0:00:00.240,0:00:01.330
- [Instructor] If I'm a child,

0:00:01.330,0:00:03.830
and if I wanted to[br]represent one of something,

0:00:03.830,0:00:05.310
I might just write, hey, one of that.

0:00:05.310,0:00:07.370
Like one stick or one twig.

0:00:07.370,0:00:08.440
And then if I wanted two,

0:00:08.440,0:00:12.050
I might just write two[br]twigs, right, one plus one.

0:00:12.050,0:00:14.570
Three, three twigs.

0:00:14.570,0:00:15.520
One next to another.

0:00:15.520,0:00:17.590
I just write them and say what I have

0:00:17.590,0:00:20.370
is just one plus one plus one, three.

0:00:20.370,0:00:24.240
And this is called additive[br]way of writing things.

0:00:24.240,0:00:28.040
I just add what's there individually

0:00:28.040,0:00:30.960
because I know this I-like[br]thing stands for one.

0:00:30.960,0:00:34.880
And then I add I, I, I, three[br]Is, which is just three.

0:00:34.880,0:00:36.840
So Roman numerals follows this idea

0:00:36.840,0:00:40.780
of additive representation of numbers.

0:00:40.780,0:00:42.950
So I'm gonna look at one, and what is one?

0:00:42.950,0:00:45.240
I'll go here, I'll look at my table.

0:00:45.240,0:00:46.810
Oh, one, I write one as I.

0:00:46.810,0:00:48.690
And so it's just I.

0:00:48.690,0:00:52.050
And then two is, I need two ones,

0:00:52.050,0:00:54.990
and I just have to write them[br]next to each, so it's II.

0:00:54.990,0:00:58.163
And three is going to[br]be, that's right, III.

0:00:58.163,0:01:01.390
Four is going to be IIII.

0:01:01.390,0:01:04.610
Not really. (chuckles)

0:01:04.610,0:01:06.850
So it seems right, right, to do this.

0:01:06.850,0:01:09.520
Write three Is for[br]three, four Is for four.

0:01:09.520,0:01:12.390
But then it turns out that[br]this is not how we do it,

0:01:12.390,0:01:14.290
at least not anymore.

0:01:14.290,0:01:16.250
So I'm gonna put four as controversial.

0:01:16.250,0:01:18.670
We're gonna talk about four more.

0:01:18.670,0:01:21.610
So after this, what about five?

0:01:21.610,0:01:22.540
I'm gonna up here.

0:01:22.540,0:01:23.933
I can maybe put IIIII.

0:01:25.690,0:01:27.990
But then, what's going on?

0:01:27.990,0:01:29.540
It's already becoming hard to read.

0:01:29.540,0:01:32.010
And we've already made up[br]a new alphabet for five.

0:01:32.010,0:01:36.230
So I'll be putting five[br]just because that's easier.

0:01:36.230,0:01:37.870
Now, what should I do for six?

0:01:37.870,0:01:39.080
It gets interesting for six

0:01:39.080,0:01:41.700
because how do I write six?

0:01:41.700,0:01:44.117
So when I look at six,[br]I'll go up here and ask,

0:01:44.117,0:01:46.127
"What is the largest thing that I have

0:01:46.127,0:01:48.030
"that's smaller than six?"

0:01:48.030,0:01:50.623
So six lies between five and 10.

0:01:51.570,0:01:56.570
So I know that I can write[br]six as five plus something.

0:01:56.630,0:02:00.390
So I'll first write my five, which is V.

0:02:00.390,0:02:02.523
And then ask, "Okay,[br]what is remaining here?"

0:02:02.523,0:02:04.670
There's just one remaining.

0:02:04.670,0:02:05.813
And one, I know how I can write it.

0:02:05.813,0:02:08.310
I'll just go write the one.

0:02:08.310,0:02:13.310
And the same thing happens[br]for seven, five plus two.

0:02:13.460,0:02:15.690
Eight is five plus three.

0:02:15.690,0:02:17.990
Nine, again, I'm gonna[br]put a question mark here

0:02:17.990,0:02:20.430
because it's a different way of writing.

0:02:20.430,0:02:24.503
We do not write VIIII with the four Is.

0:02:25.352,0:02:27.200
Right, it becomes very long.

0:02:27.200,0:02:28.950
So I'm gonna put a question mark here.

0:02:28.950,0:02:30.520
I'm gonna learn how[br]we're gonna write this.

0:02:30.520,0:02:32.803
And for 10, I'm gonna go up here.

0:02:34.020,0:02:35.777
It's exactly, I already[br]have an alphabet for it,

0:02:35.777,0:02:38.820
so I'm just gonna use that.

0:02:38.820,0:02:40.740
So let's look at an example.

0:02:40.740,0:02:45.300
If I had 23 with me, and[br]if I want to write 23,

0:02:45.300,0:02:46.840
how should I think about it?

0:02:46.840,0:02:51.840
So in my head, I'm going,[br]"Okay, 23 is more than 10.

0:02:52.017,0:02:53.997
"It's less than 50.

0:02:53.997,0:02:56.677
"So I should write it as sum times 10,

0:02:56.677,0:02:57.850
"and then I'll see what happens."

0:02:57.850,0:02:59.670
So how many 10s are there?

0:02:59.670,0:03:01.710
And I see that there are two 10s.

0:03:01.710,0:03:04.360
So I'm gonna write two 10s.

0:03:04.360,0:03:05.700
Maybe I should use a different color.

0:03:05.700,0:03:07.653
So two 10s, XX.

0:03:09.800,0:03:11.340
Is that enough?

0:03:11.340,0:03:13.300
No, now what I have is 20.

0:03:13.300,0:03:16.020
So I've taken 23, and I've made it,

0:03:16.020,0:03:19.333
imagined it to be 20 plus three.

0:03:20.300,0:03:22.023
20 plus three.

0:03:23.040,0:03:25.810
And this 20 has already[br]been written over here.

0:03:25.810,0:03:27.680
So now what about this three?

0:03:27.680,0:03:30.500
I treat this as a fresh problem,

0:03:30.500,0:03:31.840
as if I'm starting all over again.

0:03:31.840,0:03:33.690
I'll ask, "Three, where is three between?"

0:03:33.690,0:03:34.610
It's between one and five.

0:03:34.610,0:03:37.210
I already know how to write[br]that, how to write three.

0:03:37.210,0:03:40.600
So I just go up here and write[br]three as one, two, three.

0:03:40.600,0:03:44.150
So XXIII will give me 23.

0:03:44.150,0:03:45.113
And as you can see,

0:03:47.382,0:03:48.760
if I had been given this number

0:03:48.760,0:03:52.220
and asked, "What is this number?"[br]how would I have read it?

0:03:52.220,0:03:54.950
I'll have read it by[br]going X is 10, X is 10.

0:03:54.950,0:03:57.445
So 10 plus 10 is 20.

0:03:57.445,0:03:59.362
I is one, III is three.

0:04:00.409,0:04:02.520
So this is 20, this is three.

0:04:02.520,0:04:05.260
So 20 plus three is 23.

0:04:05.260,0:04:07.180
So that's exactly the backward process

0:04:07.180,0:04:08.940
that I would have used.

0:04:08.940,0:04:11.010
And you can see that in all these cases

0:04:11.010,0:04:12.037
the bigger number is what we write first

0:04:12.037,0:04:14.440
and then the smaller numbers.

0:04:14.440,0:04:17.040
So XX comes and then II.

0:04:17.040,0:04:20.360
Over here we can see that V[br]comes first and then the I.

0:04:20.360,0:04:22.750
So whenever we're writing usual numbers

0:04:22.750,0:04:24.300
it comes in this format,

0:04:24.300,0:04:26.640
the bigger one first and[br]then the smaller ones.

0:04:26.640,0:04:29.410
But if you see, in our unique as well.

0:04:29.410,0:04:32.480
So we said these two are[br]controversial, right.

0:04:32.480,0:04:33.710
Four and nine.

0:04:33.710,0:04:36.190
So what is it about four and nine?

0:04:36.190,0:04:37.840
How do we write four and nine?

0:04:37.840,0:04:40.657
So instead of thinking[br]of four as four ones

0:04:40.657,0:04:42.690
where we write IIII.

0:04:42.690,0:04:45.840
Which in fact, people used to[br]do back in the Roman times,

0:04:45.840,0:04:46.850
and then they stopped doing it

0:04:46.850,0:04:49.220
because it just took too much space.

0:04:49.220,0:04:51.880
And in important documents[br]where there's not much space,

0:04:51.880,0:04:54.730
we want a shorter way to write four.

0:04:54.730,0:04:56.710
So what is a shorter way to write four?

0:04:56.710,0:04:57.757
They thought of four as,

0:04:57.757,0:05:01.040
"Hey, I can write it[br]was one less than five."

0:05:01.040,0:05:05.680
So I can put an I, and then[br]I can write a V after that.

0:05:05.680,0:05:09.650
So this is the first time[br]you're seeing a smaller number

0:05:09.650,0:05:11.670
come before a larger number.

0:05:11.670,0:05:13.327
So they said, "Whenever[br]you see this happening,

0:05:13.327,0:05:16.327
"think of it as not one[br]plus five equal to six.

0:05:16.327,0:05:17.857
"Don't do that.

0:05:17.857,0:05:22.490
"But think of it as five[br]minus one, which is four."

0:05:22.490,0:05:24.370
The same thing goes for nine.

0:05:24.370,0:05:27.054
Think of it as one minus 10.

0:05:27.054,0:05:30.700
Actually, 10 minus one, or[br]one less than 10 for nine.

0:05:30.700,0:05:33.840
So we had an IV for four and IX for nine.

0:05:33.840,0:05:34.673
And when we do this,

0:05:34.673,0:05:37.320
this is called the subtractive notation.

0:05:37.320,0:05:39.293
I'm gonna write it here as subtractive.

0:05:40.580,0:05:41.880
Now this name is not that important,

0:05:41.880,0:05:44.600
it's just for you to realize[br]that when we're writing it here

0:05:44.600,0:05:46.530
in this way, we're actually[br]using a subtraction.

0:05:46.530,0:05:49.190
Five minus one and 10 minus one.

0:05:49.190,0:05:50.750
Now, the most important thing to know

0:05:50.750,0:05:54.410
about this subtractive notation[br]is that it's super rare.

0:05:55.470,0:05:57.680
It very, very rarely happens.

0:05:57.680,0:06:01.000
So I'm gonna fill in for[br]four and nine over here.

0:06:01.000,0:06:01.917
And then let's look at where

0:06:01.917,0:06:03.930
the subtractive notation is used.

0:06:03.930,0:06:08.720
So four is IV, and nine is IX.

0:06:08.720,0:06:11.693
So where is the subtractive notation used?

0:06:13.160,0:06:15.650
So you can see that it's[br]used very, very, very rarely.

0:06:15.650,0:06:18.150
In fact, the only special[br]cases are four and nine.

0:06:19.280,0:06:20.567
You're asking me,

0:06:20.567,0:06:22.980
"Don't we use this kind[br]of thing anywhere else?"

0:06:22.980,0:06:23.840
We do.

0:06:23.840,0:06:27.022
But just for the multiples[br]of four and nine.

0:06:27.022,0:06:31.603
Four and nine, 40, 90, 400, 900.

0:06:33.590,0:06:36.990
Only for these numbers do you[br]use this subtractive notation.

0:06:36.990,0:06:39.150
And in fact, you can forget 400 and 900,

0:06:39.150,0:06:40.440
they're two big numbers.

0:06:40.440,0:06:43.310
We can just take the[br]four, nine, 40, and 90.

0:06:43.310,0:06:46.040
So four, you know how to write it, IV.

0:06:46.040,0:06:48.670
Nine, you'll write it as IX.

0:06:48.670,0:06:50.093
What about 40 right now?

0:06:51.560,0:06:54.630
So instead of thinking of 40[br]as four 10s and writing XXXX,

0:06:56.500,0:07:00.750
what we do is that we[br]look at 40 as N minus 50.

0:07:00.750,0:07:02.423
So what must I do then?

0:07:04.070,0:07:05.870
And I'm again saying 10 minus 50,

0:07:05.870,0:07:08.070
what I really mean is 50[br]minus 10. (chuckling softly)

0:07:08.070,0:07:09.630
I should stop doing this.

0:07:09.630,0:07:12.050
So 10 less than 50.

0:07:12.050,0:07:14.440
So 10 less than 50, 50 is L.

0:07:14.440,0:07:16.220
So I'll write XL.

0:07:16.220,0:07:19.010
I want you to stop and[br]think about how to do 90.

0:07:19.010,0:07:21.240
And when you try that, and[br]once you get the answer,

0:07:21.240,0:07:22.570
look at what I'm doing.

0:07:22.570,0:07:24.210
So how do you do 90?

0:07:24.210,0:07:26.960
You will look at this number, 100,

0:07:26.960,0:07:28.320
and then subtract N from it.

0:07:28.320,0:07:33.320
So 10 less than 100, that's 90.