Comparing improper fractions and mixed numbers

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:04.090
I've got pairs of mixed
numbers and improper fractions,

00:00:04.090 --> 00:00:06.890
and I want to think about
which of the two is larger.

00:00:06.890 --> 00:00:09.674
So 1 and 7/8, 39/10.

00:00:09.674 --> 00:00:11.090
So you could do
this in your head.

00:00:11.090 --> 00:00:13.110
You could say 10
goes into 39, I'll

00:00:13.110 --> 00:00:18.020
even write it out, 10 goes
into 39 3 times, 3 times 10.

00:00:18.020 --> 00:00:20.410
And you want to find the
largest number of times 10

00:00:20.410 --> 00:00:22.740
goes into this
without going over.

00:00:22.740 --> 00:00:24.611
So you couldn't
write a 4 over here,

00:00:24.611 --> 00:00:25.860
because then that would be 40.

00:00:25.860 --> 00:00:27.510
That would go over 39.

00:00:27.510 --> 00:00:29.450
3 times 10 is 30.

00:00:29.450 --> 00:00:32.090
And then you have
a remainder of 9.

00:00:32.090 --> 00:00:35.170
So you could rewrite this
expression right over here.

00:00:35.170 --> 00:00:40.450
Instead of 39/10 you could
write it as 30/10 plus 9/10.

00:00:40.450 --> 00:00:42.790
And 30/10 is just 3.

00:00:42.790 --> 00:00:46.412
So this is equal to 3 and 9/10.

00:00:46.412 --> 00:00:47.870
And you could do
that in your head.

00:00:47.870 --> 00:00:53.740
You could say 10 goes into 39 3
times, and the remainder is 9.

00:00:53.740 --> 00:00:55.010
You have your 9/10.

00:00:55.010 --> 00:00:57.290
And that's essentially just
doing this in your head.

00:00:57.290 --> 00:00:59.290
So now we can compare,
and we can literally just

00:00:59.290 --> 00:01:00.581
look at the whole number parts.

00:01:00.581 --> 00:01:02.860
This is 1 and
something, 1 and 7/8,

00:01:02.860 --> 00:01:05.280
and we're comparing it to
essentially 3 and 9/10.

00:01:05.280 --> 00:01:07.640
3 and 9/10 is clearly
a larger number.

00:01:07.640 --> 00:01:09.980
We have a 3 out
here instead of a 1,

00:01:09.980 --> 00:01:12.180
so we will write less than.

00:01:12.180 --> 00:01:15.150
And the way I always remember
it is, the opening always

00:01:15.150 --> 00:01:16.390
faces the larger number.

00:01:18.990 --> 00:01:20.290
And the point is small.

00:01:20.290 --> 00:01:23.370
It always points to
the smaller number.

00:01:23.370 --> 00:01:25.370
Now let's do this next one.

00:01:25.370 --> 00:01:28.430
4 and 7/8 versus 49/9.

00:01:28.430 --> 00:01:31.060
So let's convert this
to a mixed number.

00:01:31.060 --> 00:01:36.660
9 goes into 49 5 times,
and 5 times 9 is 45.

00:01:36.660 --> 00:01:38.770
So the remainder
is going to be 4.

00:01:38.770 --> 00:01:42.446
The remainder is 4,
so this is 5 and 4/9.

00:01:42.446 --> 00:01:43.820
Once again, we
can literally just

00:01:43.820 --> 00:01:45.390
look at the whole number parts.

00:01:45.390 --> 00:01:49.230
5 is clearly larger than 4,
so once again, less than.

00:01:49.230 --> 00:01:51.800
Point facing the
smaller number, opening

00:01:51.800 --> 00:01:54.120
facing the larger number.

00:01:54.120 --> 00:01:57.680
Now 2 and 1/2 versus 11/10.

00:01:57.680 --> 00:02:00.377
10 goes into 11 only 1 time.

00:02:00.377 --> 00:02:02.210
And if you care about
the remainder, it's 1.

00:02:02.210 --> 00:02:04.340
So it's 1 and 1/10.

00:02:04.340 --> 00:02:06.934
Which is clearly
smaller than 2 and 1/2.

00:02:06.934 --> 00:02:08.600
You just look at the
whole number parts.

00:02:08.600 --> 00:02:10.840
2 is clearly larger than 1.

00:02:10.840 --> 00:02:14.730
So we want our opening of our
less than or greater than sign

00:02:14.730 --> 00:02:16.719
to face the larger number.

00:02:16.719 --> 00:02:18.010
So we would write it like this.

00:02:18.010 --> 00:02:22.210
And this is greater than, so 2
and 1/2 is greater than 11/10.

00:02:22.210 --> 00:02:25.480
The little point facing
the smaller number.

00:02:25.480 --> 00:02:27.610
5 and 4/9 versus 40/7.

00:02:30.140 --> 00:02:32.520
7 goes into 40, so
let me rewrite this,

00:02:32.520 --> 00:02:34.772
7 goes into 40 5 times.

00:02:34.772 --> 00:02:36.730
And then you're going to
have a remainder of 5,

00:02:36.730 --> 00:02:38.175
because 7 times 5 is 35.

00:02:38.175 --> 00:02:40.660
You have a remainder
of 5 to get to 40.

00:02:40.660 --> 00:02:42.792
So it's 5 and 5/7.

00:02:42.792 --> 00:02:45.000
And if that looks like I'm
doing some type of voodoo,

00:02:45.000 --> 00:02:46.916
just remember, I'm really
just breaking it up.

00:02:46.916 --> 00:02:49.580
I'm just really
saying that 40/7 is

00:02:49.580 --> 00:02:53.310
the same thing as 35 plus 5/7.

00:02:53.310 --> 00:02:57.200
The largest multiple of 7
that is less than this number.

00:02:57.200 --> 00:03:01.770
And this is the same
thing as 35/7 plus 5/7.

00:03:01.770 --> 00:03:03.940
And then this, 35/7 5.

00:03:03.940 --> 00:03:06.430
And 5/7 is just 5/7 there.

00:03:06.430 --> 00:03:08.980
This one is interesting because
we have the same whole number

00:03:08.980 --> 00:03:10.280
out front on our mixed numbers.

00:03:10.280 --> 00:03:12.010
5 versus 5.

00:03:12.010 --> 00:03:13.920
So now we actually do
have to pay attention

00:03:13.920 --> 00:03:16.580
to the fractional part
of our mixed number.

00:03:16.580 --> 00:03:20.944
We essentially have
to compare 4/9 to 5/7.

00:03:20.944 --> 00:03:22.610
And there's a couple
of ways to do this.

00:03:22.610 --> 00:03:24.650
You could get them to
have the same denominator.

00:03:24.650 --> 00:03:26.660
That's probably the
easiest way to do it.

00:03:26.660 --> 00:03:29.665
So you could rewrite-- so what's
the least common multiple of 9

00:03:29.665 --> 00:03:30.720
and 7?

00:03:30.720 --> 00:03:33.740
They share no factors, so
really the least common multiple

00:03:33.740 --> 00:03:35.310
is going to be their product.

00:03:35.310 --> 00:03:38.840
So if we want to rewrite
4/9 we would write 63

00:03:38.840 --> 00:03:41.030
in the denominator,
that's 9 times 7.

00:03:41.030 --> 00:03:43.980
If we multiply the
denominator by 7

00:03:43.980 --> 00:03:47.000
we also have to multiply
the numerator by 7.

00:03:47.000 --> 00:03:49.550
So that will be 28.

00:03:49.550 --> 00:03:53.120
Now 5/7, we're going to
make the denominator 63.

00:03:53.120 --> 00:03:55.860
We're multiplying the
denominator times 9.

00:03:55.860 --> 00:03:59.360
Then we have to multiply the
numerator times 9 as well.

00:03:59.360 --> 00:04:01.710
5 times 9 is 45.

00:04:01.710 --> 00:04:02.760
So here it's easy to see.

00:04:02.760 --> 00:04:07.570
45/63 is clearly
larger than 28/63.

00:04:07.570 --> 00:04:09.130
And so we could write this.

00:04:09.130 --> 00:04:11.260
And because the whole number
of parts are the same,

00:04:11.260 --> 00:04:14.700
and 5/7 is the same
thing as 45/63,

00:04:14.700 --> 00:04:17.490
and 4/9 is the same
thing as 28/63,

00:04:17.490 --> 00:04:22.660
we can write that 5 and
4/9 is less than 40/7.

00:04:22.660 --> 00:04:26.220
Another way that you could have
thought about 4/9 versus 5/7

00:04:26.220 --> 00:04:31.060
is you could have said, well,
how does 4/9 compare to 4/7?

00:04:31.060 --> 00:04:32.480
We have the same numerator.

00:04:32.480 --> 00:04:34.994
The denominator here is larger
than the denominator here.

00:04:34.994 --> 00:04:36.910
But when you have a
number in the denominator,

00:04:36.910 --> 00:04:38.900
the larger it is, the
smaller the fraction.

00:04:38.900 --> 00:04:41.220
The smaller the absolute
value of the fraction.

00:04:41.220 --> 00:04:44.390
So this right over here is
a smaller quantity than 4/7.

00:04:44.390 --> 00:04:48.100
And 4/7 is clearly a
smaller quantity than 5/7.

00:04:48.100 --> 00:04:50.660
So 4/9 is clearly
smaller than 5/7,

00:04:50.660 --> 00:04:53.510
so we would have
gotten the same result.


The why of the 3 divisibility rule

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.050
You're just walking down the street

00:00:02.050 --> 00:00:03.040
and someone comes up to you

00:00:03.040 --> 00:00:04.210
and says "Quick! Quick!--

00:00:04.210 --> 00:00:09.120
4792. Is this divisible by 3? This is an emergency!

00:00:09.120 --> 00:00:10.700
Tell me as quickly as possible!

00:00:10.700 --> 00:00:13.010
And luckily you have a little tool in your toolkit

00:00:13.010 --> 00:00:16.110
where you know how to test for divisibility by 3

00:00:16.110 --> 00:00:17.510
Well, you say I can just add up the digits

00:00:17.510 --> 00:00:19.810
If the sum of that is a multiple of 3

00:00:19.810 --> 00:00:21.590
then this whole thing is a multiple of 3

00:00:21.590 --> 00:00:27.470
So you say 4 plus 7 plus 9 plus 2

00:00:27.470 --> 00:00:30.730
That's 11. Plus 9, it's 20. Plus 2 is 22

00:00:30.730 --> 00:00:32.660
That's not divisible by 3

00:00:32.660 --> 00:00:34.860
If you're unsure, you can even add the digits of that

00:00:34.860 --> 00:00:38.360
2 plus 2 is 4. Clearly not divisible by 3

00:00:38.360 --> 00:00:44.580
So this thing right over here is not divisible by 3

00:00:44.580 --> 00:00:49.070
And so luckily that emergency was saved

00:00:49.070 --> 00:00:50.840
But then you walk down the street a little bit more

00:00:50.840 --> 00:00:56.060
and someone comes up to you--- "Quick! Quick! Quick! 386,802--

00:00:56.060 --> 00:00:57.900
Is that divisible by 3?"

00:00:57.900 --> 00:00:59.360
Well, you employ the same tactic

00:00:59.360 --> 00:01:05.800
You say, what's 3 plus 8 plus 6 plus 8 plus 0 plus 2?

00:01:05.800 --> 00:01:11.540
3 plus 8 is 11. Plus 6 is 17. Plus 8 is 25. Plus 2 is 27

00:01:11.540 --> 00:01:14.090
Well, 27 is divisible by 3

00:01:14.090 --> 00:01:16.610
And if you're unsure, you could add these digits right over here

00:01:16.610 --> 00:01:20.810
2 plus 7 is equal to 9. Clearly divisible by 3

00:01:20.810 --> 00:01:23.790
So this is divisible by 3 as well

00:01:23.790 --> 00:01:25.340
So now you feel pretty good

00:01:25.340 --> 00:01:28.230
You've helped two perfect strangers with their emergencies

00:01:28.230 --> 00:01:30.360
You figured out if these numbers were divisible by 3

00:01:30.360 --> 00:01:32.100
very very very very quickly

00:01:32.100 --> 00:01:33.510
But you have a nagging feeling

00:01:33.510 --> 00:01:35.690
Because you're not quite sure why that worked

00:01:35.690 --> 00:01:37.840
You've just kind of always known it

00:01:37.840 --> 00:01:40.740
And so, let's think about why it worked

00:01:40.740 --> 00:01:42.410
To think about it, I'll just pick a random number

00:01:42.410 --> 00:01:44.390
But we could do this really for any number

00:01:44.390 --> 00:01:45.910
But I don't want to go too puffy on it

00:01:45.910 --> 00:01:48.290
just so you can see it's pretty common sense here

00:01:48.290 --> 00:01:51.060
And the number we'll use is 498

00:01:51.060 --> 00:01:53.440
I can literally use any number in this situation

00:01:53.440 --> 00:01:56.220
And to think about why this whole little tool

00:01:56.220 --> 00:01:57.530
this little system works

00:01:57.530 --> 00:02:00.020
we just have to rewrite 498

00:02:00.020 --> 00:02:03.510
We can rewrite the 4- since it's in the hundred's place

00:02:03.510 --> 00:02:05.740
we can write that as 4 times 100

00:02:05.740 --> 00:02:11.460
Or 4 times 100, that's the same thing as 4 times 1 plus 99

00:02:11.460 --> 00:02:12.410
That's all this 4 is

00:02:12.410 --> 00:02:14.910
400, which is the same thing as 4 times 100

00:02:14.910 --> 00:02:17.360
which is the same thing as 4 times 1 plus 99

00:02:17.360 --> 00:02:19.030
And the little trick here is I want to write-

00:02:19.030 --> 00:02:21.530
instead of writing 100, I want to write this as the sum of 1

00:02:21.530 --> 00:02:24.620
plus something that is divisible by 3

00:02:24.620 --> 00:02:26.370
And 99 is divisible by 3

00:02:26.370 --> 00:02:30.050
If I add more digits here- 999, 9999--

00:02:30.050 --> 00:02:31.630
they're all divisible by 3

00:02:31.630 --> 00:02:34.950
And this is why you can do the same reasoning for divisibility by 9

00:02:34.950 --> 00:02:36.690
Because they are divisible by 9 as well

00:02:36.690 --> 00:02:39.630
Anyway, that's what the 4 in the hundred's place represents

00:02:39.630 --> 00:02:43.270
This 9 in the ten's place- well that represents 90

00:02:43.270 --> 00:02:47.460
or 9 times 10, or 9 times 1 plus 9

00:02:47.460 --> 00:02:50.780
And then finally this 8. That's in the one's place

00:02:50.780 --> 00:02:54.320
8 times 1, or we just write plus 8

00:02:54.320 --> 00:02:56.800
Now we can distribute this 4

00:02:56.800 --> 00:03:04.520
This is 4 times 1 plus 4 times 99. So it's 4 plus 4 times 99

00:03:04.520 --> 00:03:06.670
Actually let me write it like this. I'm going to write--

00:03:06.670 --> 00:03:12.640
Actually let me write it first like 4 plus 4 times 99

00:03:12.640 --> 00:03:13.960
Do the same thing over here

00:03:13.960 --> 00:03:17.290
This is the same thing as plus 9--

00:03:17.290 --> 00:03:26.100
do that magenta color- plus 9 plus 9 times 9

00:03:26.100 --> 00:03:30.390
And then finally I have this 8 right over here

00:03:30.390 --> 00:03:32.290
And I can rearrange everything

00:03:32.290 --> 00:03:36.900
These terms right over here, the 4 times 99, and the 9 times 9

00:03:36.900 --> 00:03:38.140
I can write over here

00:03:38.140 --> 00:03:41.550
4 times 99- I'll write what's like a different notation

00:03:41.550 --> 00:03:45.040
plus the 9 times the 9, that's those two terms

00:03:45.040 --> 00:03:54.190
and then we have the plus 4 plus 9 plus 8

00:03:54.190 --> 00:03:58.990
Well, can we now tell whether this is divisible by 3?

00:03:58.990 --> 00:04:02.880
These terms, these first two terms are definitely divisible by 3

00:04:02.880 --> 00:04:05.290
This's divisible by 3 because 99 is divisible by 3

00:04:05.290 --> 00:04:06.640
regardless of what we have already

00:04:06.640 --> 00:04:07.810
you don't even have to look at this

00:04:07.810 --> 00:04:09.980
This is divisible by 3, so if you're multiplying it

00:04:09.980 --> 00:04:11.340
it's still going to be divisible by 3

00:04:11.340 --> 00:04:14.140
This is divisible by 3, so if you're multiplying this whole thing

00:04:14.140 --> 00:04:15.630
it's still going to be divisible by 3

00:04:15.630 --> 00:04:17.380
If you add two things that are divisible by 3

00:04:17.380 --> 00:04:19.130
the whole thing is going to be divisible by 3

00:04:19.130 --> 00:04:21.290
So all of this is divisible by 3

00:04:21.290 --> 00:04:23.820
And if you have another digit here, you'd done the same exact thing

00:04:23.820 --> 00:04:30.270
Instead of having 1 plus 99, you'd had 1 plus 999, 1 plus 9999, etc

00:04:30.270 --> 00:04:32.480
So the only thing you have to really worry about

00:04:32.480 --> 00:04:33.990
is this part right over here

00:04:33.990 --> 00:04:35.460
you have to ask yourself

00:04:35.460 --> 00:04:37.440
in order for this whole thing to be divisible by 3

00:04:37.440 --> 00:04:40.630
this part is- well that part is, then this part

00:04:40.630 --> 00:04:43.480
in order for the whole thing has to be divisible by 3

00:04:43.480 --> 00:04:50.120
that also has to be divisible by 3

00:04:50.120 --> 00:04:51.890
But what is this right over here?

00:04:51.890 --> 00:04:53.450
These are just our original digits

00:04:53.450 --> 00:04:56.630
498. 4 and 9 and 8

00:04:56.630 --> 00:04:58.650
We just have to make sure that when we take the sum

00:04:58.650 --> 00:05:01.000
it's divisible by 3


The why of the 9 divisibility rule

WEBVTT
Kind: captions
Language: en

00:00:00.322 --> 00:00:01.906
- [Voiceover] Someone walks
up to you on the street

00:00:01.906 --> 00:00:06.031
and says, "2943, quick is
this divisible by nine?

00:00:06.031 --> 00:00:08.285
It's a matter of life and death",

00:00:08.285 --> 00:00:09.185
and you could say,

00:00:09.185 --> 00:00:10.913
"Well, I can do this fairly quickly".

00:00:10.913 --> 00:00:12.749
To figure out whether
it's divisible by nine,

00:00:12.749 --> 00:00:14.481
I just have to add up the digits

00:00:14.481 --> 00:00:16.646
and figure out if the sum of the digits

00:00:16.646 --> 00:00:21.600
is a multiple of nine or
whether it's divisible by nine.

00:00:21.600 --> 00:00:22.768
So let's do that.

00:00:22.768 --> 00:00:26.942
Two plus nine plus four plus three,

00:00:26.942 --> 00:00:30.638
two plus nine is 11, 11 plus four is 15,

00:00:30.638 --> 00:00:33.007
15 plus three is 18,

00:00:33.007 --> 00:00:35.704
and 18 is definitely divisible by nine,

00:00:35.704 --> 00:00:38.032
so this is going to be divisible by nine.

00:00:38.032 --> 00:00:39.361
And even if you are unsure of whether

00:00:39.361 --> 00:00:41.329
18 is divisible by nine you could apply

00:00:41.329 --> 00:00:42.638
the same rule again.

00:00:42.638 --> 00:00:45.422
One plus eight is equal to nine,

00:00:45.422 --> 00:00:49.165
so that's definitely,
definitely divisible by nine.

00:00:49.165 --> 00:00:51.396
And so the person can go save their lives

00:00:51.396 --> 00:00:52.883
or whoever's life they were trying to save

00:00:52.883 --> 00:00:54.351
with that information.

00:00:54.351 --> 00:00:55.735
But this might leave you thinking,

00:00:55.735 --> 00:00:57.630
well, that was nice and useful,

00:00:57.630 --> 00:00:59.136
but why did that work?

00:00:59.136 --> 00:01:00.388
Does this work for all numbers?

00:01:00.388 --> 00:01:01.754
Does it only work for nine?

00:01:01.754 --> 00:01:03.361
and I don't think this works for eight,

00:01:03.361 --> 00:01:05.988
or seven, or 11, or 17,

00:01:05.988 --> 00:01:07.910
why does it work for nine?

00:01:07.910 --> 00:01:09.443
And actually also works for three,

00:01:09.443 --> 00:01:11.284
and we'll think about
that in the future video.

00:01:11.284 --> 00:01:12.406
And to realize that,

00:01:12.406 --> 00:01:16.422
we just have to rewrite 2943.

00:01:16.422 --> 00:01:19.924
So the two in 2943, it's
in the thousands place,

00:01:19.924 --> 00:01:22.094
so we can literally rewrite it as

00:01:22.094 --> 00:01:23.882
two times,

00:01:23.882 --> 00:01:26.402
two times 1000,

00:01:26.402 --> 00:01:28.114
the nine is in the hundreds place,

00:01:28.114 --> 00:01:32.705
so we can literally write
it as nine times 100,

00:01:32.705 --> 00:01:35.005
the four is in the tens place,

00:01:35.005 --> 00:01:38.954
so it's literally the same
thing as four times 10,

00:01:38.954 --> 00:01:41.349
and then finally we have
our three in the ones place,

00:01:41.349 --> 00:01:44.759
we could write it as three
times one or just three.

00:01:44.759 --> 00:01:48.710
This is literally 2000,
900, 40, and three,

00:01:48.710 --> 00:01:52.330
2943.

00:01:52.330 --> 00:01:54.041
But now we can rewrite
each of these things,

00:01:54.041 --> 00:01:56.157
this thousand, this hundred, this ten,

00:01:56.157 --> 00:01:59.558
as a sum of one plus something
that is divisible by nine,

00:01:59.558 --> 00:02:00.714
so 1000,

00:02:00.714 --> 00:02:05.294
1000 I can rewrite as one plus 999,

00:02:05.294 --> 00:02:07.508
one plus 999,

00:02:07.508 --> 00:02:10.648
I can rewrite 100 as one plus 99,

00:02:10.648 --> 00:02:12.598
one plus 99,

00:02:12.598 --> 00:02:16.577
I can rewrite 10 as one plus 9,

00:02:16.577 --> 00:02:18.877
and so two times 1000 is the same thing

00:02:18.877 --> 00:02:22.184
as two times one plus 999,

00:02:22.184 --> 00:02:24.023
nine times 100 is the same thing

00:02:24.023 --> 00:02:27.268
as nine times one plus 99,

00:02:27.268 --> 00:02:29.515
four times 10 is the same thing

00:02:29.515 --> 00:02:31.611
as four times one plus nine,

00:02:31.611 --> 00:02:34.652
and then I have this plus three over here.

00:02:34.652 --> 00:02:36.264
But now I can distribute,

00:02:36.264 --> 00:02:38.460
I can say, well, this over
here is the same thing

00:02:38.460 --> 00:02:40.401
as two times one, which is just two,

00:02:40.401 --> 00:02:43.745
plus two times 999,

00:02:43.745 --> 00:02:46.383
this thing right over
here is the same thing

00:02:46.383 --> 00:02:48.967
as nine times one, just to
be clear what I'm doing,

00:02:48.967 --> 00:02:52.588
I'm distributing the two
over the first parenthesis,

00:02:52.588 --> 00:02:54.010
these first two terms,

00:02:54.010 --> 00:02:55.959
then the nine, I'm gonna distribute again,

00:02:55.959 --> 00:02:57.510
so it's gonna be nine,

00:02:57.510 --> 00:03:01.041
nine times one plus nine times 99,

00:03:01.041 --> 00:03:04.105
plus nine times 99,

00:03:04.105 --> 00:03:06.055
and then over here,

00:03:06.055 --> 00:03:07.912
I forgot the plus sign right over here,

00:03:07.912 --> 00:03:09.417
I'm gonna distribute the four,

00:03:09.417 --> 00:03:11.804
four times one, so plus four,

00:03:11.804 --> 00:03:13.653
and then four times nine,

00:03:13.653 --> 00:03:16.383
so plus four times nine,

00:03:16.383 --> 00:03:18.919
and then finally I have
this positive three,

00:03:18.919 --> 00:03:20.893
or plus three right over here.

00:03:20.893 --> 00:03:23.113
Now I'm just gonna
rearrange this addition,

00:03:23.113 --> 00:03:24.343
so I'm gonna take all the terms

00:03:24.343 --> 00:03:27.010
we're multiplying by 999,

00:03:27.010 --> 00:03:29.256
and I'm going to do that in orange.

00:03:29.256 --> 00:03:31.036
So I'm gonna take this term,

00:03:31.036 --> 00:03:32.341
this term,

00:03:32.341 --> 00:03:35.173
and this term right over here,

00:03:35.173 --> 00:03:38.794
and so I have two times
999, that's that there,

00:03:38.794 --> 00:03:41.726
plus nine times 99,

00:03:41.726 --> 00:03:43.962
plus four times nine,

00:03:43.962 --> 00:03:45.332
plus four times nine,

00:03:45.332 --> 00:03:46.506
so that's those three terms,

00:03:46.506 --> 00:03:48.982
and then I have plus two,

00:03:48.982 --> 00:03:50.331
plus two,

00:03:50.331 --> 00:03:52.058
plus nine,

00:03:52.058 --> 00:03:54.238
plus nine,

00:03:54.238 --> 00:03:56.041
plus four,

00:03:56.041 --> 00:03:57.361
plus four,

00:03:57.361 --> 00:03:59.356
and plus three,

00:03:59.356 --> 00:04:00.124
plus three.

00:04:00.124 --> 00:04:02.641
And this is interesting, this
is just the sum of our digits,

00:04:02.641 --> 00:04:04.508
this is just what we did up here,

00:04:04.508 --> 00:04:07.099
and you might see where
all of this is going,

00:04:07.099 --> 00:04:08.514
this orange stuff here,

00:04:08.514 --> 00:04:10.598
is this divisible by nine?

00:04:10.598 --> 00:04:12.675
What? Sure, it will definitely,

00:04:12.675 --> 00:04:14.579
999, that's divisible by nine,

00:04:14.579 --> 00:04:16.381
so anything this is multiplying by,

00:04:16.381 --> 00:04:17.216
it's divisible by nine,

00:04:17.216 --> 00:04:19.427
so this is divisible by nine,

00:04:19.427 --> 00:04:20.989
this is definitely divisible by nine,

00:04:20.989 --> 00:04:23.626
99, regardless of whether
is being multiplied by nine,

00:04:23.626 --> 00:04:25.197
whatever is multiplying by nine,

00:04:25.197 --> 00:04:29.282
whatever is multiplying 99 is
gonna be divisible by nine,

00:04:29.282 --> 00:04:31.149
because 99 is divisible by nine,

00:04:31.149 --> 00:04:33.726
so this works out, and
same thing over here,

00:04:33.726 --> 00:04:38.218
you're always going to be
multiplying by a multiple of nine,

00:04:38.218 --> 00:04:40.017
so all of this business all over here

00:04:40.017 --> 00:04:44.097
is definitely going to
be divisible by nine.

00:04:44.097 --> 00:04:45.917
And so in order for the whole thing,

00:04:45.917 --> 00:04:49.810
and all I did is I rewrote
2943 like this right over here,

00:04:49.810 --> 00:04:52.613
in order for the whole thing
to be divisible by nine,

00:04:52.613 --> 00:04:54.917
this part definitely is divisible by nine,

00:04:54.917 --> 00:04:57.260
in order for the whole thing
does the rest of this sum,

00:04:57.260 --> 00:04:59.916
it has to be divisible by nine as well.

00:04:59.916 --> 00:05:02.013
So in order for this whole thing,

00:05:02.013 --> 00:05:05.050
all of this has to be divisible,

00:05:05.050 --> 00:05:07.572
divisible by nine.


The fundamental theorem of arithmetic

WEBVTT
Kind: captions
Language: en

00:00:00.731 --> 00:00:02.875
Sal: This is Sal, I'm here with Ben Eater.

00:00:02.875 --> 00:00:03.356
Ben: Hi Sal.

00:00:03.356 --> 00:00:05.117
Sal: And you've made this module,

00:00:05.117 --> 00:00:05.413
Ben: I did.

00:00:05.413 --> 00:00:08.554
Sal: called the fundamental theorem of arithmetic!

00:00:08.554 --> 00:00:09.018
Ben: Very exciting!

00:00:09.018 --> 00:00:12.523
Sal: Very exciting, isn't this? This says that we can essentially get to

00:00:12.523 --> 00:00:16.679
any whole number greater than one

00:00:16.679 --> 00:00:17.299
Ben: Right.

00:00:17.299 --> 00:00:19.556
Sal: by taking products of prime numbers.

00:00:19.556 --> 00:00:20.386
Ben: That's right.

00:00:20.386 --> 00:00:21.136
Sal: That any number

00:00:21.136 --> 00:00:21.613
Ben: Any number.

00:00:21.613 --> 00:00:23.314
Sal: can be made by taking

00:00:23.314 --> 00:00:24.553
Ben: Just the right prime numbers

00:00:24.553 --> 00:00:26.152
Sal: by multiplying a bunch of prime numbers together.

00:00:26.152 --> 00:00:26.832
Ben: Yeah

00:00:26.832 --> 00:00:29.220
Sal: and right over here, were essential going to try to do that.

00:00:29.220 --> 00:00:30.406
Ben: and there's one and only

00:00:30.406 --> 00:00:31.457
one way to get to that number.

00:00:31.457 --> 00:00:34.354
Sal: There's only one way, there's not other

00:00:34.354 --> 00:00:34.868
kind of ways..

00:00:34.868 --> 00:00:37.251
Ben: There's only one set of prime numbers that will get you to any number

00:00:37.251 --> 00:00:39.113
Sal: Very, very interesting. So let's see, they say

00:00:39.113 --> 00:00:42.904
find the prime factorization of 42. So when their saying prime factorization,

00:00:42.904 --> 00:00:46.548
you know when your saying, since you wrote this, your saying

00:00:46.548 --> 00:00:48.903
factor this and all the numbers that when I multiply them together

00:00:48.903 --> 00:00:50.796
I get to 42. Well, the prime numbers.

00:00:50.796 --> 00:00:51.895
Ben: Prime numbers. And so we only give you

00:00:51.895 --> 00:00:52.951
prime numbers

00:00:52.951 --> 00:00:54.939
Sal: You only give the prime numbers, 2 through

00:00:54.939 --> 00:00:59.246
13. It says use arrows to change the exponent

00:00:59.246 --> 00:01:02.757
on each prime number. So the exponent is how many times you're going to

00:01:02.757 --> 00:01:06.334
multiply that,or, and we have many videos on this,

00:01:06.334 --> 00:01:09.119
going into depth on exponents, but you could also, one way to visualize it

00:01:09.119 --> 00:01:12.096
is how many times your going to multiply one times its number.

00:01:12.096 --> 00:01:12.652
Ben:Right.

00:01:12.652 --> 00:01:16.161
Sal:For at least this case right over here, we're multiplying

00:01:16.161 --> 00:01:19.206
one times two 0 times, so the answer is

00:01:19.206 --> 00:01:23.001
one there. If you mulitply it by 2 once, you get a two there.

00:01:23.001 --> 00:01:27.347
and then if you do it by 2 twice, you get 2 times 2, is 4.

00:01:27.347 --> 00:01:30.418
So the prime factorization of 42. So the way I

00:01:30.418 --> 00:01:32.589
think well, lets use your hints. Ben: OK

00:01:32.589 --> 00:01:34.050
Sal: Sometimes, you've got these hints

00:01:34.050 --> 00:01:36.934
you should think about if their.. so lets see, I'd like a hint.

00:01:36.934 --> 00:01:42.049
so let's see what it tells us to do. We can

00:01:42.049 --> 00:01:46.232
use a factor tree to break 42 into its prime factorization.

00:01:46.232 --> 00:01:49.986
Which of the pime numbers divides into 42? so.

00:01:49.986 --> 00:01:51.610
Ben: A good place to start is think of any of those prime numbers

00:01:51.610 --> 00:01:55.672
and which ones divide... Sal: And maybe start with those at the smaller end...Ben: thats what I like to do, Yeah. ..Sal:And 2 is

00:01:55.672 --> 00:01:59.284
usually the easiest one to think of. This is an even number, 2 is going to didive into it.

00:01:59.284 --> 00:02:03.050
so lets see if thats what ya'll confirm in the hints. Oh, yes, right there,

00:02:03.050 --> 00:02:06.215
2 goes in, and you can do this on paper if you want, or you should

00:02:06.215 --> 00:02:12.598
but 2 goes into 42 21 times, so I can just try doing it 21 times, but we're not

00:02:12.598 --> 00:02:16.142
done yet, because 21 is not a prime number, is not even listed here,

00:02:16.173 --> 00:02:20.681
keep listing the prime numbers, 21 is not prime,and so

00:02:20.681 --> 00:02:23.636
2 does not go into 21 anymore, so we're kinda done with 2,

00:02:23.636 --> 00:02:31.278
2's, so actually, lets see, we have 1 2 over there, but 3 does go into 21!

00:02:31.278 --> 00:02:33.166
Ben: It does. Sal: 3 does go into 21, and so lets see,

00:02:33.197 --> 00:02:36.498
so 21 can be factored into 3 times, so lets

00:02:36.498 --> 00:02:40.723
see if thats what your hint...Ben:which are both prime Sal: which are both prime!

00:02:40.723 --> 00:02:43.249
lets see if thats what the hint confirms. Yes! 3 times 7

00:02:43.249 --> 00:02:46.971
and then 7 is prime and your done Ben: Thats right!

00:02:46.971 --> 00:02:51.021
Sal: So you can say your gonna have, er, your gonna have this 2 right over here, your gonna have

00:02:51.021 --> 00:02:54.584
a 3, and then were gonna need a 7. 2 times 3 times

00:02:54.584 --> 00:03:00.200
7 is 42. And were done! And we go right over here

00:03:00.200 --> 00:03:07.119
answer, check, click the.. need to change your answer, no no already did that.

00:03:07.119 --> 00:03:09.406
so now i just check the answer, and I'm done! Correct! Next question.

00:03:09.406 --> 00:03:17.319
and I got two, we used the hints so we dont get full credit.

00:03:17.319 --> 00:03:24.757
Lets do the next one. Lets see what shows up. Oh,we already know the prime factors of 21, we just did that!

00:03:24.757 --> 00:03:29.021
So we got a three, and we got a 7 in there, and there we go!

00:03:29.021 --> 00:03:33.887
Lets do another one. 18.

00:03:33.887 --> 00:03:36.040
So 2 goes into 18. Ben: It does. Sal: 2 goes into 18.

00:03:36.040 --> 00:03:45.720
and so, if I wanna do this in my head, 2 times 9, and I dont need to do this in my head, since I have hints,

00:03:45.720 --> 00:03:46.788
Ben: You have hints!

00:03:46.788 --> 00:03:49.790
Sal: You have hints, that essentially do that.so we can use our factor tree

00:03:49.790 --> 00:03:51.558
and break 18 into its prime factorization. It's 2 times 9!

00:03:51.558 --> 00:03:55.345
and 9 is odd, but not prime.2 doesnt go into it, 3 does.

00:03:55.345 --> 00:04:03.972
so we have 3 times 3, leaving us with 3..

00:04:03.972 --> 00:04:09.157
3 is prime so were done factoring. So this is interesting, so we have a 2, and then were multiplying by 3 twice.

00:04:09.157 --> 00:04:19.058
3 times 3 is 9, and times 2 is 18. And were done.

00:04:19.565 --> 00:04:26.358
we have not been able to prove the fundamental theoem of arithmatic wrong! Ben: Thats right!

00:04:26.358 --> 00:04:30.358
(both men laughing) haven't disproved it. Sal: right, very good. well, this is very neat.


Interactive module: finding factors

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:03.470
SAL KHAN: I'm here with our
exercise guru, Ben Eater--

00:00:03.470 --> 00:00:04.330
BEN EATER: Hi, Sal.

00:00:04.330 --> 00:00:05.960
SAL KHAN: --who made this
exercise, and now we're

00:00:05.960 --> 00:00:07.293
going to try to work through it.

00:00:07.293 --> 00:00:07.950
So let's see.

00:00:07.950 --> 00:00:10.460
It says what are
the factors of 9?

00:00:10.460 --> 00:00:13.615
You can re-size this box
with 9 dots-- the box

00:00:13.615 --> 00:00:15.819
has 9 dots-- to help
you find the factors.

00:00:15.819 --> 00:00:17.360
BEN EATER: So what
you're looking at,

00:00:17.360 --> 00:00:19.540
is you've got these
9 dots on the screen.

00:00:19.540 --> 00:00:22.970
And right now, there are
just 1 row of 9 dots.

00:00:22.970 --> 00:00:24.430
And if you grab
that little thing

00:00:24.430 --> 00:00:27.390
on the right side of the box and
move that around to the left--

00:00:27.390 --> 00:00:28.120
SAL KHAN: Oh, that's neat.

00:00:28.120 --> 00:00:29.578
BEN EATER: --you
still have 9 dots,

00:00:29.578 --> 00:00:31.040
but they're arranged
differently.

00:00:31.040 --> 00:00:31.550
SAL KHAN: I see.

00:00:31.550 --> 00:00:33.350
BEN EATER: And so what you
can see is, try to get them

00:00:33.350 --> 00:00:35.050
so you have the same
number in each row.

00:00:35.050 --> 00:00:36.830
SAL KHAN: I see, so this one.

00:00:36.830 --> 00:00:39.660
So if you have-- let's
see, not the same number.

00:00:39.660 --> 00:00:40.660
BEN EATER: There you go.

00:00:40.660 --> 00:00:41.576
SAL KHAN: Oh, 3 there.

00:00:41.576 --> 00:00:43.880
And so you could get
to 9 by literally

00:00:43.880 --> 00:00:47.441
multiplying 3 rows
times 3 columns.

00:00:47.441 --> 00:00:48.440
BEN EATER: That's right.

00:00:48.440 --> 00:00:49.565
SAL KHAN: And you get to 9.

00:00:49.565 --> 00:00:51.200
BEN EATER: So 3 is a factor.

00:00:51.200 --> 00:00:52.620
SAL KHAN: And then we would
write that in this box right

00:00:52.620 --> 00:00:53.530
over here.

00:00:53.530 --> 00:00:54.238
BEN EATER: Mm-hm.

00:00:54.238 --> 00:00:57.280
SAL KHAN: So we'd write 3--
do we have to write 3 twice,

00:00:57.280 --> 00:00:59.690
because we have-- or is
just writing 3 once--

00:00:59.690 --> 00:01:00.240
BEN EATER: 3 is the factor--

00:01:00.240 --> 00:01:00.360
SAL KHAN: 3 is a factor.

00:01:00.360 --> 00:01:01.430
BEN EATER: --so you
just put it once.

00:01:01.430 --> 00:01:02.090
SAL KHAN: We don't
have to write it.

00:01:02.090 --> 00:01:04.390
If we wrote it twice
would it be wrong?

00:01:04.390 --> 00:01:05.019
BEN EATER: Uh-huh,
yes, actually.

00:01:05.019 --> 00:01:05.430
SAL KHAN: It would?

00:01:05.430 --> 00:01:05.870
OK, so--

00:01:05.870 --> 00:01:06.990
BEN EATER: We do
mark that wrong.

00:01:06.990 --> 00:01:08.000
SAL KHAN: --you actually
do mark that wrong.

00:01:08.000 --> 00:01:09.220
OK, so there's 3 and 3.

00:01:09.220 --> 00:01:11.720
But there was also, right when
you started, you had 9 and 1.

00:01:11.720 --> 00:01:12.920
BEN EATER: That's right.

00:01:12.920 --> 00:01:13.690
Both of those are factors.

00:01:13.690 --> 00:01:14.650
SAL KHAN: And I
don't have to write

00:01:14.650 --> 00:01:15.910
these in any special order?

00:01:15.910 --> 00:01:16.660
BEN EATER: No, no.

00:01:16.660 --> 00:01:17.400
SAL KHAN: 1 and 9.

00:01:17.400 --> 00:01:18.899
BEN EATER: Just got
to get them all.

00:01:18.899 --> 00:01:21.200
SAL KHAN: And I believe
that's all there is.

00:01:21.200 --> 00:01:22.700
And we could look
at the other ones,

00:01:22.700 --> 00:01:25.817
and we can see, look
2 and 8 definitely--

00:01:25.817 --> 00:01:26.400
BEN EATER: No.

00:01:26.400 --> 00:01:28.170
SAL KHAN: No, that's
not working out.

00:01:28.170 --> 00:01:29.670
It's interesting
for people to think

00:01:29.670 --> 00:01:30.900
about why these
aren't working out.

00:01:30.900 --> 00:01:31.608
BEN EATER: Right.

00:01:31.608 --> 00:01:33.340
Yeah, you can't divide 2 into 9.

00:01:36.259 --> 00:01:38.300
SAL KHAN: And this just
gets us to 1 and 9 again,

00:01:38.300 --> 00:01:39.510
and we don't want to repeat it.

00:01:39.510 --> 00:01:40.180
BEN EATER: Those are
the same factors.

00:01:40.180 --> 00:01:40.640
SAL KHAN: Exactly.

00:01:40.640 --> 00:01:41.690
So let's check our answer.

00:01:41.690 --> 00:01:42.360
Let's see how we're doing.

00:01:42.360 --> 00:01:43.230
BEN EATER: OK.

00:01:43.230 --> 00:01:44.230
SAL KHAN: Let's see.

00:01:44.230 --> 00:01:44.907
There we go.

00:01:44.907 --> 00:01:45.740
We got it all right.

00:01:45.740 --> 00:01:46.577
Let's do one more.

00:01:46.577 --> 00:01:47.160
BEN EATER: OK.

00:01:47.160 --> 00:01:48.410
SAL KHAN: Let's do one more.

00:01:48.410 --> 00:01:48.910
Correct.

00:01:48.910 --> 00:01:49.660
The next question.

00:01:52.110 --> 00:01:53.900
You can resize this
box with 16 dots

00:01:53.900 --> 00:01:55.650
to help you find the factor.

00:01:55.650 --> 00:01:56.690
It's 1 and 16.

00:01:56.690 --> 00:01:57.710
BEN EATER: 1 and 16
are factors, sure.

00:01:57.710 --> 00:01:58.600
SAL KHAN: So me write those.

00:01:58.600 --> 00:01:58.840
BEN EATER: Those
are all its factors.

00:01:58.840 --> 00:02:00.630
SAL KHAN: So 1 and 16.

00:02:00.630 --> 00:02:04.030
So let's play with
this box here.

00:02:04.030 --> 00:02:05.820
That's not working.

00:02:05.820 --> 00:02:07.340
That's not working.

00:02:07.340 --> 00:02:10.500
So I'm trying to figure
out-- so this is essentially

00:02:10.500 --> 00:02:14.470
going 2 times-- 2 might be
a factor if I could just--

00:02:14.470 --> 00:02:15.120
oh look, it is.

00:02:15.120 --> 00:02:15.860
BEN EATER: There it is, yeah.

00:02:15.860 --> 00:02:16.600
2 is a factor

00:02:16.600 --> 00:02:17.600
SAL KHAN: 2 is a factor.

00:02:17.600 --> 00:02:18.810
BEN EATER: And so is 8.

00:02:18.810 --> 00:02:19.935
SAL KHAN: Exactly, exactly.

00:02:19.935 --> 00:02:21.200
2 rows of 8 works.

00:02:21.200 --> 00:02:24.959
And so we could say
2 and 8 are factors.

00:02:24.959 --> 00:02:26.500
BEN EATER: And if
you think about it,

00:02:26.500 --> 00:02:27.975
2 and 8 both divide into 16.

00:02:27.975 --> 00:02:28.850
SAL KHAN: Absolutely.

00:02:28.850 --> 00:02:29.840
BEN EATER: 2
divides eight times.

00:02:29.840 --> 00:02:30.130
SAL KHAN: Yes.

00:02:30.130 --> 00:02:31.360
BEN EATER: Or 8
divides two times.

00:02:31.360 --> 00:02:31.690
SAL KHAN: Yes, you can that.

00:02:31.690 --> 00:02:32.440
You can construct.

00:02:32.440 --> 00:02:33.148
BEN EATER: Right.

00:02:33.148 --> 00:02:35.800
SAL KHAN: 16 dots, you can
divide them into 2 groups of 8.

00:02:35.800 --> 00:02:36.010
BEN EATER: That's right.

00:02:36.010 --> 00:02:37.620
SAL KHAN: And I suspect
also 8 groups of 2.

00:02:37.620 --> 00:02:38.700
BEN EATER: Well,
you can see that.

00:02:38.700 --> 00:02:40.325
SAL KHAN: Let's see,
so that's 2 groups

00:02:40.325 --> 00:02:41.950
of 8, really, or 2 rows of 8.

00:02:41.950 --> 00:02:44.550
Let's see, or oh, look at that.

00:02:44.550 --> 00:02:45.380
4 times 4.

00:02:45.380 --> 00:02:45.770
BEN EATER: 4 times 4.

00:02:45.770 --> 00:02:46.800
SAL KHAN: And I'll
just write the 4 once.

00:02:46.800 --> 00:02:47.966
BEN EATER: So 4 is a factor.

00:02:47.966 --> 00:02:49.160
SAL KHAN: 4 is a factor.

00:02:49.160 --> 00:02:53.500
And then I can-- and I've
probably-- well, I've

00:02:53.500 --> 00:02:54.620
already gotten 8 and 2.

00:02:54.620 --> 00:02:54.700
BEN EATER: Already got that.

00:02:54.700 --> 00:02:56.780
SAL KHAN: Now I have 8 groups
of 2 instead of 2 groups of 8,

00:02:56.780 --> 00:02:58.071
but they're still both factors.

00:02:58.071 --> 00:02:59.970
Then you go back to 16 and 1.

00:02:59.970 --> 00:03:00.230
BEN EATER: That's right.

00:03:00.230 --> 00:03:01.120
SAL KHAN: So I think we're done.

00:03:01.120 --> 00:03:01.640
BEN EATER: I think so.

00:03:01.640 --> 00:03:02.630
SAL KHAN: Let's
check our answer.

00:03:02.630 --> 00:03:02.890
Oh--

00:03:02.890 --> 00:03:03.210
BEN EATER: Very good.

00:03:03.210 --> 00:03:04.040
SAL KHAN: --very
good, very good.

00:03:04.040 --> 00:03:04.950
Well, thank you.

00:03:04.950 --> 00:03:06.070
That was fun.

00:03:06.070 --> 00:03:06.340
BEN EATER: Thank you.

00:03:06.340 --> 00:03:07.740
SAL KHAN: This was very
neat, very fun to play with.

00:03:07.740 --> 00:03:09.290
BEN EATER: Well, thanks.


Hyperbolic functions and the unit hyperbola

WEBVTT
Kind: captions
Language: en

00:00:00.121 --> 00:00:02.171
We know that if we take all of the points

00:00:02.171 --> 00:00:08.104
in the *X, Y planewherex^2 + y^2 = 1*,

00:00:08.104 --> 00:00:10.913
we get ourselves the unit circle.

00:00:10.913 --> 00:00:13.247
Let me draw the unit circle.

00:00:13.247 --> 00:00:18.997
That's my y-axis; this is my x-axis.

00:00:18.997 --> 00:00:22.831
And the unit circle has the circle with the radius one.

00:00:22.831 --> 00:00:30.971
So that's x=1, that's x=-1, that's y=1, that's y=-1

00:00:30.971 --> 00:00:34.503
the unit circle looks something... let me draw it...

00:00:34.503 --> 00:00:35.997
something like this, I think you get the point

00:00:35.997 --> 00:00:39.103
Let's see if I can fill it in a little bit better.

00:00:39.103 --> 00:00:41.564
So you realize that it's not a dotted circle.

00:00:41.564 --> 00:00:43.170
There.

00:00:43.170 --> 00:00:47.438
That's my best attempt at drawing the unit circle.

00:00:47.438 --> 00:00:51.081
And we also know that the traditional trig functions,

00:00:51.081 --> 00:00:54.080
or maybe we should call them the circular trig functions

00:00:54.080 --> 00:00:58.504
are actually defined so that if you parameterize

00:00:58.504 --> 00:01:03.703
so if you were to take *x=cos t*

00:01:03.703 --> 00:01:07.413
and *y=sin t*

00:01:07.413 --> 00:01:10.372
and you pick any t, right over here

00:01:10.372 --> 00:01:13.838
and by definition it's going to sit on the unit circle

00:01:13.838 --> 00:01:18.438
by definition, *x^2 + y^2 = 1*

00:01:18.438 --> 00:01:21.556
so if you pick any t it's going to sit some place

00:01:21.571 --> 00:01:22.238
on this unit circle.

00:01:22.238 --> 00:01:24.532
Or another way to think of it is

00:01:24.532 --> 00:01:27.771
if you vary t it's going to start tracing out this circle

00:01:27.771 --> 00:01:31.329
And we know that t corresponds to the angle

00:01:31.329 --> 00:01:34.913
with a positive x-axis, in this case,

00:01:34.913 --> 00:01:37.246
that right over there is t.

00:01:37.246 --> 00:01:40.080
Now wouldn't it be neat if there were a similar

00:01:40.080 --> 00:01:42.914
analogy for, not the unit circle, but

00:01:42.914 --> 00:01:45.170
something we could call the unit hyperbola?

00:01:45.170 --> 00:01:49.581
So that's our little review of trigonometry right there;

00:01:49.581 --> 00:01:51.582
our traditional trigonometry, now let's think about

00:01:51.582 --> 00:01:53.830
the unit hyperbola.

00:01:53.830 --> 00:01:57.771
Well, *x^2 + y^2 = 1* is a unit circle, I'll say that

00:01:57.771 --> 00:02:07.771
*x^2 - y^2 = 1, I'm going to call this myunit hyperbola*.

00:02:07.771 --> 00:02:12.248
Or a unit rectangular hyperbola. Hyperbola.

00:02:12.248 --> 00:02:15.171
This is just a little bit of review from *Conic Sections*,

00:02:15.171 --> 00:02:16.970
but it would look something like this:

00:02:16.970 --> 00:02:28.170
It would look... something... that's my y-axis,

00:02:28.170 --> 00:02:33.838
this is my x-axis, and then we can say,

00:02:33.838 --> 00:02:36.771
well if y is 0, x can be ±1,

00:02:36.771 --> 00:02:40.246
so you can think of that as the unit part

00:02:40.246 --> 00:02:45.369
where it intersects the x-axis; that's +1, that's -1

00:02:45.369 --> 00:02:48.704
and it has asymptotes, y=x and y=-x

00:02:48.704 --> 00:02:52.039
We go through the intuition on that in the

00:02:52.039 --> 00:02:58.413
Conic Section videos, y=x is that dotted line,

00:02:58.413 --> 00:03:04.611
y=-x is that dotted line, right over there,

00:03:04.611 --> 00:03:08.079
and then this thing is going to look like this.

00:03:08.079 --> 00:03:11.503
It's going to have a right half that does something like this,

00:03:11.503 --> 00:03:14.103
and does something like this, all a review of

00:03:14.103 --> 00:03:18.039
*Conic Sections*, it gets closer to its asymptotes.

00:03:18.039 --> 00:03:22.039
To y=x or y=-x

00:03:22.039 --> 00:03:23.663
and the same thing on the left-hand side.

00:03:23.663 --> 00:03:28.837
It's going to do something like that.

00:03:28.837 --> 00:03:33.837
Wouldn't it be neat if we could parameterize

00:03:33.837 --> 00:03:39.703
x and y with analogous functions so that

00:03:39.703 --> 00:03:42.170
we get a similar type of property?

00:03:42.170 --> 00:03:44.569
And you might guess what those functions are,

00:03:44.569 --> 00:03:46.036
but let's actually try to verify it.

00:03:46.036 --> 00:03:53.330
What would happen if x is equal to our hyperbolic

00:03:53.330 --> 00:03:57.104
*cosine of t*, which is the same thing as

00:03:57.104 --> 00:04:05.504
*e^t + e^(-t)*, all of that over 2

00:04:05.504 --> 00:04:07.504
and y were to be equal to

00:04:07.504 --> 00:04:11.580
*hyperbolic sine* of t, which is equal to

00:04:11.580 --> 00:04:17.247
*e^t - e^(-t)* over 2. Wouldn't it be neat if there

00:04:17.247 --> 00:04:21.848
were an analogy here; over here you pick any t

00:04:21.848 --> 00:04:26.247
based on our circular trig functions, you ended up

00:04:26.247 --> 00:04:29.413
with a point on the unit circle. Wouldn't it be amazing

00:04:29.413 --> 00:04:31.663
if for any point t you ended up with a point on our,

00:04:31.663 --> 00:04:34.504
what we're calling our *unit hyperbola*?

00:04:34.504 --> 00:04:37.503
Well, in order for that to be true, with this parameterization

00:04:37.503 --> 00:04:40.704
*x^2 - y^2* would need to be equal to one.

00:04:40.704 --> 00:04:42.438
Let's see if that is the case!

00:04:42.438 --> 00:04:52.172
So *x^2 - y^2* is equal to, well let's square this business

00:04:52.172 --> 00:04:55.772
it's equal to e^(2t) plus

00:04:55.772 --> 00:04:58.105
two times the product of these two things

00:04:58.105 --> 00:05:05.313
*2e^t • e^(-t)*, this is e^0 here which is 1.

00:05:05.313 --> 00:05:13.171
Plus e^(-2t), e^(-t)^2, all of that over 4

00:05:13.171 --> 00:05:15.979
And then from that we will subtract y^2.

00:05:15.979 --> 00:05:20.838
Minus, so the numerator's going to be

00:05:20.838 --> 00:05:35.730
*e^(2t) - 2e^t • e^(-t) + e^(-2t)*, all of that over 4

00:05:35.730 --> 00:05:39.571
So, immediately, a couple of simplifications here.

00:05:39.571 --> 00:05:44.838
*e^t • e^(-t)*, that's just e^(t-t)

00:05:44.838 --> 00:05:48.064
which is equal to e^0, which is equal to 1

00:05:48.064 --> 00:05:51.396
This is going to be one, that's going to be one,

00:05:51.396 --> 00:05:54.439
so we're going to have a 2 in either of those cases

00:05:54.439 --> 00:05:57.902
and if we were to simplify it, all of this stuff over here

00:05:57.902 --> 00:06:01.440
I'll do a numerator, so this is going to be equal to

00:06:01.440 --> 00:06:05.372
over our [denominator] of 4

00:06:05.372 --> 00:06:19.146
*e^(2t) + 2 + e^(-2t) - e^(2t)*

00:06:19.146 --> 00:06:19.646
just distributing the negative sign

00:06:19.646 --> 00:06:26.099
Plus two, and then minus e^(-2t)

00:06:26.099 --> 00:06:30.103
Well this is convenient!

00:06:30.103 --> 00:06:33.302
(Oh, I was writing it in black, a hard color to see)

00:06:33.302 --> 00:06:39.104
This cancels with this,

00:06:39.104 --> 00:06:42.104
This and this also add up to zero and you're left with

00:06:42.104 --> 00:06:43.396
two plus two over four

00:06:43.396 --> 00:06:47.969
which is indeed equal to one!

00:06:47.969 --> 00:06:51.565
So this is a pretty good reason to call these two functions

00:06:51.657 --> 00:06:53.889
hyperbolic trig functions.

00:06:53.889 --> 00:06:55.588
These are the circular trig functions,

00:06:55.588 --> 00:06:57.238
you give me a t on these parameterizations

00:06:57.238 --> 00:06:58.313
we end up on the unit circle!

00:06:58.313 --> 00:07:00.897
You vary t, you trace out the unit circle.

00:07:00.897 --> 00:07:04.570
Here, for any real t, we're going to assume we're

00:07:04.570 --> 00:07:05.647
dealing with real numbers,

00:07:05.647 --> 00:07:08.238
for any real t we're going to end up on

00:07:08.238 --> 00:07:11.480
the unit hyperbola right over here

00:07:11.480 --> 00:07:13.370
and in particular we're going to end up on the right

00:07:13.370 --> 00:07:17.063
so it's not exactly... over here pretty much any of these

00:07:17.063 --> 00:07:18.646
points could be parameterized right here

00:07:18.646 --> 00:07:20.656
over here we're going to end up

00:07:20.656 --> 00:07:22.572
on a point on the right side of the unit hyperbola.

00:07:22.572 --> 00:07:23.982
The reason why it's the right side

00:07:23.982 --> 00:07:26.171
is... you go straight to the definition of

00:07:26.171 --> 00:07:29.504
*cosh t*, this thing can only be positive

00:07:29.504 --> 00:07:32.313
This thing can only be positive.

00:07:32.313 --> 00:07:35.646
e^t can only be positive, e^-t can only be positive

00:07:35.646 --> 00:07:41.171
so this is only positive.

00:07:41.171 --> 00:07:44.312
But you give any t you will end up on this hyperbola!

00:07:44.312 --> 00:07:46.770
Specifically the right side, if you want points on

00:07:46.770 --> 00:07:48.638
the left hand side, you'd have to take the

00:07:48.638 --> 00:07:55.646
*-cosh tand thesinh t*

00:07:55.646 --> 00:07:56.771
to end up right over there.

00:07:56.771 --> 00:08:00.238
But it's a pretty neat analogy.

00:08:00.238 --> 00:08:01.397
We're looking at Euler's identity and we

00:08:01.397 --> 00:08:04.396
kind of said, "oh, let's just start playing with these things!"

00:08:04.396 --> 00:08:07.312
There seems to be a similarity here if we were to

00:08:07.312 --> 00:08:09.063
remove the i 's and, all of a sudden, we've discovered

00:08:09.063 --> 00:08:11.314
another thing! That there is this relationship

00:08:11.314 --> 00:08:13.838
here there is this relationship between these trig functions

00:08:13.838 --> 00:08:17.904
and the unit circle, here between our newly defined

00:08:17.904 --> 00:08:21.771
hyperbolic trig functions and the unit hyperbola.

00:08:21.771 --> 00:08:24.239
And you'd also find if you were to vary t it's going

00:08:24.239 --> 00:08:26.731
to trace out... just as if you were to vary t here it

00:08:26.731 --> 00:08:29.297
traces out the unit circle... if you trace t here it will trace

00:08:29.297 --> 00:08:32.561
out the right-hand side, the right-hand side

00:08:32.561 --> 00:08:34.172
of the unit hyperbola.

00:08:34.172 --> 00:08:37.067
For this parameterization right here.


Hyperbolic function inspiration

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.220
In the calculus
playlist, we came up

00:00:02.220 --> 00:00:04.335
with a rationale
for Euler's formula.

00:00:06.875 --> 00:00:09.250
And I guess that video has
gotten somewhat famous with me

00:00:09.250 --> 00:00:12.680
talking about minds being blown
and people not having emotion.

00:00:12.680 --> 00:00:14.915
But just as a reminder,
Euler's formula

00:00:14.915 --> 00:00:17.670
is that e to the i theta--

00:00:17.670 --> 00:00:19.720
Just the formula by
itself is pretty amazing,

00:00:19.720 --> 00:00:21.120
but it becomes
especially amazing

00:00:21.120 --> 00:00:26.600
when you substitute
pi or tau for theta.

00:00:26.600 --> 00:00:29.140
And if you don't know what tau
is, there are videos on that.

00:00:29.140 --> 00:00:31.790
But Euler's formula tells
us that e to the i theta--

00:00:31.790 --> 00:00:34.320
there's a very good rationale
for saying that this is going

00:00:34.320 --> 00:00:40.140
to be equal to cosine
of theta plus--

00:00:40.140 --> 00:00:45.932
and I'll just change the color
for fun-- plus i sine of theta.

00:00:45.932 --> 00:00:48.390
And so one thing I want to do
is explore this a little bit.

00:00:48.390 --> 00:00:50.705
Can I express-- because
of Euler's formula-- can

00:00:50.705 --> 00:00:52.880
I now express cosine
of theta in terms

00:00:52.880 --> 00:00:54.860
of some combination
of the other stuff?

00:00:54.860 --> 00:00:56.870
And can I now
express sine of theta

00:00:56.870 --> 00:01:00.160
in terms of some other
combination of the other stuff?

00:01:00.160 --> 00:01:03.130
Well, let's think about how
we can isolate a cosine theta.

00:01:03.130 --> 00:01:05.239
If this is e to
the i theta, let's

00:01:05.239 --> 00:01:11.190
see what e to the
negative i theta would be.

00:01:16.390 --> 00:01:19.980
Actually, let me do it over
here on the right-hand side.

00:01:19.980 --> 00:01:21.680
So e to the negative i theta.

00:01:25.530 --> 00:01:29.150
This is the same thing
as e to the i times

00:01:29.150 --> 00:01:33.410
negative theta, which--
by Euler's formula--

00:01:33.410 --> 00:01:38.590
it would be equal to cosine
of negative theta plus i

00:01:38.590 --> 00:01:39.920
sine of negative theta.

00:01:46.110 --> 00:01:48.460
Well, cosine of
negative theta-- this

00:01:48.460 --> 00:01:50.340
is the same thing
as cosine of theta.

00:01:53.360 --> 00:01:55.320
And sine of negative
theta is the same thing

00:01:55.320 --> 00:01:56.920
as negative sine of theta.

00:01:56.920 --> 00:02:02.270
So this whole thing simplifies
to negative i sine of theta.

00:02:02.270 --> 00:02:06.530
So we can write e to
the negative i theta.

00:02:06.530 --> 00:02:08.820
Let me do that pink color.

00:02:08.820 --> 00:02:11.780
We can write e to
the negative i theta

00:02:11.780 --> 00:02:18.600
as being equal to cosine of
theta minus i sine of theta.

00:02:23.140 --> 00:02:25.920
And now we have something
pretty interesting going on.

00:02:25.920 --> 00:02:29.180
If we add these two expressions,
these two equalities

00:02:29.180 --> 00:02:32.010
right over here,
the left-hand side--

00:02:32.010 --> 00:02:33.109
so let's just add them up.

00:02:33.109 --> 00:02:34.650
The left-hand side,
we're going to be

00:02:34.650 --> 00:02:43.430
left with e to the i theta
plus e to the negative i theta.

00:02:43.430 --> 00:02:45.960
And on the right-hand
side, these i

00:02:45.960 --> 00:02:50.500
sine thetas are going to
cancel out with each other.

00:02:50.500 --> 00:02:53.020
And we're going to be left
with 2 times cosine of theta.

00:02:55.276 --> 00:02:56.900
And so now it's pretty
straightforward.

00:02:56.900 --> 00:03:00.950
I just isolate the cosine
theta, divide both sides by 2.

00:03:00.950 --> 00:03:03.500
Divide both sides
by 2, and we get

00:03:03.500 --> 00:03:05.150
something pretty interesting.

00:03:05.150 --> 00:03:08.660
We get that cosine of theta
is equal to this business-- e

00:03:08.660 --> 00:03:12.800
to the i theta plus e to
the negative i theta, all

00:03:12.800 --> 00:03:14.956
of that over 2.

00:03:14.956 --> 00:03:16.580
And we could do a
similar type of thing

00:03:16.580 --> 00:03:18.870
to try to isolate--
let me rewrite it over.

00:03:18.870 --> 00:03:21.150
So let me rewrite this
little discovery we made.

00:03:21.150 --> 00:03:26.300
Cosine of theta can be
rewritten as e to the i theta

00:03:26.300 --> 00:03:32.680
plus e to the negative i
theta all of that over 2.

00:03:32.680 --> 00:03:33.930
So that's kind of interesting.

00:03:33.930 --> 00:03:35.950
We can now express
cosine of theta

00:03:35.950 --> 00:03:38.580
in terms of a bunch of
exponentials right over here.

00:03:38.580 --> 00:03:42.050
Let's see if we could do the
same thing for sine of theta.

00:03:42.050 --> 00:03:45.047
Let's imagine e to the i
theta-- and let's try to do it

00:03:45.047 --> 00:03:45.630
the other way.

00:03:45.630 --> 00:03:48.280
There's always these
weird kind of parallels

00:03:48.280 --> 00:03:49.420
between cosine and sine.

00:03:49.420 --> 00:03:50.920
So let's just see
what happens if we

00:03:50.920 --> 00:03:52.930
were to subtract this from that.

00:03:52.930 --> 00:03:54.530
So let's try doing that.

00:03:54.530 --> 00:03:57.250
If we were to take
e to the i theta--

00:03:57.250 --> 00:03:59.550
let me just rewrite it over
again. e to the i theta

00:03:59.550 --> 00:04:03.130
is equal to cosine of
theta plus i sine of theta.

00:04:07.560 --> 00:04:13.570
And from that, let's subtract
e to the negative i theta.

00:04:13.570 --> 00:04:15.600
Well, now we're going
to subtract this.

00:04:15.600 --> 00:04:19.766
So we're going to get
negative cosine theta.

00:04:19.766 --> 00:04:21.140
And then, we're
subtracting this.

00:04:21.140 --> 00:04:22.720
So plus i sine theta.

00:04:29.260 --> 00:04:34.360
And so we are left with--
when we do the subtraction--

00:04:34.360 --> 00:04:39.270
the left-hand side, we
have e to the i theta

00:04:39.270 --> 00:04:46.360
minus e to the negative i
theta is equal to-- this

00:04:46.360 --> 00:04:53.180
cancels out-- and we're left
with 2i times sine theta.

00:04:53.180 --> 00:04:56.030
If we want to solve
for sine theta,

00:04:56.030 --> 00:04:57.860
we divide both sides by 2i.

00:05:02.330 --> 00:05:05.360
And we are going to be
left with sine theta is

00:05:05.360 --> 00:05:07.770
equal to this business
right over here.

00:05:07.770 --> 00:05:10.190
So let's do this.

00:05:10.190 --> 00:05:12.610
So this is our other
discovery-- sine theta

00:05:12.610 --> 00:05:16.560
can be written as
e to the i theta

00:05:16.560 --> 00:05:22.907
minus e to the negative i
theta, all of that over 2i.

00:05:22.907 --> 00:05:25.240
So there's something interesting
about these structures.

00:05:25.240 --> 00:05:27.760
I can write cosine of theta
is the sum of these two

00:05:27.760 --> 00:05:30.670
exponentials that are dealing
with imaginary numbers, sine

00:05:30.670 --> 00:05:33.860
of theta as the difference of
these two exponential numbers

00:05:33.860 --> 00:05:35.550
dealing with imaginary numbers.

00:05:35.550 --> 00:05:36.510
Actually, over
here, it's actually

00:05:36.510 --> 00:05:37.885
the average of
the two functions.

00:05:37.885 --> 00:05:38.870
You're dividing by 2.

00:05:38.870 --> 00:05:41.050
Here we're dividing by 2i.

00:05:41.050 --> 00:05:43.370
And if you don't like
that i in the denominator,

00:05:43.370 --> 00:05:46.454
you can multiply the numerator
and the denominator by i.

00:05:46.454 --> 00:05:47.620
This will become a negative.

00:05:47.620 --> 00:05:49.630
You'll have the i
in the numerator.

00:05:49.630 --> 00:05:52.090
But either way, you will
have the same expression.

00:05:52.090 --> 00:05:54.400
So this is all I
can say about this.

00:05:54.400 --> 00:05:55.870
This is just kind
of interesting.

00:05:55.870 --> 00:05:58.710
It's just kind of a fun
exploration of mathematics.

00:05:58.710 --> 00:06:01.630
But maybe there's something
neat about the structure

00:06:01.630 --> 00:06:02.730
of these functions.

00:06:02.730 --> 00:06:05.220
What happens if we were to
take functions like this,

00:06:05.220 --> 00:06:08.350
but if we were to essentially
get rid of the i's right

00:06:08.350 --> 00:06:09.050
over here?

00:06:09.050 --> 00:06:10.800
They seem like there
might be some kind

00:06:10.800 --> 00:06:14.280
of an analog between cosine
of theta and sine of theta.

00:06:14.280 --> 00:06:16.000
And so let's define
those functions.

00:06:16.000 --> 00:06:16.930
And in the next
few videos, we'll

00:06:16.930 --> 00:06:18.388
explore them and
see if they really

00:06:18.388 --> 00:06:21.980
are analogous in some bizarre,
strange, and beautiful way

00:06:21.980 --> 00:06:23.922
to cosine of theta
and sine theta.

00:06:23.922 --> 00:06:26.130
So we're just going to make
new function definitions,

00:06:26.130 --> 00:06:29.030
and they're just being
inspired by these new ways

00:06:29.030 --> 00:06:33.250
that we discovered to write
cosine of theta and sine theta.

00:06:33.250 --> 00:06:38.080
And I'm going to call
the first function cosh.

00:06:38.080 --> 00:06:41.380
And it really does come
from hyperbolic cosine,

00:06:41.380 --> 00:06:45.684
and we'll explore more why
we call it hyperbolic cosine.

00:06:45.684 --> 00:06:46.850
But let's just call it cosh.

00:06:46.850 --> 00:06:48.770
I'm just experimenting
right over here.

00:06:48.770 --> 00:06:55.835
And we'll just say cosh of x
is equal to-- it's going to be

00:06:55.835 --> 00:06:57.960
inspired by this, but we're
going to get rid of all

00:06:57.960 --> 00:06:58.620
of the i's.

00:06:58.620 --> 00:07:02.000
So it's going to
be e to the x plus

00:07:02.000 --> 00:07:05.860
e to the negative x,
all of that over 2.

00:07:05.860 --> 00:07:08.270
Once again, just inspired
by this right over here.

00:07:08.270 --> 00:07:10.290
Getting rid of all of the i's.

00:07:10.290 --> 00:07:14.610
And then, let's define
another one-- hyperbolic sine.

00:07:14.610 --> 00:07:16.720
So I'll call it sinh.

00:07:16.720 --> 00:07:18.440
Sinh for hyperbolic sine.

00:07:18.440 --> 00:07:21.390
Sinh-- I guess you
could also do-- of x.

00:07:21.390 --> 00:07:23.140
And I'm just to get
rid of all of the i's.

00:07:23.140 --> 00:07:25.290
I'm just being inspired by this.

00:07:25.290 --> 00:07:33.686
So e to the x minus e to the
negative x, all of that over 2.

00:07:33.686 --> 00:07:35.060
And in the next
few videos, we're

00:07:35.060 --> 00:07:36.320
going to start exploring this.

00:07:36.320 --> 00:07:37.070
And we're going
to start realizing

00:07:37.070 --> 00:07:39.540
that there is this
strange, bizarre parallel

00:07:39.540 --> 00:07:44.690
between these things and our
traditional trigonometric

00:07:44.690 --> 00:07:45.460
functions.

00:07:45.460 --> 00:07:48.060
Now, these will show up in
some of your mathematics,

00:07:48.060 --> 00:07:49.090
but it is important.

00:07:49.090 --> 00:07:50.860
These aren't as
core-- especially

00:07:50.860 --> 00:07:54.100
in traditional mathematics-- as
these two characters are here,

00:07:54.100 --> 00:07:56.738
but it is a fun exploration.


Introduction to economics

WEBVTT
Kind: captions
Language: en

00:00:00.436 --> 00:00:03.569
As we begin our journey into the world of economics,

00:00:03.569 --> 00:00:08.902
I thought I would begin with a quote from one of the most famous economists of all time,

00:00:08.902 --> 00:00:11.514
the Scottish philosopher Adam Smith.

00:00:11.514 --> 00:00:15.035
And he really is kind of the first real economist

00:00:15.035 --> 00:00:17.037
in the way that we view it now.

00:00:17.037 --> 00:00:19.309
And this is from his The Wealth of Nations,

00:00:19.309 --> 00:00:24.369
published in 1776, coincidentally, the same year as

00:00:24.369 --> 00:00:28.594
the American Declaration of Independence, and it's one of his most-famous excerpts.

00:00:28.594 --> 00:00:34.104
He generally indeed, he being an economic actor, neither intends to promote the public interest,

00:00:34.104 --> 00:00:36.263
nor knows how much he is promoting it.

00:00:36.263 --> 00:00:39.769
By directing that industry, so that the industry in control of that individual actor

00:00:39.769 --> 00:00:44.042
in such a manner, as its produce may be of the greatest value,

00:00:44.042 --> 00:00:46.260
he intends only his own gain.

00:00:46.260 --> 00:00:49.102
'He intends only his own gain'.

00:00:49.102 --> 00:00:53.770
And he is in this, as in many other cases, led by an invisible hand

00:00:53.770 --> 00:00:56.770
to promote an end which was no part of his intention.

00:00:56.770 --> 00:00:59.435
And this term "the invisible hand" is famous.

00:00:59.435 --> 00:01:04.932
Led by an invisible hand to promote an end which was no part of his intention.

00:01:04.932 --> 00:01:08.180
He is saying, look, when individual actors just act in their own self-interest,

00:01:08.180 --> 00:01:12.300
that often in aggregate leads to things that each of those individual actors did not intend.

00:01:12.300 --> 00:01:15.641
Then he says: nor is it always the worst for society

00:01:15.641 --> 00:01:17.175
that it was no part of it.

00:01:17.175 --> 00:01:18.929
So, it was not necessarily a bad thing.

00:01:18.929 --> 00:01:23.436
By pursuing his own interest, he frequently promotes

00:01:23.436 --> 00:01:28.702
that of the society more effectually than when he really intends to promote it.

00:01:28.702 --> 00:01:30.590
So, this is really a pretty strong statement.

00:01:30.590 --> 00:01:32.597
It's really at the core of capitalism.

00:01:32.597 --> 00:01:35.179
And that's why I point out that it was published

00:01:35.179 --> 00:01:36.970
in the same year as the American Declaration of Independence,

00:01:36.970 --> 00:01:39.905
because obviously America, the Founding Fathers,

00:01:41.043 --> 00:01:42.182
they wrote the Declaration of Independence, the Constitution,

00:01:42.182 --> 00:01:45.176
that really talks about what it means to be a democratic country,

00:01:45.176 --> 00:01:46.775
what are the rights of its citizens.

00:01:46.775 --> 00:01:52.041
But the United States, with its overall experience of an American,

00:01:52.041 --> 00:01:54.967
is at least as influenced by the work of Adam Smith,

00:01:54.967 --> 00:01:58.569
by this kind of foundational ideas of capitalism.

00:01:58.569 --> 00:02:02.265
And they just both happened to happen around the same time.

00:02:02.265 --> 00:02:08.102
But this idea is not always that intuitive. Individual actors, by essentially pursuing

00:02:08.102 --> 00:02:12.836
their own self-interested ends might be doing more for society than

00:02:12.836 --> 00:02:18.104
than if any of them actually tried to promote the overall well-being of society.

00:02:18.104 --> 00:02:22.907
And I don't think that Adam Smith would say that it's always good

00:02:22.907 --> 00:02:24.843
for someone to act self-interested, or that it's never good for people to actually

00:02:24.843 --> 00:02:29.900
think about the implications of what they are doing in an aggregate sense,

00:02:29.900 --> 00:02:34.514
but he is saying that frequently .. frequently, this self-interested action

00:02:34.514 --> 00:02:38.436
*could* lead to the greater good. Could lead to more innovation.

00:02:38.436 --> 00:02:44.309
Could lead to better investment. Could lead to more productivity. Could lead to more wealth,

00:02:44.309 --> 00:02:47.438
more, a larger pie for everyone.

00:02:47.438 --> 00:02:53.435
And now Economics is frequently .. and when he makes a statement, he is actually making

00:02:53.435 --> 00:02:56.101
a mix of micro-economic and macro-economic statements.

00:02:56.101 --> 00:03:00.102
Micro is that people, individual actors are acting out of their own self-interest.

00:03:00.102 --> 00:03:04.262
And the macro is that it might be good for the economy, or the nation as a whole.

00:03:04.262 --> 00:03:08.767
And so, now, modern economists tend to divide themselves into these two schools,

00:03:08.767 --> 00:03:16.233
or into these two subjects: microeconomics, which is the study of individual actors.

00:03:16.233 --> 00:03:21.565
Microeconomics .. and those actors could be firms, could be people, it could be households.

00:03:21.565 --> 00:03:25.840
And you have macro-economics, which is the study of the economy in aggregate.

00:03:25.840 --> 00:03:30.107
Macro-economics. And you get it from the words.

00:03:30.107 --> 00:03:34.929
Micro -- the prefix refers to very small things. Macro refers to the larger,

00:03:34.929 --> 00:03:36.347
to the bigger picture.

00:03:36.347 --> 00:03:42.588
And so, micro-economics is essentially how actors .. actors make decisions

00:03:42.588 --> 00:03:49.310
or, you could actually say 'allocations', allocations .. decisions or allocations.

00:03:49.310 --> 00:03:52.102
Allocation ..

00:03:52.102 --> 00:03:55.636
of scarce resources. And you hear the words scarce resources a lot

00:03:55.636 --> 00:03:57.969
when people talk about economics.

00:03:57.969 --> 00:04:01.304
And a scarce resource is one you don't have an infinite amount of.

00:04:01.304 --> 00:04:06.377
For example, love might not be a scarce resource. You might have an infinite amount of love.

00:04:06.377 --> 00:04:13.925
But a resource that would be scarce is something like food, or water, or money, or time, or labor.

00:04:13.925 --> 00:04:19.682
These are all scarce resources. And so microeconomics is how do people decide

00:04:19.682 --> 00:04:23.703
where to put those scarce resource, how do they decide where to deploy them.

00:04:23.703 --> 00:04:27.775
And how does that .. does that affect prices and markets, and whatever else.

00:04:27.775 --> 00:04:32.708
Macro-economics is the study of what happens at the aggregate to an economy.

00:04:32.708 --> 00:04:38.601
So, 'aggregate', what happens in aggregate to an economy, from the millions of individual actors.

00:04:38.601 --> 00:04:45.770
Aggregate economy. We now have millions of actors.

00:04:45.770 --> 00:04:49.900
And often focuses on policy-related questions.

00:04:49.900 --> 00:04:54.701
SO, do you raise or lower taxes. Or, what's going to happen when you raise or lower taxes.

00:04:54.701 --> 00:05:00.036
Do you regulate or de-regulate? How does that affect the overall productivity

00:05:00.036 --> 00:05:06.926
when you do this. So, it's policy, top-down .. 'top-down' questions.

00:05:06.926 --> 00:05:12.434
And in both macro- and micro-economics, there is especially in the modern sense of it,

00:05:12.434 --> 00:05:16.236
there is an attempt to make them rigorous, to make them mathematical.

00:05:16.236 --> 00:05:20.845
So, in either case you could start with some of the ideas, some of the philosophical ideas,

00:05:20.845 --> 00:05:24.569
so of the logical ideas, to say someone like Adam Smith might have.

00:05:24.569 --> 00:05:31.596
So, you have these basic ideas about how people think, how people make decisions.

00:05:31.596 --> 00:05:38.102
So, philosophy, 'philosophy' of people, of decision-making,

00:05:38.102 --> 00:05:42.569
in the case of micro-economics -- 'decision-making'

00:05:42.569 --> 00:05:45.634
And then you make some assumptions about it.

00:05:45.634 --> 00:05:49.593
Or you simplify it .. let me write this .. you simplify it.

00:05:49.593 --> 00:05:53.846
And you really are simplifying. You say "oh, all people are rational",

00:05:53.846 --> 00:05:57.570
"all people are gonna act in their own self-interest, or all people are going to maximize their gain",

00:05:57.570 --> 00:06:01.601
which isn't true -- human beings are motivated by a whole bunch of things.

00:06:01.601 --> 00:06:06.272
We simplify things, so we can start to deal with it kind of a mathematical way.

00:06:06.272 --> 00:06:11.368
SO you simplify it, so you can start dealing with it in a mathematical sense.

00:06:11.368 --> 00:06:14.930
So, this is valuable to clarify your thinking.

00:06:14.930 --> 00:06:17.513
It can allow you to prove things based on your assumptions.

00:06:17.513 --> 00:06:21.522
And so, you can start to visualize things mathematically, with charts and graphs

00:06:21.522 --> 00:06:23.841
and think about what would actually happen with markets.

00:06:23.841 --> 00:06:27.436
So it's very valuable to have this mathematical, rigorous, thinking.

00:06:27.436 --> 00:06:32.677
But at the same time, it could be a little bit dangerous, because you are making

00:06:32.677 --> 00:06:37.702
these huge simplifications, and sometimes the math might lead you to some very strong conclusions.

00:06:37.702 --> 00:06:43.341
Conclusions, which you might feel very strongly about, because it looks like you've proven them

00:06:43.341 --> 00:06:46.977
in the same way that you might prove relativity, but they were based

00:06:46.977 --> 00:06:50.430
on some assumptions that either might be wrong, or might be over-simplifications,

00:06:50.430 --> 00:06:55.511
or might not be relevant to the context that you're trying to make conclusions about.

00:06:55.511 --> 00:06:59.036
So it's very very very important to take it all with a grain of salt,

00:06:59.036 --> 00:07:02.369
to remember that it's all based on some simplifying assumption.

00:07:02.369 --> 00:07:05.675
And macro-economics is probably more guilty of it.

00:07:05.675 --> 00:07:09.367
In micro-economics you are taking these deeply complicated things that are

00:07:09.367 --> 00:07:11.926
the human brain, how people act and respond to each other, and then

00:07:11.926 --> 00:07:17.511
you are aggregating it over millions of people, so it's ultra-complicated.

00:07:17.511 --> 00:07:21.509
You've millions of these infinitely complicated people, all interacting with each other.

00:07:21.509 --> 00:07:25.512
SO, it's very complicated. Many millions of interactions, and

00:07:25.512 --> 00:07:31.186
fundamentally unpredictable interactions, and then trying to make assumptions on those,

00:07:31.186 --> 00:07:37.369
trying to make assumptions and then doing math with that --

00:07:37.369 --> 00:07:41.570
that could lead you to some conclusions or might be leading you to some predictions.

00:07:41.570 --> 00:07:47.702
And, once again, this is very important. This is valuable, it is valuable to make these mathematical models,

00:07:47.702 --> 00:07:50.969
with these mathematical assumptions for these mathematical conclusions,

00:07:50.969 --> 00:07:53.260
but it always need to be taken with a grain of salt.

00:07:53.260 --> 00:07:58.256
So, then you have a proper grain of salt, so that you are always focused on the true intuition.

00:07:58.256 --> 00:08:02.641
And that's really the most important thing to get from a course on economics.

00:08:02.641 --> 00:08:05.338
So you can truly reason through what's likely to happen,

00:08:05.338 --> 00:08:06.969
maybe even without the mathematics.

00:08:06.969 --> 00:08:11.103
I'll leave you with two quotes. And thse quotes are a little bit .. a little bit funny,

00:08:11.103 --> 00:08:15.268
but they're really I think helpful things to keep in mind,

00:08:15.268 --> 00:08:17.842
especially when you go deep into the mathematical side of economics.

00:08:17.842 --> 00:08:24.680
So, this right over here is a quote by Aflred Knopf, who was publisher in the 1900s.

00:08:24.680 --> 00:08:28.847
"An economist is a man who states the obvious in terms of the incomprehensible."

00:08:28.847 --> 00:08:34.238
And I'm assuming what he is talking about as the incomprehensible, he is referring to some of the 'mathy' stuff

00:08:34.238 --> 00:08:37.767
that you see in economics, and hopefully we're going to make this

00:08:37.767 --> 00:08:39.507
as comprehensible as possible.

00:08:39.507 --> 00:08:41.172
You'll see there is value in this.

00:08:41.172 --> 00:08:43.256
But it's a very important statement he is making.

00:08:43.256 --> 00:08:45.436
Oftentimes, it's taking a common-sense thing.

00:08:45.436 --> 00:08:49.103
It's taking something that's obvious .. that's obvious.

00:08:49.103 --> 00:08:53.181
And it's very important to always keep that in mind, to always make sure that

00:08:53.181 --> 00:08:56.301
you have the intuition for what's happening in the math,

00:08:56.301 --> 00:08:59.176
or to know when the math is going into a direction that might be strange

00:08:59.176 --> 00:09:02.370
based on over-simplifications or wrong assumptions.

00:09:02.370 --> 00:09:05.644
And then you have this quote here by Lawrence J. Peter,

00:09:05.644 --> 00:09:08.569
most famous for Peter's Principals, a professor at USC.

00:09:08.569 --> 00:09:12.436
"An economist is an expert know will know tomorrow why the things

00:09:12.436 --> 00:09:14.597
he predicted yesterday didn't happen today."

00:09:14.597 --> 00:09:17.436
And once again -- important to keep in the back of one's mind,

00:09:17.436 --> 00:09:21.436
because especially relevant to macro-economics, because in macro-economics

00:09:21.436 --> 00:09:26.108
there is always all sorts of prediction about the state of the economy:

00:09:26.108 --> 00:09:30.847
about what need to be done, about how long the recession will last, what will be the economic growth next year,

00:09:30.847 --> 00:09:34.264
what will inflation do ... and they often prove to be wrong.

00:09:34.264 --> 00:09:38.036
In fact, few economists even tend to agree on many of these things.

00:09:38.036 --> 00:09:42.037
And it's very important to realize that, because oftentimes when you are deep

00:09:42.037 --> 00:09:46.436
in the mathematics, economics might *seem* to be a science, like physics,

00:09:46.436 --> 00:09:50.768
but it's not a science like physics. It is open .. it is open to subjectivity,

00:09:50.768 --> 00:09:54.768
and a lot of that subjectivity is all around the assumptions that you choose to make.


Divergence theorem proof (part 5)

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:04.365
Now we can work on the triple
integral part of our problem

00:00:04.365 --> 00:00:06.600
or our proof, this
right over here.

00:00:06.600 --> 00:00:08.760
And so I can rewrite that.

00:00:08.760 --> 00:00:14.880
So this is the triple integral
over our region, which

00:00:14.880 --> 00:00:17.491
we're assuming is a type I
region, of the partial of R

00:00:17.491 --> 00:00:19.490
with respect to Z, which
we can write like this.

00:00:19.490 --> 00:00:20.410
It doesn't matter.

00:00:20.410 --> 00:00:22.570
Partial of r with
respect to do z.

00:00:22.570 --> 00:00:25.080
And I'll dV.

00:00:25.080 --> 00:00:27.840
And we can rewrite
this as-- we can

00:00:27.840 --> 00:00:30.840
assume we're going to integrate
with respect to Z first.

00:00:30.840 --> 00:00:34.657
So I'm going to integrate
with respect to Z.

00:00:34.657 --> 00:00:35.990
Let me do that in another color.

00:00:35.990 --> 00:00:38.670
I'm going integrate
with respect to Z first.

00:00:38.670 --> 00:00:42.890
The lower bound on Z in our
type I region, the lower bound

00:00:42.890 --> 00:00:43.880
is f1.

00:00:43.880 --> 00:00:46.260
The upper bound is f2.

00:00:46.260 --> 00:00:54.084
So we're going to integrate
from f1 of x, y to f2 of x, y.

00:00:57.930 --> 00:01:02.030
And I'm going to integrate the
partial of R with respect to Z.

00:01:02.030 --> 00:01:04.000
So let me do that in
that same yellow color--

00:01:04.000 --> 00:01:07.430
partial of R with respect to Z.

00:01:07.430 --> 00:01:09.270
And then I have
dZ, and then I'll

00:01:09.270 --> 00:01:12.020
have to integrate with respect
to y and x or with respect

00:01:12.020 --> 00:01:12.770
to x and y.

00:01:12.770 --> 00:01:14.940
So it's dx dy or dy dz.

00:01:14.940 --> 00:01:18.084
I can just write that as dA.

00:01:18.084 --> 00:01:19.500
So what you could
think of it-- we

00:01:19.500 --> 00:01:20.650
can evaluate the yellow part.

00:01:20.650 --> 00:01:22.858
And then we're just going
to take the double integral

00:01:22.858 --> 00:01:25.070
over the x, y domain.

00:01:25.070 --> 00:01:28.270
So this is just going to
be over the x, y domain.

00:01:28.270 --> 00:01:30.680
Let me put some
brackets here just

00:01:30.680 --> 00:01:33.130
to make it clear what
we're going to do.

00:01:33.130 --> 00:01:35.800
So all we're doing is we're
integrating with respect to Z

00:01:35.800 --> 00:01:37.529
first, and we have
the bounds there.

00:01:37.529 --> 00:01:39.070
Well, this is pretty
straightforward.

00:01:39.070 --> 00:01:40.980
This is all going to
be equal to-- I'll

00:01:40.980 --> 00:01:43.790
write the outside first--
the double integral

00:01:43.790 --> 00:01:45.570
over the domain.

00:01:45.570 --> 00:01:48.750
And I have the dA
right over here.

00:01:48.750 --> 00:01:51.490
Actually, let me give myself
some real estate-- dA.

00:01:51.490 --> 00:01:53.425
Well, what's the
antiderivative of this?

00:01:53.425 --> 00:01:57.676
This is just R, and this
is just R, or R of x, y,

00:01:57.676 --> 00:02:01.870
z evaluated when Z is
f1-- or when Z is f2.

00:02:01.870 --> 00:02:04.890
And from that, we
evaluate when Z is f1.

00:02:04.890 --> 00:02:07.104
So this is just going
to be R of x, y and z,

00:02:07.104 --> 00:02:08.770
and we evaluate when
Z is equal to that.

00:02:08.770 --> 00:02:11.430
And from that, we subtract
when Z is equal to that.

00:02:11.430 --> 00:02:15.185
So that's going to be
equal to-- so R of x, y

00:02:15.185 --> 00:02:24.660
z evaluated when Z is equal to
that is R of x, y, f2 of x, y.

00:02:24.660 --> 00:02:27.480
And from that, we
need to subtract R

00:02:27.480 --> 00:02:36.540
when Z is this-- minus
R of x, y f1 of x, y,

00:02:36.540 --> 00:02:38.825
and then make sure that
we got our parentheses.

00:02:42.430 --> 00:02:45.480
Now, this is exactly what
we saw in the last video.

00:02:45.480 --> 00:02:50.850
It is exactly that, which shows
that this is exactly this.

00:02:50.850 --> 00:02:53.080
So when we assumed it
was a type I region,

00:02:53.080 --> 00:02:55.760
we got that this is
exactly equal to this.

00:02:55.760 --> 00:02:58.630
You do the exact same argument
with the type II region

00:02:58.630 --> 00:03:00.950
to show that this is equal
to this, type III region

00:03:00.950 --> 00:03:03.020
to show this is equal
to that, and you

00:03:03.020 --> 00:03:05.710
have your divergence
theorem proved.

00:03:05.710 --> 00:03:11.590
And we can consider
ourselves done.


Divergence theorem proof (part 4)

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.790
Sometimes when you're doing
a large multipart proof

00:00:02.790 --> 00:00:04.760
like this, it's easy
to lose your bearings.

00:00:04.760 --> 00:00:06.710
We're trying to prove
the divergence theorem.

00:00:06.710 --> 00:00:08.900
Well, we started off
just rewriting the flux

00:00:08.900 --> 00:00:12.660
across the surface and
rewriting the triple integral

00:00:12.660 --> 00:00:13.634
of the divergence.

00:00:13.634 --> 00:00:15.050
And we said, well,
if we can prove

00:00:15.050 --> 00:00:17.300
that each of these components
are equal to each other,

00:00:17.300 --> 00:00:18.530
we will have our proof done.

00:00:18.530 --> 00:00:20.710
And what I said is that
we're going to use the fact

00:00:20.710 --> 00:00:22.940
this is a Type 1 region
to prove this part,

00:00:22.940 --> 00:00:25.191
the fact that it's a Type 2
region to prove this part,

00:00:25.191 --> 00:00:27.606
and the fact that it's a Type
3 region to prove this part.

00:00:27.606 --> 00:00:29.400
In particular, I'm
going to show the fact

00:00:29.400 --> 00:00:31.950
that, if it is a Type 1
region, we can prove this.

00:00:31.950 --> 00:00:33.660
And you can use the
exact same argument

00:00:33.660 --> 00:00:35.560
to prove the other
two-- in which case,

00:00:35.560 --> 00:00:37.680
the divergence theorem
would be correct.

00:00:37.680 --> 00:00:39.520
So we're focused on
this part right here.

00:00:39.520 --> 00:00:42.980
And in particular, we're focused
on evaluating this surface

00:00:42.980 --> 00:00:43.607
integral.

00:00:43.607 --> 00:00:45.190
And to evaluate that
surface integral,

00:00:45.190 --> 00:00:48.220
we broke up the entire
surface into three surfaces--

00:00:48.220 --> 00:00:50.720
that upper bound on z,
the lower bound on z,

00:00:50.720 --> 00:00:53.060
and kind of the side
of-- if you imagine

00:00:53.060 --> 00:00:54.530
some type of a funky cylinder.

00:00:54.530 --> 00:00:56.885
These don't have to be
flat tops and bottoms.

00:00:56.885 --> 00:00:58.510
And for a Type 1
region, you don't even

00:00:58.510 --> 00:00:59.801
have to have this side surface.

00:00:59.801 --> 00:01:03.010
This is only for the case where
the two surfaces don't touch

00:01:03.010 --> 00:01:06.130
each other within this
domain right over here.

00:01:06.130 --> 00:01:07.830
So we broke it up
into three surfaces.

00:01:07.830 --> 00:01:10.160
But we said, look,
the normal vector

00:01:10.160 --> 00:01:11.940
along the side right
over here is never

00:01:11.940 --> 00:01:15.300
going to have a k component.

00:01:15.300 --> 00:01:18.570
And so when you take the dot
product with the k unit vector,

00:01:18.570 --> 00:01:20.560
this is just going to cross out.

00:01:20.560 --> 00:01:22.080
So our surface
integral simplified

00:01:22.080 --> 00:01:26.950
to the surface integral over s2
and the surface integral of s1.

00:01:26.950 --> 00:01:29.897
And in the last video, we
evaluated the surface integral

00:01:29.897 --> 00:01:32.230
of s2-- or at least, we turned
it into a double integral

00:01:32.230 --> 00:01:33.550
over the domain.

00:01:33.550 --> 00:01:36.960
Now, I'm going to do the
same exact thing with s1.

00:01:36.960 --> 00:01:39.850
So let's just remind ourselves
what we are concerned with.

00:01:39.850 --> 00:01:42.120
We want to re-express s1.

00:01:46.170 --> 00:01:48.000
This surface
integral-- I should say

00:01:48.000 --> 00:01:50.730
we want to re-express the
surface integral over s1.

00:01:50.730 --> 00:01:53.864
It was r of x, y, and z.

00:01:53.864 --> 00:01:56.155
Up here we just wrote r, but
this is making it explicit

00:01:56.155 --> 00:01:59.710
that r is a function
of x, y, and z.

00:01:59.710 --> 00:02:06.450
Times k dot n ds.

00:02:06.450 --> 00:02:08.840
And the way we evaluate
any surface integral

00:02:08.840 --> 00:02:11.960
is we want to have
a parameterization

00:02:11.960 --> 00:02:13.690
for our actual surface.

00:02:13.690 --> 00:02:15.440
So let's introduce
a parameterization

00:02:15.440 --> 00:02:16.560
for our surface.

00:02:16.560 --> 00:02:21.560
Let's say that
s1-- and I'll use--

00:02:21.560 --> 00:02:23.440
the letter O is kind
of a weird one to use.

00:02:23.440 --> 00:02:24.939
It looks like a 0,
and we're already

00:02:24.939 --> 00:02:27.090
using it for our normal vector.

00:02:27.090 --> 00:02:28.030
Let's use e.

00:02:28.030 --> 00:02:32.690
So the parameterization of
our surface could be x times i

00:02:32.690 --> 00:02:37.460
plus y times j plus-- and
since it's a Type 2 region,

00:02:37.460 --> 00:02:42.120
our surface is a function
of x and y-- plus f1--

00:02:42.120 --> 00:02:44.770
which is a function
of xy-- times k.

00:02:44.770 --> 00:02:48.420
We see f1 right over here.

00:02:48.420 --> 00:02:51.240
That's our lower
bound on our region.

00:02:51.240 --> 00:02:55.240
And then, this is for
all of the xy pairs that

00:02:55.240 --> 00:02:59.590
are a member of our
domain in question.

00:02:59.590 --> 00:03:01.120
So we have our parameterization.

00:03:01.120 --> 00:03:04.800
Now, we can think
about how we can

00:03:04.800 --> 00:03:08.390
write this right
over here, the nds.

00:03:08.390 --> 00:03:11.652
And so n times ds-- so
we've done this multiple,

00:03:11.652 --> 00:03:14.070
multiple, multiple
times-- which is actually

00:03:14.070 --> 00:03:17.610
the same thing as ds.

00:03:17.610 --> 00:03:19.320
We've done this multiple times.

00:03:19.320 --> 00:03:23.100
This is equal to the cross
product of the parameterization

00:03:23.100 --> 00:03:25.740
in one direction with
respect to one parameter.

00:03:25.740 --> 00:03:28.990
And then, that cross with
respect to the other parameter.

00:03:28.990 --> 00:03:31.300
And then, that times da.

00:03:31.300 --> 00:03:33.850
But we want to make sure
we get the order right.

00:03:33.850 --> 00:03:35.360
So I'm going to
claim that this is

00:03:35.360 --> 00:03:39.180
going to be the partial of our
parameterization with respect

00:03:39.180 --> 00:03:42.810
to y crossed with the partial
of our parameterization

00:03:42.810 --> 00:03:44.540
with respect to x.

00:03:44.540 --> 00:03:46.307
And then, we have times da.

00:03:46.307 --> 00:03:48.640
But we need to make sure this
has the right orientation.

00:03:48.640 --> 00:03:51.120
Because, for this bottom
surface, remember,

00:03:51.120 --> 00:03:57.090
we need to be pointed straight
down, outward from the region.

00:03:57.090 --> 00:03:59.070
So if we're going
in the y direction,

00:03:59.070 --> 00:04:01.210
the partial with respect
to y is like that.

00:04:01.210 --> 00:04:04.550
The partial with respect
to x is like that.

00:04:04.550 --> 00:04:06.040
And if you use the
right-hand rule,

00:04:06.040 --> 00:04:08.060
your thumb will point
downward, like that.

00:04:08.060 --> 00:04:09.800
I could draw my right hand.

00:04:09.800 --> 00:04:11.850
My index finger
could go like that.

00:04:11.850 --> 00:04:13.560
My middle finger
would bend like this.

00:04:13.560 --> 00:04:15.720
I don't really care what
my other two fingers do.

00:04:15.720 --> 00:04:18.010
And then, my thumb
would go downwards.

00:04:18.010 --> 00:04:20.425
So this is the right order,
which is a different order,

00:04:20.425 --> 00:04:22.550
or it's the opposite
ordering, as we did last time.

00:04:22.550 --> 00:04:24.383
You'll see, we'll just
get a negative value,

00:04:24.383 --> 00:04:26.240
but let's just work
it through just

00:04:26.240 --> 00:04:28.750
to be a little bit
more convincing.

00:04:28.750 --> 00:04:31.060
So this business
right over here is

00:04:31.060 --> 00:04:40.150
going to be equal to-- let's
write our i, j, and k unit

00:04:40.150 --> 00:04:41.310
vectors.

00:04:41.310 --> 00:04:44.160
It's going to be all
of that times da.

00:04:44.160 --> 00:04:49.350
So the partial of
e with respect to y

00:04:49.350 --> 00:04:57.640
is 0, 1, partial of
f1 with respect to y.

00:04:57.640 --> 00:04:59.920
Partial with respect
to x is going

00:04:59.920 --> 00:05:05.419
to be 0, 1, partial of
f1 with respect to x.

00:05:05.419 --> 00:05:07.335
And then, when you
evaluate this whole thing--

00:05:07.335 --> 00:05:09.251
and so let me draw a
little dotted line here--

00:05:09.251 --> 00:05:12.850
this is going to be equal
to some business times i

00:05:12.850 --> 00:05:15.470
minus some other
business times j.

00:05:15.470 --> 00:05:17.120
And on our k
component, you're going

00:05:17.120 --> 00:05:20.230
to have 0 times 0
minus 1 times k.

00:05:20.230 --> 00:05:25.400
So minus k, and then,
all of that times da.

00:05:25.400 --> 00:05:27.910
And now, the reason why I didn't
take even worry about what

00:05:27.910 --> 00:05:29.540
these things are
going to be is I'm

00:05:29.540 --> 00:05:31.840
going to have to take the
dropped product with k.

00:05:31.840 --> 00:05:34.280
So this whole thing--
all of this business

00:05:34.280 --> 00:05:37.550
right over here-- I can now
express in the xy domain.

00:05:37.550 --> 00:05:42.450
I can now write is going to be
equal to the double integral.

00:05:42.450 --> 00:05:44.200
And let me do it in
that same purple color

00:05:44.200 --> 00:05:45.960
because that was the original
color for that surface

00:05:45.960 --> 00:05:46.460
integral.

00:05:46.460 --> 00:05:50.110
It's going to be the double
integral over our parameter's

00:05:50.110 --> 00:05:56.560
domain, in the xy
plane of r of x--

00:05:56.560 --> 00:06:01.090
let me write this a little
bit cleaner-- r of xy.

00:06:01.090 --> 00:06:04.590
Instead of z, I'm going to
write z over that surface is

00:06:04.590 --> 00:06:07.470
f1 of x and y, so that
we have everything

00:06:07.470 --> 00:06:09.250
in terms of our parameters.

00:06:09.250 --> 00:06:12.670
Times all of this
business k dotted this.

00:06:12.670 --> 00:06:14.740
Well, what is k dotted this?

00:06:14.740 --> 00:06:17.500
Well, the dot product of k
and negative k is negative 1.

00:06:17.500 --> 00:06:20.340
So we're just going to
be left with negative da.

00:06:20.340 --> 00:06:24.550
So we'll put the
da out front here.

00:06:24.550 --> 00:06:27.860
And we have a negative
da right over here.

00:06:27.860 --> 00:06:35.090
So we have now expressed
this surface integral

00:06:35.090 --> 00:06:37.030
as the sum of two
double integrals.

00:06:40.150 --> 00:06:41.150
Let me find it.

00:06:41.150 --> 00:06:47.670
The sum of this, the sum of
that and that right over there.

00:06:47.670 --> 00:06:50.030
And actually, let
me just rewrite it

00:06:50.030 --> 00:06:52.340
just so that we can
make everything clear.

00:06:52.340 --> 00:06:54.942
So this surface integral.

00:06:54.942 --> 00:06:56.400
So let me write it
right over here.

00:06:56.400 --> 00:07:01.440
The surface integral over our
entire surface of r times k

00:07:01.440 --> 00:07:06.895
dot n ds is equal to
the double integral--

00:07:06.895 --> 00:07:08.870
and I'll do this
in a new color--

00:07:08.870 --> 00:07:12.170
is equal to the double
integral in the domain

00:07:12.170 --> 00:07:18.500
d of this thing
minus this thing da.

00:07:18.500 --> 00:07:26.830
So I'll write r of
xy and f2 of xy,

00:07:26.830 --> 00:07:37.220
and that's that minus this--
minus r of xy and f1--

00:07:37.220 --> 00:07:41.719
be careful here-- f1 of xy.

00:07:41.719 --> 00:07:43.135
That's this thing
right over here.

00:07:46.208 --> 00:07:49.110
All of that times da.

00:07:49.110 --> 00:07:51.610
Now, we just showed
this is equal to this.

00:07:51.610 --> 00:07:54.760
All we have to show now
is that this is also

00:07:54.760 --> 00:07:56.010
equal to that same expression.

00:07:56.010 --> 00:07:58.602
And we will have proven
this for the Type 1 case.

00:07:58.602 --> 00:08:00.810
And we can use the exact
same argument for the Type 2

00:08:00.810 --> 00:08:04.360
and Type 3 cases and feel good
that the divergence theorem is

00:08:04.360 --> 00:08:08.270
about to be proven for
a simple solid region.