’t many people doing math in Medieval Europe — counting and sums, sure, but not much of anything more complicated for that. So a weird system like this is fine, and once you get used to it, you can probably read those IX and XL and CM’s with no trouble. You can still add and subtract quickly with it, so what else do you need?
Thus, the Romans were logical and lazy, but to get really weird numbers you need a bunch of monks locked away in monasteries together and no television.
We’ve now talked about the Egyptians and the Romans. Other civilizations that use base-ten include the ancient Chinese and the ancient Indians. In fact, our modern number system with only ten digits comes from ancient Indian, in the Hindu civilization. The system then followed ancient trade routes through Arabia and the Middle East and then finally arrived in Europe. Hence, our current number system is called the Hindu Arabic numerals.
Those of you with a keen eye might have noticed a famous, ancient civilization that I haven’t yet mentioned, a civilization full of crazy mathematicians who get credit for way more than they ought to.
That’s right, it’s time to talk about the Greeks!
The story I learned in school said that because the Greeks didn’t need to spend all their time on agriculture, they had more time to think, and that’s why they invented more stuff than the civilizations that pre-dated them. That’s baloney, and we’ll talk more about that later, but the Greeks definitely had time enough to come up with an overly complicated number system.
You remember how I listed two options for number systems? Well, the Greeks found a third option, which is basically a combination of options one and two.
They had distinct symbols for digits one through nine, and then a digit for ten. That doesn’t sound too weird. But they also had distinct digits for twenty, thirty, forty, and so on to ninety. And since when you’re on a roll you like to just keep going, they also had distinct symbols for one hundred, two hundred, three hundred, and so on.
That’s a lot of memorizing. But here’s the crazy part of the Greek numbers. The symbols they used for numbers were the exact same symbols they used for letters. You know, like alpha, beta, gamma, and so on. (Including pi, because I must always mention pi if it comes up. I love pi. I also love pie, and on Pi Day I eat at least pi over three pie. Yum.)
Once you look at how the Greeks set up their numbers, it gets a little less crazy, because you can’t write out a number and accidentally also spell a word. Although, it would be hilarious if there was a special number that had mystical significance to the Greeks that also spelled out a rude word.
If the Greeks wanted to write the number 578, they wrote phi – omicron – eta, where phi is five hundred, omicron is seventy, and eta is eight. If I was going to pronounce that, it would sound like PHO—EH (bearing in mind that I learned Greek for math, not for linguistics or history, and thus I’m sure my pronunciations are awful).
Now, the Romans used I, V, X, L, and so on as letters too, but at least they didn’t turn their entire alphabet into numbers. For those of you paying attention and counting, you’ll also notice that the Greek alphabet only has twenty-six letters, but we obviously need more than twenty-six numbers in order to count Greek-style. So the higher numbers use different symbols.
Earlier, I said that the Hindu-Arabic numerals (ie, our modern numbers) eventually reached Europe via first India and then the Middle East. The arrival of those numbers to Europe exactly pre-date when Europeans really started doing math again.
As a good teacher, I will remind you not to jump to conclusions. The numerals reached Europe around the late Middle Ages, when Europe was starting to once again do some actual scholarship. So the two events coincided for sure, but you can’t say we owe math to the new number system. (I mean, sure, you could, but we try not to ascribe credit (or blame) unless we’re sure.)
To finish with base-ten numerals, let’s talk briefly about the ancient Chinese. Like the Greeks, they used an amalgamated system, but with fewer digits to remember. They had symbols for numbers one through nine, and then a symbol for ten. They also had symbols for hundred, thousand, ten-thousand, and so on. That means the Chinese system matches up pretty well with how we read numbers in English. To write 15487 in the Chinese system, I’d write one – ten-thousand 一万, five – thousand 五千, four – hundred 四百, eight – ten 八十, and seven 七. I’ve put a picture here of what that looks like:
一万五千四百八十七
I get students who complain that there’s no creativity in math. I get that complaint — math is all about logic and patterns. But there is creativity if you know where to look for it. Number systems are definitely one place. Even using a base-ten system, we’ve seen a bunch of different ways you can put that together and get numbers.
I ask a philosophical question to my higher-level students: is math created or discovered? Number systems, and specifically the symbols we draw and how we put those symbols together, are certainly created.
