Where did our numbers come from?

How do we end up with these shapes for our numbers? Is there a better system out there? And what does how you count say about where you're from? Find out now in another Rob Words. On this channel we talk a lot about letters, you know, the alphabet. We've talked about punctuation too. But we have never discussed the other little symbols that we use all the time when we write. These. So let's do that because they have a fascinating and surprising history that is very different to that of our alphabet. Completely different, in fact. Like languages, numbers are a part of our cultural identity. Just do me a favour, right? Count to 10 on your hands. If you're in a place where that's going to make you look like a weirdo, just imagine yourself doing it. Go on, count to 10. Now how did you do it? Did you start like this with your index finger then count up from there? If you did, there's a good chance you're watching from North America. That's how it tends to be done in Canada and the United States. The way I do it is like this, starting with my thumb. One, two, three, four, five. And that is the more typical way of doing it in Europe, in the UK. And it's sometimes called the French or the German method. You've perhaps seen that scene in Inglorious Bastards where the fellow gives himself away as not German by showing the number three using three fingers instead of two fingers and a thumb. Actually, in Britain we're a bit weird. We'd generally show three like that too, even though we count up from the thumb. Whatever. But these are by no means the only options. In the Middle East and other places like the Philippines, I think, they'll start with the little finger and count up like that. In Japan, the whole thing seems to me to be completely backwards. That is zero and you bring down fingers to count up to five. And I think you open your hand up again to count the rest of the way to ten. A system I think is really cool is used in part of India where you count up to five on your left hand. One, two, three, four, five. And then keep track of how many times you've done it on the right. So that's one. One, two, three, four, five. Two. One, two, three, four, five. Three. You see, I've counted up to 15 there. So I've counted beyond ten. And in China, they count to five like Americans. One, two, three, four, five. But for each of the numbers six to ten, they have a specific gesture using the same hand. Hold on, I've been practicing these. So you've got six, which is the phone. Seven, which is the pinch. Eight is the gun. Nine is the hook. And ten is the cross. They're supposed to, I think, kind of resemble the Chinese symbols for those numbers. If you've got a different way of doing it where you are, do let me know in the comments because I'd be fascinated to know. But you might be wondering why I'm going on about this. Well, it is to make the point that the way we handle numbers does depend on our culture. As it is for counting on our hands, so it is for writing numbers down. Now, do you know that I've got a free newsletter? Every week I send out a dose of word facts and language fun that I've researched and written myself. And every week I make it using Opera Browser. It has so many useful functions like tab islands, which allow me to keep all my regular newsletter tabs grouped together and then collapse them all when I'm not using them and doing something else. Plus these handy tab traces let me see which sites I've recently used. The darker the line, the more recently I looked at it. It also has a brilliant split screen feature, which means I can easily have my source material open on one side of the window and be typing away on the other. I simply drag a tab down and that's it. No need for multiple browser windows. 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You know something that I struggle to get my head around? It's the fact that we had words for numbers way, way, way before we had symbols to represent them. We were saying the words 1, 2, 3, your ancient equivalents of them, for thousands of years before we could write 1, 2, 3. And the English language was particularly slow in getting the message that it'd be handy to have figures to represent our numbers. When English sprung into existence in the form of Old English, the language of the Anglo-Saxons, some 1,500 years ago, it was, like other Germanic languages, being written out in runes, not the Latin alphabet that we use today. And there were no runes for numbers. If you wanted to write a number in runes, you just had to spell the word out like we sometimes still do today. It wasn't until Christian monks started writing out Old English using the Latin alphabet, the system left over by the Romans, that English gained a shorthand way of writing numbers, Roman numerals. But you know full well that we don't use these Roman numerals anymore, except for some special occasions, like on fancy clocks, for numbering the Super Bowl, and for some reason at the end of TV programmes. Or is that just a British thing? No, we use these numbers instead. So where did these come from? Well, you might have heard them referred to as Arabic numerals, which is a big, if slightly misleading, clue. A more accurate name for them is Hindu-Arabic numerals, because these did not originate in the Arab world, but in India. Allow me to explain. These are the numbers 1, 2, 9, or at least they were in the Brahmi writing system, being used around 300 BC in part of India. I actually rather like them. Look at the numbers 1 to 3. Could they be more intuitive? Chinese does something similar with 1 to 3 now. But more importantly, look at the numbers for 6 and 7. Look familiar? Now have a look at these numerals, being used around a thousand years later, also in India. Notice the shapes for 1, 2, 3. Again, familiar, right? Do you see where I'm going with this? Plus there is an extremely important addition to these, because they're not just 1 to 9, they include an extra one, don't they? Wonder what that is. It's around this time that Arabic mathematicians took a shine to these Indian numerals and started to use them themselves. And in Western Arabic regions, they morphed into these. Check these out. They're basically our numbers, aren't they? So how did we end up with them? Well, the short answer is trade. Arab merchants traded with Southern Europeans, and so Europeans became acquainted with these shapes. But how did they come to replace the Roman numerals that were, for the best part of two millennia, the dominant means of writing numbers in Europe? Well, to explain that, I need to introduce you to this guy, the 9th century mathematician, Al-Khwarizmi. Though he was actually Persian, Al-Khwarizmi did his best work in the Arab world. In Baghdad, to be precise. And his influence on modern mathematics can't be overstated. For example, he pioneered a mathematical discipline that he called restoration, or in Arabic, al-Jawar, a term that we still use today, algebra. What's more, the Latin rendering of his name became our word, algorithm, originally used to describe doing arithmetic with numbers rather than an abacus. But his biggest contribution, at least for the sake of this video, is spreading awareness among European mathematicians that these numbers were vastly superior to the Roman system that they were currently using. This isn't an active thing that Al-Khwarizmi did. It actually happened a few hundred years after his death, when Europeans started to pay closer attention to his works. Works that revealed the virtues of the Arabs' system, so-called place-value system, where any number you could think of could be expressed using the same set of 10 digits, arranged by order of factors of 10 from left to right. If you wanted to express 1,346, you could write 1 in the thousands position, 3 in the hundreds position, 4 in the tens position, and 6 in the units position. Lovely. Efficient. And what's more, this system had a handy placeholder for when there was no value in one of the positions. Want to write 2021? Well, use the numeral cipher to show that there aren't any hundreds. That word cipher in European mouths morphed into zero, and also the word cipher. These Hindu-Arabic numerals were clearly miles better than Roman numerals. For starters, the Romans didn't have a zero, the losers. Plus look at how messy a number like 3884 is. M-M-M-D-C-C-C-L-X-X-X-I-V, compared with the lovely, snappy Arabic version. In the Arabic system, generally, the longer the number, the bigger. Definitely not the case with the Romans. There are many other technical reasons why Hindu-Arabic numerals are an improvement on the Roman ones, but they are more than my tiny, non-mathematical brain can make sense of. The point is, the improvement was significant enough that they replaced Roman numerals across Europe, and we're still using them today. So how have we ended up with the best possible system? Well, the fact Hindu-Arabic numerals have been taken up across much of the world suggests we might well have. But can we be sure? Well, let's look at how other civilizations have tackled the task of expressing numbers throughout history, and see if our system might have missed something. Let's start way back with the ancient Egyptians. We know that they had these funky pictures, hieroglyphics, to write out words. But what about numbers? Because the Egyptians were no fools when it came to mathematics. You can't build a successful pyramid without doing some sums first. Well, the ancient Egyptians had glyphs for numbers too, and you would just repeat them to increase the size of the number. One to nine were just lines, then when you got to ten, you'd use a new glyph. A hundred had another glyph, and a thousand, another. Note that the Egyptians, like the Romans, didn't have a symbol for zero. Their counting system didn't need it. And for the sake of counting things out, if you have none of something, what is there to count? If you think about it, we actually don't have a hand gesture for zero, do we? I mean, I suppose you could do that, but I don't think I've ever actually done that. Also note that to the ancient Egyptians, ten was particularly important. You get to ten of something, and the pattern resets, and the glyphs are all factors of ten. This is base ten counting, which is what we do. The best theory for why we and the Egyptians and others use base ten counting is that we are used to counting on our fingers and thumbs, of which we have ten. That's why base ten exists. Other cultures have base twenty counting, which also makes sense if you factor in counting on your toes. But the reason for the next civilization we're going to look at, System, is an even greater mystery. Let's have a look at the Babylonians. The Babylonians used a style of writing called cuneiform. They didn't write by applying a die to a flat surface like the Egyptians and their papyrus, but by pressing their characters into soft clay, using something like a trimmed reed as a stylus. This meant that the shapes you could make were really limited, meaning all cuneiform writing looked sort of like that. And these are the Babylonians' cuneiform numbers. These are the numbers one to sixty, and, you know, they're pretty straightforward, really. The lines that look like they're pointing down represent single digits, and the ones that look like they're pointing to the left represent the tens. Easy. And it is easy, unless you want to count any higher than that. You see, Babylonian counting was not base ten. It wasn't even base twenty. It was base sixty. And that makes counting above sixty, to our eyes at least, really complicated. So for our system, another way you can write out the number, say, one thousand two hundred and thirty four, is one times a thousand plus two times a hundred plus three times ten plus four times one. This can also be expressed like this, with the power of ten decreasing as you work your way along the number. Now imagine that instead of each of these being ten to the power of something, they're actually sixty to the power of something. Because that is how Babylonian numbers worked. So in the Babylonian system, you could write out their equivalents of the digits one, two, three, four, like that, but it would not equal one thousand two hundred and thirty four. Because each place expresses a power of sixty, not ten, that number would actually equal one times sixty to the power of three plus two times sixty to the power of two plus three times sixty to the power of one plus four times sixty to the power of zero, or two hundred and twenty three thousand three hundred and eighty four. Yikes. Confused? That's fine, because that's sort of my point. It is confusing to us because we're used to base ten. But to the Babylonians, for goodness knows what reason, chose base sixty. Nobody has sixty fingers and toes. To them, it made perfect sense. The Babylonians are another civilization that managed without a zero. But let's take a look at a people who were ahead of their time when it came to notating nothingness. Let's cross to the Americas and meet the Maya. This is the Maya counting system, and it is amazing. The Maya found a way to be able to write every number they could ever need using a total of just three symbols, a dot, a line, and this one that often gets described as a shell. The dots were units. You'd use them to count up to four, and then when you hit five, convert them to a line. You could have up to three lines and four dots, taking up to nineteen and making this a base twenty system. And as for the shell, that was the Mayan zero. The Maya had a place value system like ours, but it worked vertically rather than horizontally. So the taller a number was, the bigger it was. And also like our system, their zero figure, the shell, could be used as a placeholder value. For example, here's how the Maya wrote 101. In the twenties row, because remember this is base twenty rather than base ten, the line represents five. Five times twenty is a hundred. Then in the row below, we put the one. But what if we'd wanted to just write one hundred? Well, instead of that one in the bottom row, we'd put the Mayan shell to say that there's nothing there. Nada. Zero. And that is groundbreaking. Now, all the systems we looked at so far have something in common. They are written systems of representing numbers. But for some highly successful civilizations, that was never an option because they had no written language. Civilizations like the Inca. The Inca, of course, had language. They had spoken language, but they did not write anything down. Hence, they did not have a written numerical system. Instead, they had something else. This. The quipu. When the Inca wanted to express a number in a form other than just saying it out loud, they would say it with string by tying the number into a cord a bit like this, I guess. And by tying the right kinds of knots in the right places, they could express any number they wanted. So let's, for example, try to tie the number four. I've been having a bit of a practice. Let's see how it goes. So you firstly make a loop like that. You see that? Then you take this bit and you go around four times with it. One. Two. Three. Four. Pull that tight. And voila! One. Two. Three. Four. Pretty cool. Not bad. I was a bit close there. Now, despite the pretty obvious differences from our own numerical system, the Inca counted in a way that is surprisingly similar to how we do it. They counted in base 10 and with a place value system based around increasing factors of 10 each time. But instead of working along the number like we do it, working your way up the string. So if you want to turn our four into 24, you move above the four, create another loop, and then tie a two. Once around. Twice around. Pull it tight. And there we have it. Two above a four. 24. And if we wanted to turn that 24 into 204, for example, we'd get rid of the knot here and tie it further up in the hundreds position, leaving a gap in the tens position, making 204. 204. The Incan equivalent of zero is a blank bit of string. To further demonstrate that, have a look at one that I prepared earlier. Let's see if we can work out what this is. I actually put bigger gaps this time because I'm not very good at controlling the gap sizes. But in the thousand position, we have one knot. In the hundreds position, we have no knot. In the tens position, we have one knot. And in the units position, we have, well, this is a nine. What number have we created? 1019. Pretty cool and weirdly easy, right? I honestly think that the Incan quipu, despite looking the least like our Hindu Arabic system, might be the closest to it in spirit. It just makes sense. However, we should also take a closer look at the only other numerical system that the English language has ever used. Roman numerals. How do they compare with all these others? Well, in my view, not terribly well. I mean, they're a bit baffling. After all, they're just Roman letters that have been repurposed. And with no obvious logic, I obviously get the use of I for one and for two and for three, like tally marks. And it seems like the C must be short for centum, the Latin for 100, and M must be short for mille, the Latin for 1000. But then what are the V, D and L short for? Nobody knows. Maybe they're short for nothing. The great innovation that the Romans came up with for their numbers was to introduce subtraction as well as addition. A character placed before another character of higher value is taken away from it, meaning four is written by a one before a five and six is a one after a five. This means that numbers can be shorter. So rather than writing 94 as L-X-X-X-X-I-I-I-I-50 plus 10 plus 10 plus 10 plus 10 plus one plus one plus one plus one, you can write it simply as X-C-I-V. 10 before 100 plus one before five. But neither, of course, is as good as just writing a nine and a four. So I for one am relieved that Arab traders delivered us their system of writing numbers, the system that they in turn borrowed from India. So what do you reckon? Do you prefer any of the other systems I've mentioned? What about the modern systems? Are there any that you think are superior to the one that we're using? Let me know in the comments. We didn't have time to get into the weird ways in which different cultures say numbers. That is for another wordy, nerdy video. Thanks for watching this one. Thanks to Opera Browser for sponsoring it. And I'll see you in whatever you choose to watch next. Take care.