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- [Instructor] If I'm a child,

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and if I wanted to
represent one of something,

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I might just write, hey, one of that.

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Like one stick or one twig.

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And then if I wanted two,

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I might just write two
twigs, right, one plus one.

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Three, three twigs.

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One next to another.

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I just write them and say what I have

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is just one plus one plus one, three.

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And this is called additive
way of writing things.

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I just add what's there individually

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because I know this I-like
thing stands for one.

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And then I add I, I, I, three
Is, which is just three.

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So Roman numerals follows this idea

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of additive representation of numbers.

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So I'm gonna look at one, and what is one?

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I'll go here, I'll look at my table.

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Oh, one, I write one as I.

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And so it's just I.

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And then two is, I need two ones,

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and I just have to write them
next to each, so it's II.

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And three is going to
be, that's right, III.

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Four is going to be IIII.

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Not really. (chuckles)

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So it seems right, right, to do this.

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Write three Is for
three, four Is for four.

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But then it turns out that
this is not how we do it,

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at least not anymore.

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So I'm gonna put four as controversial.

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We're gonna talk about four more.

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So after this, what about five?

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I'm gonna up here.

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I can maybe put IIIII.

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But then, what's going on?

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It's already becoming hard to read.

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And we've already made up
a new alphabet for five.

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So I'll be putting five
just because that's easier.

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Now, what should I do for six?

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It gets interesting for six

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because how do I write six?

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So when I look at six,
I'll go up here and ask,

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"What is the largest thing that I have

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"that's smaller than six?"

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So six lies between five and 10.

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So I know that I can write
six as five plus something.

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So I'll first write my five, which is V.

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And then ask, "Okay,
what is remaining here?"

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There's just one remaining.

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And one, I know how I can write it.

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I'll just go write the one.

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And the same thing happens
for seven, five plus two.

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Eight is five plus three.

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Nine, again, I'm gonna
put a question mark here

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because it's a different way of writing.

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We do not write VIIII with the four Is.

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Right, it becomes very long.

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So I'm gonna put a question mark here.

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I'm gonna learn how
we're gonna write this.

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And for 10, I'm gonna go up here.

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It's exactly, I already
have an alphabet for it,

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so I'm just gonna use that.

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So let's look at an example.

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If I had 23 with me, and
if I want to write 23,

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how should I think about it?

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So in my head, I'm going,
"Okay, 23 is more than 10.

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"It's less than 50.

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"So I should write it as sum times 10,

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"and then I'll see what happens."

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So how many 10s are there?

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And I see that there are two 10s.

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So I'm gonna write two 10s.

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Maybe I should use a different color.

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So two 10s, XX.

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Is that enough?

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No, now what I have is 20.

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So I've taken 23, and I've made it,

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imagined it to be 20 plus three.

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20 plus three.

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And this 20 has already
been written over here.

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So now what about this three?

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I treat this as a fresh problem,

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as if I'm starting all over again.

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I'll ask, "Three, where is three between?"

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It's between one and five.

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I already know how to write
that, how to write three.

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So I just go up here and write
three as one, two, three.

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So XXIII will give me 23.

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And as you can see,

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if I had been given this number

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and asked, "What is this number?"
how would I have read it?

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I'll have read it by
going X is 10, X is 10.

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So 10 plus 10 is 20.

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I is one, III is three.

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So this is 20, this is three.

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So 20 plus three is 23.

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So that's exactly the backward process

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that I would have used.

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And you can see that in all these cases

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the bigger number is what we write first

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and then the smaller numbers.

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So XX comes and then II.

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Over here we can see that V
comes first and then the I.

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So whenever we're writing usual numbers

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it comes in this format,

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the bigger one first and
then the smaller ones.

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But if you see, in our unique as well.

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So we said these two are
controversial, right.

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Four and nine.

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So what is it about four and nine?

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How do we write four and nine?

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So instead of thinking
of four as four ones

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where we write IIII.

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Which in fact, people used to
do back in the Roman times,

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and then they stopped doing it

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because it just took too much space.

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And in important documents
where there's not much space,

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we want a shorter way to write four.

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So what is a shorter way to write four?

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They thought of four as,

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"Hey, I can write it
was one less than five."

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So I can put an I, and then
I can write a V after that.

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So this is the first time
you're seeing a smaller number

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come before a larger number.

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So they said, "Whenever
you see this happening,

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"think of it as not one
plus five equal to six.

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"Don't do that.

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"But think of it as five
minus one, which is four."

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The same thing goes for nine.

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Think of it as one minus 10.

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Actually, 10 minus one, or
one less than 10 for nine.

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So we had an IV for four and IX for nine.

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And when we do this,

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this is called the subtractive notation.

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I'm gonna write it here as subtractive.

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Now this name is not that important,

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it's just for you to realize
that when we're writing it here

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in this way, we're actually
using a subtraction.

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Five minus one and 10 minus one.

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Now, the most important thing to know

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about this subtractive notation
is that it's super rare.

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It very, very rarely happens.

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So I'm gonna fill in for
four and nine over here.

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And then let's look at where

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the subtractive notation is used.

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So four is IV, and nine is IX.

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So where is the subtractive notation used?

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So you can see that it's
used very, very, very rarely.

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In fact, the only special
cases are four and nine.

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You're asking me,

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"Don't we use this kind
of thing anywhere else?"

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We do.

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But just for the multiples
of four and nine.

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Four and nine, 40, 90, 400, 900.

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Only for these numbers do you
use this subtractive notation.

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And in fact, you can forget 400 and 900,

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they're two big numbers.

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We can just take the
four, nine, 40, and 90.

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So four, you know how to write it, IV.

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Nine, you'll write it as IX.

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What about 40 right now?

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So instead of thinking of 40
as four 10s and writing XXXX,

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what we do is that we
look at 40 as N minus 50.

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So what must I do then?

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And I'm again saying 10 minus 50,

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what I really mean is 50
minus 10. (chuckling softly)

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I should stop doing this.

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So 10 less than 50.

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So 10 less than 50, 50 is L.

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So I'll write XL.

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I want you to stop and
think about how to do 90.

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And when you try that, and
once you get the answer,

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look at what I'm doing.

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So how do you do 90?

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You will look at this number, 100,

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and then subtract N from it.

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So 10 less than 100, that's 90.