Reading and writing Roman numerals

- [Instructor] If I'm a child, and if I wanted to represent one of something, I might just write, hey, one of that. Like one stick or one twig. And then if I wanted two, I might just write two twigs, right, one plus one. Three, three twigs. One next to another. I just write them and say what I have is just one plus one plus one, three. And this is called additive way of writing things. I just add what's there individually because I know this I-like thing stands for one. And then I add I, I, I, three Is, which is just three. So Roman numerals follows this idea of additive representation of numbers. So I'm gonna look at one, and what is one? I'll go here, I'll look at my table. Oh, one, I write one as I. And so it's just I. And then two is, I need two ones, and I just have to write them next to each, so it's II. And three is going to be, that's right, III. Four is going to be IIII. Not really. (chuckles) So it seems right, right, to do this. Write three Is for three, four Is for four. But then it turns out that this is not how we do it, at least not anymore. So I'm gonna put four as controversial. We're gonna talk about four more. So after this, what about five? I'm gonna up here. I can maybe put IIIII. But then, what's going on? It's already becoming hard to read. And we've already made up a new alphabet for five. So I'll be putting five just because that's easier. Now, what should I do for six? It gets interesting for six because how do I write six? So when I look at six, I'll go up here and ask, "What is the largest thing that I have "that's smaller than six?" So six lies between five and 10. So I know that I can write six as five plus something. So I'll first write my five, which is V. And then ask, "Okay, what is remaining here?" There's just one remaining. And one, I know how I can write it. I'll just go write the one. And the same thing happens for seven, five plus two. Eight is five plus three. Nine, again, I'm gonna put a question mark here because it's a different way of writing. We do not write VIIII with the four Is. Right, it becomes very long. So I'm gonna put a question mark here. I'm gonna learn how we're gonna write this. And for 10, I'm gonna go up here. It's exactly, I already have an alphabet for it, so I'm just gonna use that. So let's look at an example. If I had 23 with me, and if I want to write 23, how should I think about it? So in my head, I'm going, "Okay, 23 is more than 10. "It's less than 50. "So I should write it as sum times 10, "and then I'll see what happens." So how many 10s are there? And I see that there are two 10s. So I'm gonna write two 10s. Maybe I should use a different color. So two 10s, XX. Is that enough? No, now what I have is 20. So I've taken 23, and I've made it, imagined it to be 20 plus three. 20 plus three. And this 20 has already been written over here. So now what about this three? I treat this as a fresh problem, as if I'm starting all over again. I'll ask, "Three, where is three between?" It's between one and five. I already know how to write that, how to write three. So I just go up here and write three as one, two, three. So XXIII will give me 23. And as you can see, if I had been given this number and asked, "What is this number?" how would I have read it? I'll have read it by going X is 10, X is 10. So 10 plus 10 is 20. I is one, III is three. So this is 20, this is three. So 20 plus three is 23. So that's exactly the backward process that I would have used. And you can see that in all these cases the bigger number is what we write first and then the smaller numbers. So XX comes and then II. Over here we can see that V comes first and then the I. So whenever we're writing usual numbers it comes in this format, the bigger one first and then the smaller ones. But if you see, in our unique as well. So we said these two are controversial, right. Four and nine. So what is it about four and nine? How do we write four and nine? So instead of thinking of four as four ones where we write IIII. Which in fact, people used to do back in the Roman times, and then they stopped doing it because it just took too much space. And in important documents where there's not much space, we want a shorter way to write four. So what is a shorter way to write four? They thought of four as, "Hey, I can write it was one less than five." So I can put an I, and then I can write a V after that. So this is the first time you're seeing a smaller number come before a larger number. So they said, "Whenever you see this happening, "think of it as not one plus five equal to six. "Don't do that. "But think of it as five minus one, which is four." The same thing goes for nine. Think of it as one minus 10. Actually, 10 minus one, or one less than 10 for nine. So we had an IV for four and IX for nine. And when we do this, this is called the subtractive notation. I'm gonna write it here as subtractive. Now this name is not that important, it's just for you to realize that when we're writing it here in this way, we're actually using a subtraction. Five minus one and 10 minus one. Now, the most important thing to know about this subtractive notation is that it's super rare. It very, very rarely happens. So I'm gonna fill in for four and nine over here. And then let's look at where the subtractive notation is used. So four is IV, and nine is IX. So where is the subtractive notation used? So you can see that it's used very, very, very rarely. In fact, the only special cases are four and nine. You're asking me, "Don't we use this kind of thing anywhere else?" We do. But just for the multiples of four and nine. Four and nine, 40, 90, 400, 900. Only for these numbers do you use this subtractive notation. And in fact, you can forget 400 and 900, they're two big numbers. We can just take the four, nine, 40, and 90. So four, you know how to write it, IV. Nine, you'll write it as IX. What about 40 right now? So instead of thinking of 40 as four 10s and writing XXXX, what we do is that we look at 40 as N minus 50. So what must I do then? And I'm again saying 10 minus 50, what I really mean is 50 minus 10. (chuckling softly) I should stop doing this. So 10 less than 50. So 10 less than 50, 50 is L. So I'll write XL. I want you to stop and think about how to do 90. And when you try that, and once you get the answer, look at what I'm doing. So how do you do 90? You will look at this number, 100, and then subtract N from it. So 10 less than 100, that's 90.