Integration with partial fractions

WEBVTT
Kind: captions
Language: en

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- [Instructor] We are
asked to find the value

00:00:01.680 --> 00:00:03.440
of this indefinite integral.

00:00:03.440 --> 00:00:06.210
And some of you, in attempting
this, might try to say,

00:00:06.210 --> 00:00:09.370
all right, is the numerator
here the derivative

00:00:09.370 --> 00:00:12.700
or a constant multiple of the
derivative of the denominator?

00:00:12.700 --> 00:00:15.170
In which case, u-substitution might apply,

00:00:15.170 --> 00:00:16.700
but it's not the case here.

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So what do we do?

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And my hint to you would be
partial fraction decomposition,

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which might invoke some memories
from a precalculus class

00:00:25.300 --> 00:00:27.270
or maybe from an algebra 2 class,

00:00:27.270 --> 00:00:30.170
but it's a technique to break
up this rational expression

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into the sum of two rational expressions.

00:00:32.460 --> 00:00:35.160
And a good hint there is the fact that

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this denominator here is factorable

00:00:37.380 --> 00:00:39.320
into these two expressions.

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So what we're gonna try to do

00:00:40.320 --> 00:00:42.730
with partial fraction
decomposition is say,

00:00:42.730 --> 00:00:45.673
can we express x-5 over (2x-3)(x-1),

00:00:49.650 --> 00:00:52.960
can we express it as a sum
of two rational expressions,

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where the denominator of the
first rational expression

00:00:55.550 --> 00:00:57.620
is 2x-3,

00:00:57.620 --> 00:01:00.270
and the denominator of the
second rational expression

00:01:00.270 --> 00:01:05.270
is x-1, and I don't have to
put parentheses there, is x-1.

00:01:05.330 --> 00:01:07.300
Now we don't know the numerators.

00:01:07.300 --> 00:01:09.060
As you might have learned before,

00:01:09.060 --> 00:01:10.390
and I encourage you,
if this the first time

00:01:10.390 --> 00:01:12.820
you're ever seeing partial
fraction decomposition,

00:01:12.820 --> 00:01:14.370
look that up on Khan Academy.

00:01:14.370 --> 00:01:16.110
We have many videos on it.

00:01:16.110 --> 00:01:17.300
But the general principle here

00:01:17.300 --> 00:01:19.120
is that your numerator is
gonna be one degree less

00:01:19.120 --> 00:01:19.953
than your denominator.

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So our denominators here are first degree,

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so our numerator's gonna be
zero degree, or just constants.

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So it's gonna be some
unknown constant data.

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Let's call that A.

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And some unknown constant B.

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And our goal is to solve for A and B.

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And this is all a review.

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This isn't really calculus.

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This is more precalculus now, or algebra.

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How do we solve for A and B?

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Well, let's add them

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as if we're just adding two fractions

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with unlike denominators.

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So we wanna have a common denominator.

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So what we wanna do

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is multiply this first rational expression

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by x-1 in the numerator
and the denominator.

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So you could call this A(x-1)

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over (2x-3)(x-1).

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And then for this second
rational expression,

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we'd multiply the numerator
and the denominator

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by 2x-3.

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So (2x-3)B

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over (2x-3)(x-1).

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And now, since I have the same
denominator, I can add them.

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And the goal is when I add 'em,

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and I think about the
numerator, I will try to say,

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well how do the A's and B's line up

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to what we have already, right over here?

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So this is going to be
equal to, same denominator,

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so the denominator's (2x-3)(x-1),

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and then in the numerators,

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actually, let me do it over here.

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This is the same thing as
Ax-A, just distributing the A,

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and if I distribute the B here,

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this is the same thing as 2Bx-3B.

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And so if we add the numerators,
we can add the x terms,

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Ax+2Bx, so we could call that (A+2B)x.

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And then if we add these
constant terms, -A, -3B.

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And since we have a minus right over here,

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I'm gonna try to pattern match,

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so I'm gonna have minus,

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and I'll write that as -(A+3B).

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If you distribute this negative sign,

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then you're gonna get -A and -3B.

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So these are equivalent.

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And now we can see the pattern.

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We say, all right, our
denominators are now the same,

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so this thing in our numerator has to be

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the same thing as x-5.

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So whatever our coefficient here is on x,

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that's gotta be equal to 1,
that's the coefficient on x.

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And this thing that we're subtracting,

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that's gotta be equal to 5.

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And we're gonna set up a system

00:03:43.250 --> 00:03:46.910
of two equations with two
unknowns to solve for A and B.

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So we know that A+, so
let me write that down,

00:03:49.480 --> 00:03:54.480
we know that A+2B needs to be equal to 1.

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And we know that A+3B, or A+3B,

00:04:00.540 --> 00:04:03.670
is going to be equal to 5.

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And so now, to solve for A and B,

00:04:06.440 --> 00:04:08.770
well, we could do that by elimination.

00:04:08.770 --> 00:04:13.170
So let's see, what if we
multiply this top equation by -1.

00:04:13.170 --> 00:04:18.170
So that'd be -A, -2B, -1,
and now we add them together.

00:04:19.810 --> 00:04:21.620
The whole point was to cancel out the A's,

00:04:21.620 --> 00:04:25.810
so we're left with -2B+3B,
that's just going to be B,

00:04:25.810 --> 00:04:27.780
is equal to 4,

00:04:27.780 --> 00:04:31.290
and then we can substitute
back in to solve for A.

00:04:31.290 --> 00:04:34.840
So let's say right over
here, we know that,

00:04:34.840 --> 00:04:37.633
we know that A+3x4,

00:04:38.720 --> 00:04:40.330
so that's gonna be 12,

00:04:40.330 --> 00:04:42.240
I'm using this equation right over here,

00:04:42.240 --> 00:04:44.340
is going to be equal to 5.

00:04:44.340 --> 00:04:49.340
Subtract 12 from both sides,
you get A is equal to -7.

00:04:49.640 --> 00:04:53.500
So just like that, we can
rewrite this entire integral.

00:04:53.500 --> 00:04:56.130
We can say this is going to be equal to

00:04:56.130 --> 00:04:59.540
the indefinite integral
of, open parentheses,

00:04:59.540 --> 00:05:01.170
A over 2x-3.

00:05:01.170 --> 00:05:03.520
We now know that A is -7,

00:05:03.520 --> 00:05:06.970
so it's -7 over 2x-3,

00:05:08.880 --> 00:05:13.650
and then we're going have +B, B is 4,

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so, +4 over x-1,

00:05:18.240 --> 00:05:22.860
over x-1, and close parentheses, dx.

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Now if you are so inspired,

00:05:24.830 --> 00:05:26.240
I encourage you to pause the video

00:05:26.240 --> 00:05:28.110
and try to run with it from this point,

00:05:28.110 --> 00:05:30.640
because we have seen the
techniques to solve integrals

00:05:30.640 --> 00:05:33.000
like this before, but
I'll do it step by step.

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This is going to be the same thing as,

00:05:35.110 --> 00:05:36.260
well actually let me just,

00:05:36.260 --> 00:05:37.610
so this is going to be the same thing as

00:05:37.610 --> 00:05:41.160
the integral of, so 2x-3.

00:05:41.160 --> 00:05:42.840
I could write the -7 here,

00:05:42.840 --> 00:05:45.200
but I'm gonna take the
constant out of the integral.

00:05:45.200 --> 00:05:46.910
So I'll put a -7 here.

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And to help us solve this,
and this could be a 1,

00:05:50.340 --> 00:05:51.320
but to help us solve this,

00:05:51.320 --> 00:05:52.870
it would be nice if we had a 2 here.

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Why is that?

00:05:53.720 --> 00:05:56.760
Because 2 is the derivative of 2x-3,

00:05:56.760 --> 00:05:59.130
and so then we can do our u-substitution,

00:05:59.130 --> 00:06:01.410
which we have, sometimes,
gotten practiced doing

00:06:01.410 --> 00:06:02.610
a little bit in our head.

00:06:02.610 --> 00:06:04.990
And so if we want this to be a 2,

00:06:04.990 --> 00:06:06.410
we can't just multiply by 2.

00:06:06.410 --> 00:06:08.140
We've also gotta divide by 2.

00:06:08.140 --> 00:06:10.820
And I can do that outside,
'cause that's just a constant

00:06:10.820 --> 00:06:15.210
dx+, well the derivative of x-1 is 1,

00:06:15.210 --> 00:06:16.720
so we just want a 1 up in the numerator.

00:06:16.720 --> 00:06:18.920
So we can take the 4 out of the integral.

00:06:18.920 --> 00:06:23.920
4 times the integral of 1 over x-1, dx.

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So this is going to be equal to,

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we just have our constant out front, -7/2,

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and since we have this
thing in the denominator,

00:06:33.610 --> 00:06:34.860
and we have its derivative,

00:06:34.860 --> 00:06:36.840
we can really just think
about this as integrating

00:06:36.840 --> 00:06:39.230
with respect to this
thing in the denominator.

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You can sometimes view u-substitution,

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which I'm not explicitly going to do here,

00:06:42.970 --> 00:06:44.570
as the reverse chain rule.

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The antiderivative of 1
over x is the natural log

00:06:47.230 --> 00:06:48.780
of the absolute value of x,

00:06:48.780 --> 00:06:51.010
but here, this is going,
the antiderivative of this

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is going to be the natural log

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of the absolute value of 2x-3.

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And then for this part, it's gonna be +4,

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times the antiderivative here

00:07:03.160 --> 00:07:07.100
is the natural log of the
absolute value of x-1.

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Once again, I can do that,

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'cause the derivative of x-1 is just 1.

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And of course, since we're
taking an indefinite integral,

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we do not want to forget our +C.

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And we're done.


Worked example: Merging definite integrals over adjacent intervals

WEBVTT
Kind: captions
Language: en

00:00:00.250 --> 00:00:01.300
- [Instructor] What we have here

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is a graph of y is equal to f of x,

00:00:04.400 --> 00:00:09.160
and these numbers are the
areas of these shaded regions,

00:00:09.160 --> 00:00:12.530
these regions between
our curve and the x-axis.

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What we're going to do in
this video is do some examples

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of evaluating definite
integrals using this information

00:00:17.970 --> 00:00:21.670
and some knowledge of
definite integral properties.

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So let's start with an example.

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Let's say we want to evaluate
the definite integral

00:00:25.650 --> 00:00:30.650
going from negative four
to negative two of f of x

00:00:31.120 --> 00:00:34.360
d of x plus the definite
integral going from

00:00:34.360 --> 00:00:38.760
negative two to zero of f of x dx.

00:00:38.760 --> 00:00:40.090
Pause this video, and see if you can

00:00:40.090 --> 00:00:41.993
evaluate this entire expression.

00:00:42.920 --> 00:00:45.470
So this first part of our expression,

00:00:45.470 --> 00:00:46.880
the definite integral from negative four

00:00:46.880 --> 00:00:49.020
to negative two of f of x dx,

00:00:49.020 --> 00:00:50.590
we're going from x equals negative four

00:00:50.590 --> 00:00:52.660
to x equals negative two.

00:00:52.660 --> 00:00:55.820
And so this would evaluate as

00:00:55.820 --> 00:00:56.960
this area

00:00:56.960 --> 00:01:00.360
between our curve and our x-axis,

00:01:00.360 --> 00:01:02.330
but it would be the negative of that area

00:01:02.330 --> 00:01:05.630
because our curve is below the x-axis.

00:01:05.630 --> 00:01:07.040
And we could try to estimate it

00:01:07.040 --> 00:01:08.520
based on the information they've given us,

00:01:08.520 --> 00:01:10.890
but they haven't given
us exactly that value.

00:01:10.890 --> 00:01:15.180
But we also need to figure
out this right over here.

00:01:15.180 --> 00:01:18.630
And here we're going from x
equals negative two to zero

00:01:18.630 --> 00:01:21.623
of f of x d of x, so that's
going to be this area.

00:01:22.960 --> 00:01:24.690
So if you're looking at the sum of these

00:01:24.690 --> 00:01:27.970
two definite integrals, and
notice the upper bound here

00:01:27.970 --> 00:01:31.400
is the lower bound here,
you're really thinking about,

00:01:31.400 --> 00:01:34.260
this is really going to
be the same thing as,

00:01:34.260 --> 00:01:37.600
this is equal to the definite integral

00:01:37.600 --> 00:01:39.830
going from x equals negative four

00:01:39.830 --> 00:01:44.640
all the way to x equals zero of f of x dx.

00:01:44.640 --> 00:01:47.480
And this is indeed one of
our integration properties.

00:01:47.480 --> 00:01:50.500
If our upper bound here is the
same as our lower bound here

00:01:50.500 --> 00:01:53.100
and we are integrating the same thing,

00:01:53.100 --> 00:01:54.800
well, then you can merge these two

00:01:54.800 --> 00:01:56.350
definite integrals in this way.

00:01:56.350 --> 00:01:59.710
And this is just going
to be this entire area.

00:01:59.710 --> 00:02:02.060
But because we are below the x-axis

00:02:02.060 --> 00:02:03.070
and above our curve here,

00:02:03.070 --> 00:02:04.800
it would be the negative of that area.

00:02:04.800 --> 00:02:08.670
So this is going to be
equal to negative seven.

00:02:08.670 --> 00:02:10.880
Let's do another example.

00:02:10.880 --> 00:02:12.590
Let's say someone were to ask you,

00:02:12.590 --> 00:02:15.260
walk up you on the street and say, quick,

00:02:15.260 --> 00:02:19.700
here's a graph, what is
the value of the expression

00:02:19.700 --> 00:02:20.700
that I'm about to write down,

00:02:20.700 --> 00:02:25.360
the definite integral going
from zero to four of f of x dx

00:02:26.230 --> 00:02:30.530
plus the definite integral
going from four to six

00:02:30.530 --> 00:02:33.280
of f of x dx?

00:02:33.280 --> 00:02:36.020
Pause this video, and see
if you can figure that out.

00:02:36.020 --> 00:02:39.690
Well, once again, this
first part right over here,

00:02:39.690 --> 00:02:43.150
going from zero to four,

00:02:43.150 --> 00:02:45.360
so what would be is it would be this area.

00:02:45.360 --> 00:02:47.960
It would be this five right over here,

00:02:47.960 --> 00:02:50.410
but then we would need
to subtract this area

00:02:50.410 --> 00:02:53.790
because this area is below our
x-axis and above our curve.

00:02:53.790 --> 00:02:56.700
We don't know exactly what this is.

00:02:56.700 --> 00:02:58.080
But luckily we also need,

00:02:58.080 --> 00:03:02.760
we need to take the sum of
everything I just showed,

00:03:02.760 --> 00:03:04.400
but plus,

00:03:04.400 --> 00:03:07.190
plus this right over here.

00:03:07.190 --> 00:03:10.060
And this, we're going from four to six,

00:03:10.060 --> 00:03:11.913
so it's going to be this area.

00:03:12.880 --> 00:03:15.360
Well, once again, when
you look at it this way,

00:03:15.360 --> 00:03:17.670
you can see that this expression

00:03:17.670 --> 00:03:20.700
is going to be equivalent to
taking the definite integral

00:03:20.700 --> 00:03:23.210
all the way from zero to six,

00:03:23.210 --> 00:03:26.820
zero to six of f of x dx.

00:03:26.820 --> 00:03:28.950
And once again, even if
you didn't see the graph,

00:03:28.950 --> 00:03:30.530
you would know that
because, in both cases,

00:03:30.530 --> 00:03:33.400
you're getting the definite
integral of f of x dx.

00:03:33.400 --> 00:03:36.660
And our upper bound here
is our lower bound here.

00:03:36.660 --> 00:03:38.680
So once again, we're able
to merge the integrals.

00:03:38.680 --> 00:03:41.280
And what is this going to be equal to?

00:03:41.280 --> 00:03:44.680
Well, we have this area
here, which is five.

00:03:44.680 --> 00:03:46.960
And then we have this area, which is six.

00:03:46.960 --> 00:03:48.270
That was given to us.

00:03:48.270 --> 00:03:51.200
But since it's below the
x-axis and above our curve,

00:03:51.200 --> 00:03:53.010
when we evaluate it as
a definite integral,

00:03:53.010 --> 00:03:54.660
it would evaluate as a negative six.

00:03:54.660 --> 00:03:57.910
So this is going to be
five plus negative six,

00:03:57.910 --> 00:04:00.120
which is equal to negative one.

00:04:00.120 --> 00:04:00.953
And we're done.


Functions defined by definite integrals (accumulation functions)

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:01.660
- [Instructor] You've
already spent a lot of

00:00:01.660 --> 00:00:04.650
your mathematical lives
talking about functions.

00:00:04.650 --> 00:00:08.270
The basic idea is give a
valid input into a function,

00:00:08.270 --> 00:00:11.180
so a member of that function's domain,

00:00:11.180 --> 00:00:15.950
and then the function is going
to tell you for that input

00:00:15.950 --> 00:00:18.610
what is going to be the
corresponding output.

00:00:18.610 --> 00:00:21.800
And we call that
corresponding output f of x.

00:00:21.800 --> 00:00:24.960
So, for example, there's many
ways of defining functions.

00:00:24.960 --> 00:00:29.960
You could say something like
f of x is equal to x squared.

00:00:30.460 --> 00:00:32.500
So that means that whatever x,

00:00:32.500 --> 00:00:34.310
whatever you input into the function,

00:00:34.310 --> 00:00:37.910
the output is going to
be that input squared.

00:00:37.910 --> 00:00:39.790
You could have something
defined like this.

00:00:39.790 --> 00:00:43.080
F of x is equal to

00:00:43.080 --> 00:00:44.150
x squared

00:00:45.689 --> 00:00:48.070
if x odd.

00:00:48.070 --> 00:00:53.070
And you could say it's equal
to x to the third otherwise,

00:00:53.250 --> 00:00:54.083
otherwise.

00:00:55.450 --> 00:00:57.980
So if it's an odd integer,

00:00:57.980 --> 00:00:59.790
it's an odd integer, you just square it.

00:00:59.790 --> 00:01:02.100
But otherwise, for any other real number,

00:01:02.100 --> 00:01:03.870
you take it to the third power.

00:01:03.870 --> 00:01:06.320
This is a valid way of
defining a function.

00:01:06.320 --> 00:01:08.980
What we're going to do in this
video is explore a new way

00:01:08.980 --> 00:01:11.760
or potentially a new way for
you of defining a function.

00:01:11.760 --> 00:01:14.650
And that's by using a definite integral,

00:01:14.650 --> 00:01:17.200
but it's the same general idea.

00:01:17.200 --> 00:01:20.020
So what we have graphed
here, this is the t-axis,

00:01:20.020 --> 00:01:23.540
this is the y-axis, and we have
the graph of the function f,

00:01:23.540 --> 00:01:26.940
or you could view this as the graph of

00:01:26.940 --> 00:01:28.870
y is equal to

00:01:28.870 --> 00:01:30.420
f of t.

00:01:30.420 --> 00:01:31.330
Now, what I want to,

00:01:31.330 --> 00:01:33.840
and this is another way of representing

00:01:33.840 --> 00:01:35.870
what outputs you might
get for a given input.

00:01:35.870 --> 00:01:38.720
Here, if t is one, f of t is five.

00:01:38.720 --> 00:01:41.440
If t is four, f of t is three.

00:01:41.440 --> 00:01:43.640
But I'm now going to define a new function

00:01:43.640 --> 00:01:47.000
based on a definite integral of f of t.

00:01:47.000 --> 00:01:48.920
Let's define our new function.

00:01:48.920 --> 00:01:51.080
Let's say g,

00:01:51.080 --> 00:01:53.370
let's call it g of x.

00:01:53.370 --> 00:01:55.940
Let's make it equal to
the definite integral

00:01:55.940 --> 00:01:57.760
from negative two

00:01:57.760 --> 00:01:59.480
to x

00:01:59.480 --> 00:02:02.120
of f of t

00:02:02.120 --> 00:02:02.953
dt.

00:02:04.240 --> 00:02:05.930
Now, pause this video,
really take a look at it.

00:02:05.930 --> 00:02:09.540
This might look really fancy,
but what's happening here is,

00:02:09.540 --> 00:02:13.810
given an input x, g of x
is going to be based on

00:02:13.810 --> 00:02:17.660
what the definite integral
here would be for that x.

00:02:17.660 --> 00:02:19.480
And so we can set up a little table here

00:02:19.480 --> 00:02:22.540
to think about some potential values.

00:02:22.540 --> 00:02:27.540
So let's say x, and let's
say g of x right over here.

00:02:27.620 --> 00:02:30.470
So if x is one,

00:02:30.470 --> 00:02:33.413
what is g of x going to be equal to?

00:02:34.390 --> 00:02:37.200
All right, so g of one

00:02:37.200 --> 00:02:39.960
is going to be equal to
the definite integral,

00:02:39.960 --> 00:02:41.520
going from negative two.

00:02:41.520 --> 00:02:43.750
Now x is going to be equal
to one in this situation.

00:02:43.750 --> 00:02:45.740
That's what we're inputting
into the function.

00:02:45.740 --> 00:02:49.510
So one is our upper bound of f of t dt.

00:02:49.510 --> 00:02:51.380
And what is that equal to?

00:02:51.380 --> 00:02:52.660
Well, that's going to be the area

00:02:52.660 --> 00:02:55.490
under the curve and above the t-axis,

00:02:55.490 --> 00:03:00.060
between t equals negative
two and t is equal to one.

00:03:00.060 --> 00:03:02.470
So it's going to be this area here.

00:03:02.470 --> 00:03:04.420
And we, since it's on a grid,

00:03:04.420 --> 00:03:05.730
we can actually figure this out.

00:03:05.730 --> 00:03:07.930
We can actually break
this up into two sections.

00:03:07.930 --> 00:03:12.140
This rectangular section is
three wide and five high,

00:03:12.140 --> 00:03:14.530
so it has an area of 15 square units.

00:03:14.530 --> 00:03:16.290
And this little triangular section up here

00:03:16.290 --> 00:03:18.580
is two wide and one high.

00:03:18.580 --> 00:03:19.940
Two times one times one half,

00:03:19.940 --> 00:03:22.730
area of a triangle, this
is going to be another one.

00:03:22.730 --> 00:03:25.723
So that area is going to be equal to 16.

00:03:26.580 --> 00:03:27.750
What if

00:03:27.750 --> 00:03:30.540
x is equal to

00:03:30.540 --> 00:03:31.373
two?

00:03:31.373 --> 00:03:33.960
What is g of two going to be equal to?

00:03:33.960 --> 00:03:36.210
Pause this video, and
try to figure that out.

00:03:37.310 --> 00:03:40.990
Well, g of two is going to be
equal to the definite integral

00:03:40.990 --> 00:03:44.050
from negative two, and now
our upper bound's going to be

00:03:44.050 --> 00:03:48.180
our input into the function
to two, of f of t dt.

00:03:50.030 --> 00:03:52.890
So that's going to be going from here,

00:03:52.890 --> 00:03:55.370
all the way now to here.

00:03:55.370 --> 00:03:57.610
And so it's the area we just calculated.

00:03:57.610 --> 00:03:58.930
It's all of this stuff,

00:03:58.930 --> 00:04:01.230
which we figured out was 16 square units,

00:04:01.230 --> 00:04:04.930
plus another one, two, three,
four, five square units.

00:04:04.930 --> 00:04:08.790
So 16 plus five, this is
going to be equal to 21.

00:04:08.790 --> 00:04:10.140
So hopefully that helps,

00:04:10.140 --> 00:04:12.950
and the key thing to appreciate
here is that we can define

00:04:12.950 --> 00:04:16.667
valid functions by using
definite integrals.


Congressional elections

WEBVTT
Kind: captions
Language: en

00:00:00.290 --> 00:00:03.200
- [Instructor] Here's a statistic
that might surprise you.

00:00:03.200 --> 00:00:08.200
95% of elected officials
who seek reelection

00:00:09.460 --> 00:00:12.283
in the United States get reelected.

00:00:14.620 --> 00:00:19.240
Just look at this graph that
shows the rate of reelection

00:00:19.240 --> 00:00:21.810
for members of the US
House of Representatives.

00:00:21.810 --> 00:00:24.530
It hovers all the way up here.

00:00:24.530 --> 00:00:29.530
It stands to reason that
if you have two candidates

00:00:29.624 --> 00:00:33.850
for every office, one from each party,

00:00:33.850 --> 00:00:37.610
it would be about 50% of
the time the candidates

00:00:37.610 --> 00:00:42.070
would be reelected, but
it's so much more than that.

00:00:42.070 --> 00:00:44.390
In this video on Congressional elections,

00:00:44.390 --> 00:00:47.910
I want to talk about some
of the structural aspects.

00:00:47.910 --> 00:00:51.570
That is, the way things
are set up in Congressional

00:00:51.570 --> 00:00:54.650
elections that leads to
this really high number.

00:00:54.650 --> 00:00:59.650
What are the advantages
of being an incumbent,

00:01:00.540 --> 00:01:05.540
that is, the person who already
holds a political office?

00:01:06.740 --> 00:01:11.740
So what kind of advantages
does an incumbent have

00:01:12.420 --> 00:01:15.850
simply for already occupying an office?

00:01:15.850 --> 00:01:20.850
Well, say maybe there's a
Senate race between Candidate

00:01:21.690 --> 00:01:26.690
Green, call her Amy
Green, and Candidate Rose.

00:01:29.390 --> 00:01:32.310
Maybe he's Ken Rose.

00:01:32.310 --> 00:01:36.574
Now what sort of advantages Amy Green

00:01:36.574 --> 00:01:39.660
just because she already
holds the Senate seat?

00:01:39.660 --> 00:01:43.450
Well, one of her advantages
is that she has a staff,

00:01:43.450 --> 00:01:48.380
she has people whose job it is to answer

00:01:48.380 --> 00:01:50.590
the phone calls of constituents,

00:01:50.590 --> 00:01:52.660
make them feel heard.

00:01:52.660 --> 00:01:55.330
They can do all sorts
of things to help her.

00:01:55.330 --> 00:01:58.850
She also has the advantage of mail.

00:01:58.850 --> 00:02:02.530
Members of Congress can
send out mail for free,

00:02:02.530 --> 00:02:07.530
whereas Candidate Rose over
here has to pay for staff

00:02:09.124 --> 00:02:12.730
and pay for mail.

00:02:12.730 --> 00:02:14.280
Already, that's a lot of money.

00:02:14.280 --> 00:02:17.220
The other thing that Candidate
Green has going for her

00:02:17.220 --> 00:02:18.760
is name recognition.

00:02:18.760 --> 00:02:23.740
She's already been a successful
candidate, which shows

00:02:23.740 --> 00:02:27.330
that she knows how to win an election

00:02:27.330 --> 00:02:30.930
and there's a lot of advantage
to having name recognition.

00:02:30.930 --> 00:02:33.380
It's its own kind of advertising.

00:02:33.380 --> 00:02:34.700
Are you going to vote for someone

00:02:34.700 --> 00:02:37.630
you've never heard of before or someone

00:02:37.630 --> 00:02:41.200
who frequently appears in the media

00:02:41.200 --> 00:02:43.239
who could be going to local

00:02:43.239 --> 00:02:47.970
or state events, school openings.

00:02:47.970 --> 00:02:52.970
You get a lot of face time
simply from occupying a seat,

00:02:53.690 --> 00:02:55.860
and that's free advertising as well.

00:02:55.860 --> 00:02:58.940
The last aspect is money.

00:02:58.940 --> 00:03:01.460
Because of this incumbency advantage,

00:03:01.460 --> 00:03:04.940
and this is little bit
of a repeating cycle,

00:03:04.940 --> 00:03:09.834
interest groups, lobbyists,
and others are far more likely

00:03:09.834 --> 00:03:13.330
to give money to the
candidate who is already

00:03:13.330 --> 00:03:15.640
in a position because they know

00:03:15.640 --> 00:03:19.270
that candidate is likely
to win and therefore,

00:03:19.270 --> 00:03:21.988
their money is going to get them access

00:03:21.988 --> 00:03:26.540
to that candidate who will
hear out their concerns

00:03:26.540 --> 00:03:28.530
and perhaps look upon them favorably.

00:03:28.530 --> 00:03:31.710
So compared to all of this, Candidate Rose

00:03:31.710 --> 00:03:34.020
has a really uphill battle.

00:03:34.020 --> 00:03:35.930
He's got to pay for a lot of things

00:03:35.930 --> 00:03:38.090
that the incumbent doesn't have to,

00:03:38.090 --> 00:03:41.590
he doesn't have the name
recognition or the automatic

00:03:41.590 --> 00:03:45.340
media attention, and he's
viewed less favorably

00:03:45.340 --> 00:03:47.370
by the people who might give money.

00:03:47.370 --> 00:03:49.940
He's a bit of an unknown quantity.

00:03:49.940 --> 00:03:53.060
Another advantage that incumbents have

00:03:53.060 --> 00:03:56.530
is that they get to draw voting districts

00:03:56.530 --> 00:04:00.430
at least every 10 years in
response to the US Census

00:04:00.430 --> 00:04:03.630
and changes in populations,
the districts and states

00:04:03.630 --> 00:04:05.190
must be redrawn.

00:04:05.190 --> 00:04:09.121
Politicians do that redrawing
and they tend to use

00:04:09.121 --> 00:04:14.121
this opportunity to create
safe districts for themselves,

00:04:15.030 --> 00:04:18.598
safe being districts where the voters

00:04:18.598 --> 00:04:22.750
who are arranged
geographically will be likely

00:04:22.750 --> 00:04:25.900
to vote them back into power.

00:04:25.900 --> 00:04:29.159
Those in power will also
generally use this opportunity

00:04:29.159 --> 00:04:33.430
to attempt to harm their opponents

00:04:33.430 --> 00:04:35.723
in a process known as gerrymandering.

00:04:37.520 --> 00:04:40.600
There are two other major
factors that contribute

00:04:40.600 --> 00:04:43.610
to whether a Congress member is reelected.

00:04:43.610 --> 00:04:47.030
One of them is in each state, the form

00:04:47.030 --> 00:04:48.980
that their primary takes.

00:04:48.980 --> 00:04:53.980
Some states like Iowa have
caucuses where party members

00:04:54.770 --> 00:04:59.270
gather to vote on who
their nominee should be.

00:04:59.270 --> 00:05:02.030
These are very tightly
controlled by the party.

00:05:02.030 --> 00:05:07.030
There are closed primaries where only

00:05:07.137 --> 00:05:09.237
the registered members of a party may vote

00:05:10.572 --> 00:05:11.500
on their party's nominees for election,

00:05:11.500 --> 00:05:14.120
and then there are open primaries,

00:05:14.120 --> 00:05:19.120
where any registered voter
can vote for a party nominee,

00:05:19.310 --> 00:05:22.070
even if they belong to the opposing party.

00:05:22.070 --> 00:05:27.070
So the form estates primary
takes as a strong effect

00:05:27.490 --> 00:05:32.490
on how tight the party
control is over that process.

00:05:32.890 --> 00:05:37.120
The last factor affecting
Congressional elections

00:05:37.120 --> 00:05:39.490
is whether or not they coincide

00:05:39.490 --> 00:05:41.150
with a presidential election.

00:05:41.150 --> 00:05:44.560
There's much higher voter
turnout for presidential

00:05:44.560 --> 00:05:47.110
elections since there's a
great deal of interest in them

00:05:47.110 --> 00:05:50.020
and money spent on them, and that means

00:05:50.020 --> 00:05:53.980
that you tend to get a broader
sample of the population.

00:05:53.980 --> 00:05:57.360
More people are likely to vote,
so you'll have a better idea

00:05:57.360 --> 00:06:01.490
of how the American voting
public is feeling more generally,

00:06:01.490 --> 00:06:05.240
but in midterm elections,
which are those elections

00:06:05.240 --> 00:06:08.070
that don't fall at the same time

00:06:08.070 --> 00:06:12.160
as a presidential election,
voter turnout is lower,

00:06:12.160 --> 00:06:16.460
and that means that it's gonna
be the really engaged voters

00:06:16.460 --> 00:06:19.380
who are often strongly
aligned with a party

00:06:19.380 --> 00:06:20.573
that will come out.

00:06:22.920 --> 00:06:27.920
Midterm elections also tend
to punish the party in power,

00:06:28.640 --> 00:06:32.940
so when the president is highly regarded

00:06:32.940 --> 00:06:37.270
and doing well, his or her
party tends to gain seats,

00:06:37.270 --> 00:06:40.630
but if the president is
doing poorly or has a low

00:06:40.630 --> 00:06:43.770
public opinion rating,
then Congressional seats

00:06:43.770 --> 00:06:46.180
will often go to the opposite party.

00:06:46.180 --> 00:06:49.780
You can see here this
chart of net gain and loss

00:06:49.780 --> 00:06:53.530
of the president's
party, and in most cases,

00:06:53.530 --> 00:06:57.840
the president will gain a few followers

00:06:57.840 --> 00:07:00.530
in the first midterm election.

00:07:00.530 --> 00:07:04.020
See here, George W. Bush, 2002.

00:07:04.020 --> 00:07:07.660
His party gained both
House and Senate seats,

00:07:07.660 --> 00:07:12.660
but often in the sixth year
of a two-term presidency,

00:07:13.380 --> 00:07:16.410
voters harshly punish
the president's party.

00:07:16.410 --> 00:07:20.080
You can see that in George
W. Bush's sixth year,

00:07:20.080 --> 00:07:22.730
the Republicans lost 30 House seats

00:07:22.730 --> 00:07:26.330
and six Senate seats, and
a similar effect happened

00:07:26.330 --> 00:07:28.350
with Barack Obama in 2014.

00:07:28.350 --> 00:07:33.350
The Democrats lost 13 House
seats and nine Senate seats.

00:07:33.560 --> 00:07:38.070
So there's a strong correlation
with the president's party

00:07:38.070 --> 00:07:40.480
and the fates of members
of Congress who belong

00:07:40.480 --> 00:07:41.353
to that party.


Z-score introduction

WEBVTT
Kind: captions
Language: en

00:00:00.170 --> 00:00:02.360
- [Instructor] One of the
most commonly used tools

00:00:02.360 --> 00:00:06.340
in all of statistics is
the notion of a Z-score.

00:00:06.340 --> 00:00:08.330
And one way to think about a Z-score

00:00:08.330 --> 00:00:11.520
is it's just the number
of standard deviations

00:00:11.520 --> 00:00:15.420
away from the mean that
a certain data point is.

00:00:15.420 --> 00:00:16.630
So let me write that down.

00:00:16.630 --> 00:00:19.510
Number of standard deviations.

00:00:19.510 --> 00:00:20.343
I'll write it like this.

00:00:20.343 --> 00:00:25.343
Number of standard deviations
from our population mean

00:00:25.840 --> 00:00:30.840
for a particular, particular data point.

00:00:33.110 --> 00:00:35.710
Now let's make that a little bit concrete.

00:00:35.710 --> 00:00:38.550
Let's say that you're some
type of marine biologist

00:00:38.550 --> 00:00:42.480
and you've discovered a new
species of winged turtles

00:00:42.480 --> 00:00:45.810
and there's a total of
seven winged turtles,

00:00:45.810 --> 00:00:49.520
the entire population of
these winged turtles is seven.

00:00:49.520 --> 00:00:51.540
And so you go and you're actually able

00:00:51.540 --> 00:00:54.170
to measure all the winged turtles

00:00:54.170 --> 00:00:56.120
and you care about their
length and you also

00:00:56.120 --> 00:00:58.540
wanna care about, how are
those lengths distributed?

00:00:58.540 --> 00:01:02.170
Lengths of winged turtles.

00:01:06.010 --> 00:01:10.210
All right, and let's say, and
this is all in centimeters.

00:01:10.210 --> 00:01:12.130
These are very small turtles.

00:01:12.130 --> 00:01:14.210
So you discover, and these are all adults.

00:01:14.210 --> 00:01:15.930
So there's a two centimeter one,

00:01:15.930 --> 00:01:17.560
there's another two centimeter one.

00:01:17.560 --> 00:01:19.470
There's a three centimeter one.

00:01:19.470 --> 00:01:21.570
There's another two centimeter one.

00:01:21.570 --> 00:01:24.980
There's a five centimeter
one, a one centimeter one,

00:01:24.980 --> 00:01:27.400
and a six centimeter one.

00:01:27.400 --> 00:01:30.610
So we have seven data
points and from this,

00:01:30.610 --> 00:01:33.100
and I encourage you at
any point if you want.

00:01:33.100 --> 00:01:34.940
Pause this video and see
if you wanna calculate,

00:01:34.940 --> 00:01:36.330
what is the population mean here?

00:01:36.330 --> 00:01:38.230
We're assuming that this is the population

00:01:38.230 --> 00:01:39.933
of all the winged turtles.

00:01:40.770 --> 00:01:45.580
Well, the mean in this situation
is going to be equal to,

00:01:45.580 --> 00:01:48.820
you could add up all these
numbers and divide by seven

00:01:48.820 --> 00:01:51.170
and you would then get three.

00:01:51.170 --> 00:01:54.080
And then using these
data points and the mean

00:01:54.080 --> 00:01:57.470
you can calculate the
population standard deviation.

00:01:57.470 --> 00:01:59.920
And once again, as review
I always encourage you

00:01:59.920 --> 00:02:02.370
to pause this video and see
if you can do it on your own.

00:02:02.370 --> 00:02:04.350
But I've calculated that ahead of time.

00:02:04.350 --> 00:02:08.040
The population standard
deviation in this situation

00:02:08.040 --> 00:02:13.040
is approximately, I'll round
to the hundredth place, 1.69.

00:02:15.750 --> 00:02:19.230
So with this information you
should be able to calculate

00:02:19.230 --> 00:02:22.420
the Z-score for each of these data points.

00:02:22.420 --> 00:02:25.000
Pause this video and
see if you can do that.

00:02:25.000 --> 00:02:26.460
So let me make a new column here.

00:02:26.460 --> 00:02:29.173
So here I'm gonna put our Z-score.

00:02:30.080 --> 00:02:31.770
And if you just look at the definition

00:02:31.770 --> 00:02:33.540
what you're going to do for
each of these data points,

00:02:33.540 --> 00:02:35.580
let's say each data point is x,

00:02:35.580 --> 00:02:38.190
you're going to subtract
from that the mean

00:02:39.870 --> 00:02:41.510
and then you're going to divide that

00:02:41.510 --> 00:02:43.113
by the standard deviation.

00:02:44.230 --> 00:02:46.110
The numerator right over
here's gonna tell you

00:02:46.110 --> 00:02:48.640
how far you are above or below the mean,

00:02:48.640 --> 00:02:51.030
but you wanna know how
many standard deviations

00:02:51.030 --> 00:02:53.670
you are from the mean,
so then you'll divide

00:02:53.670 --> 00:02:56.050
by the population standard deviation.

00:02:56.050 --> 00:03:00.270
So for example, this first
data point right over here

00:03:00.270 --> 00:03:03.670
if I wanna calculate the
Z-score I will take two.

00:03:03.670 --> 00:03:05.970
From that I will subtract three

00:03:05.970 --> 00:03:09.010
and then I will divide by 1.69.

00:03:09.010 --> 00:03:11.263
I will divide by 1.69.

00:03:12.550 --> 00:03:14.140
And if you've got a calculator out

00:03:14.140 --> 00:03:18.110
this is going to be -1 divided by 1.69

00:03:18.110 --> 00:03:20.670
and if you use a calculator you would get,

00:03:20.670 --> 00:03:24.883
this is going to be approximately -0.59.

00:03:26.690 --> 00:03:30.360
And the Z-score for this data
point is going to be the same.

00:03:30.360 --> 00:03:33.563
That is also going to be -0.59.

00:03:34.850 --> 00:03:36.410
One way to interpret this is,

00:03:36.410 --> 00:03:40.210
this is a little bit more
than half a standard deviation

00:03:40.210 --> 00:03:44.120
below the mean, and we could
do a similar calculation

00:03:44.120 --> 00:03:46.040
for data points that are above the mean.

00:03:46.040 --> 00:03:47.990
Let's say this data point right over here.

00:03:47.990 --> 00:03:49.320
What is its Z-score?

00:03:49.320 --> 00:03:52.130
Pause this video and see
if you can figure that out.

00:03:52.130 --> 00:03:57.130
Well, it's going to be six
minus our mean, so minus three.

00:03:57.200 --> 00:04:00.160
All of that over the standard deviation.

00:04:00.160 --> 00:04:04.610
All of that over 1.69 and
this, if you have a calculator,

00:04:04.610 --> 00:04:06.420
and I calculated it ahead of time,

00:04:06.420 --> 00:04:11.160
this is going to be approximately 1.77.

00:04:11.160 --> 00:04:15.310
So more than one, but less
than two standard deviations

00:04:15.310 --> 00:04:17.920
above the mean.

00:04:17.920 --> 00:04:19.240
I encourage you to pause this video

00:04:19.240 --> 00:04:20.660
and now try to figure out the Z-scores

00:04:20.660 --> 00:04:23.210
for these other data points.

00:04:23.210 --> 00:04:26.160
Now, an obvious question that
some of you might be asking

00:04:26.160 --> 00:04:30.430
is why, why do we care how
many standard deviations

00:04:30.430 --> 00:04:33.640
above or below the mean a data point is?

00:04:33.640 --> 00:04:35.680
In your future statistical life,

00:04:35.680 --> 00:04:38.030
Z-scores are gonna be a really useful way

00:04:38.030 --> 00:04:41.240
to think about how usual or how unusual

00:04:41.240 --> 00:04:42.750
a certain data point is.

00:04:42.750 --> 00:04:44.750
And that's going to be really valuable

00:04:44.750 --> 00:04:48.380
once we start making
inferences based on our data.

00:04:48.380 --> 00:04:49.720
So I will leave you there.

00:04:49.720 --> 00:04:52.200
Just keep in mind it's a very useful idea,

00:04:52.200 --> 00:04:54.320
but at the heart of it
a fairly simple one.

00:04:54.320 --> 00:04:56.580
If you know the mean you
know the standard deviation.

00:04:56.580 --> 00:04:59.940
Take your data point, subtract
the mean from the data point

00:04:59.940 --> 00:05:02.730
and then divide by your
standard deviation.

00:05:02.730 --> 00:05:04.383
That gives you your Z-score.


Constitutional compromises: The Three-Fifths Compromise

WEBVTT
Kind: captions
Language: en

00:00:00.140 --> 00:00:01.020
- [Instructor] In the last video,

00:00:01.020 --> 00:00:03.720
we discussed one of the compromises made

00:00:03.720 --> 00:00:05.810
at the Constitutional Convention,

00:00:05.810 --> 00:00:08.460
the compromise of the electoral college.

00:00:08.460 --> 00:00:09.979
In this video, I want to discuss

00:00:09.979 --> 00:00:12.610
a different compromise.

00:00:12.610 --> 00:00:14.740
The compromise over slavery.

00:00:14.740 --> 00:00:16.387
Now, you'll remember
that one of the issues

00:00:16.387 --> 00:00:19.390
that the electoral college
was trying to solve

00:00:19.390 --> 00:00:21.737
was the idea that perhaps the revolution

00:00:21.737 --> 00:00:26.410
and the concept of
democracy had gone too far

00:00:26.410 --> 00:00:27.540
in the United States

00:00:27.540 --> 00:00:31.510
and needed to be reined
in by the more elite class

00:00:31.510 --> 00:00:33.100
of American citizens

00:00:33.100 --> 00:00:36.730
who would be better able to
make political decisions.

00:00:36.730 --> 00:00:38.010
But the flip side of this

00:00:38.010 --> 00:00:40.139
was whether the revolution had perhaps

00:00:40.139 --> 00:00:42.400
not gone far enough.

00:00:42.400 --> 00:00:46.750
In that, it didn't abolish
the institution of slavery.

00:00:46.750 --> 00:00:48.773
Now, the delegates at the
Constitutional Convention

00:00:48.773 --> 00:00:52.607
had sharply divided opinions over slavery.

00:00:52.607 --> 00:00:55.120
Those who came from southern states

00:00:55.120 --> 00:00:57.470
tended to be elite white men,

00:00:57.470 --> 00:00:59.634
who were themselves slave owners

00:00:59.634 --> 00:01:02.820
their own fortunes deeply tied into

00:01:02.820 --> 00:01:04.570
the institution of slavery.

00:01:04.570 --> 00:01:06.620
In the aftermath of the revolution,

00:01:06.620 --> 00:01:10.530
many northern states began to either

00:01:10.530 --> 00:01:14.630
outlaw or phase out slavery,

00:01:16.120 --> 00:01:19.200
recognizing that it was incompatible

00:01:19.200 --> 00:01:21.189
with the system of government defined

00:01:21.189 --> 00:01:24.320
around the concept that
all men are created equal.

00:01:24.320 --> 00:01:26.580
But if they were going to replace

00:01:26.580 --> 00:01:28.570
the articles of confederation,

00:01:28.570 --> 00:01:31.060
they were going to have
to find a way forward.

00:01:31.060 --> 00:01:34.900
And I would say overall, the
slaveowners got their way

00:01:34.900 --> 00:01:35.980
more than not.

00:01:35.980 --> 00:01:39.860
Now, one anti-slavery
aspect of the constitution

00:01:39.860 --> 00:01:43.480
was that it outlawed the
international slave trade,

00:01:43.480 --> 00:01:46.260
starting in 1808.

00:01:46.260 --> 00:01:47.970
So, here in article one,

00:01:47.970 --> 00:01:50.860
it states that the
migration or importation

00:01:50.860 --> 00:01:54.010
of such persons as any of
the states now existing

00:01:54.010 --> 00:01:55.770
shall think proper to admit

00:01:55.770 --> 00:01:58.120
shall not be prohibited by the congress

00:01:58.120 --> 00:02:01.290
prior to the year 1808,

00:02:01.290 --> 00:02:04.519
but a tax or duty may be
imposed on such importation

00:02:04.519 --> 00:02:07.090
not exceeding ten dollars for each person.

00:02:07.090 --> 00:02:11.145
So they say, we'll give the
international slave trade

00:02:11.145 --> 00:02:14.270
20 years and then we'll get rid of it.

00:02:14.270 --> 00:02:16.830
And indeed, on January 1st, 1808,

00:02:16.830 --> 00:02:19.190
the international slave trade was outlawed

00:02:19.190 --> 00:02:20.260
in the United States.

00:02:20.260 --> 00:02:24.163
So, this middle passage
in which enslaved Africans

00:02:24.163 --> 00:02:27.470
were taken from west Africa

00:02:27.470 --> 00:02:29.230
and brought across the Atlantic,

00:02:29.230 --> 00:02:32.600
the end would never be
in the United States.

00:02:32.600 --> 00:02:34.381
But this doesn't mean that
they outlawed slavery,

00:02:34.381 --> 00:02:36.430
and it doesn't mean that they outlawed

00:02:36.430 --> 00:02:38.280
the domestic slave trade,

00:02:38.280 --> 00:02:43.120
the trade in slaves between
states or within states.

00:02:43.120 --> 00:02:45.710
In fact, up until about 1850,

00:02:45.710 --> 00:02:48.950
one of the largest slave
markets in the United States

00:02:48.950 --> 00:02:51.477
was just around the corner
from the White House

00:02:51.477 --> 00:02:53.050
and the US Capitol.

00:02:53.050 --> 00:02:56.850
So, imagine walking on the
streets of Washington DC

00:02:56.850 --> 00:02:58.050
and seeing these buildings

00:02:58.050 --> 00:03:00.844
where democratic ideals are enshrined

00:03:00.844 --> 00:03:02.810
and then going around the corner

00:03:02.810 --> 00:03:05.940
and seeing women and
children and men being sold

00:03:05.940 --> 00:03:07.770
and families being torn apart.

00:03:07.770 --> 00:03:09.156
It's a very powerful image.

00:03:09.156 --> 00:03:11.340
But, although the framers did agree

00:03:11.340 --> 00:03:14.480
to phase out the
international slave trade,

00:03:14.480 --> 00:03:16.980
they made another compromise

00:03:16.980 --> 00:03:19.782
that was much more
favorable to slaveholders.

00:03:19.782 --> 00:03:21.820
The Three-Fifths Compromise.

00:03:21.820 --> 00:03:24.548
Now, you'll remember that in deciding

00:03:24.548 --> 00:03:29.030
how the legislative branch
would represent the population,

00:03:29.030 --> 00:03:33.000
in the Great Compromise,
or Connecticut Compromise,

00:03:33.000 --> 00:03:36.070
they agreed that in the
House of Representatives,

00:03:36.070 --> 00:03:38.220
the proportion of representatives

00:03:38.220 --> 00:03:39.850
would be based on population,

00:03:39.850 --> 00:03:41.200
whereas in the Senate,

00:03:41.200 --> 00:03:43.140
every state would have two senators

00:03:43.140 --> 00:03:44.740
regardless of its size.

00:03:44.740 --> 00:03:46.880
Well, the big question for this

00:03:46.880 --> 00:03:50.000
is who counts as part of the population?

00:03:50.000 --> 00:03:51.280
Is it just white men?

00:03:51.280 --> 00:03:53.990
Or do the large enslaved populations

00:03:53.990 --> 00:03:56.650
of southern states also count?

00:03:56.650 --> 00:03:58.623
Now, if you were a southern slave holder,

00:03:58.623 --> 00:04:01.530
you would have been strongly in favor

00:04:01.530 --> 00:04:03.260
of counting this population,

00:04:03.260 --> 00:04:05.630
because it means you
get more representation,

00:04:05.630 --> 00:04:07.940
and thus more power in the
House of Representatives.

00:04:07.940 --> 00:04:10.045
If you were against slavery

00:04:10.045 --> 00:04:13.200
or from a small state, or both,

00:04:13.200 --> 00:04:16.280
you would've been bitterly
opposed to the notion

00:04:16.280 --> 00:04:19.225
that people who have no rights as citizens

00:04:19.225 --> 00:04:22.720
should be counted as
citizens to give those states

00:04:22.720 --> 00:04:24.410
more power in Congress.

00:04:24.410 --> 00:04:25.630
So, here's what they decided,

00:04:25.630 --> 00:04:27.520
according to article one.

00:04:27.520 --> 00:04:29.810
Representatives and direct taxes

00:04:29.810 --> 00:04:32.180
shall be apportioned
among the several states

00:04:32.180 --> 00:04:34.310
which may be included within this union

00:04:34.310 --> 00:04:36.128
according to their respective numbers,

00:04:36.128 --> 00:04:37.690
which shall be determined

00:04:37.690 --> 00:04:40.800
by adding to the whole
number of free persons,

00:04:40.800 --> 00:04:44.230
including those bound to
service for a term of years,

00:04:44.230 --> 00:04:46.870
and excluding Indians not taxed,

00:04:46.870 --> 00:04:48.741
three fifths of all other persons.

00:04:48.741 --> 00:04:52.166
This three fifths of all other persons

00:04:52.166 --> 00:04:54.638
really means enslaved Africans.

00:04:54.638 --> 00:04:57.810
And you'll notice that the framers

00:04:57.810 --> 00:05:01.260
are really talking around slavery.

00:05:01.260 --> 00:05:03.380
In the part about the
international slave trade,

00:05:03.380 --> 00:05:06.300
they said migration of such persons.

00:05:06.300 --> 00:05:09.640
Now they say three fifths
of all other persons.

00:05:09.640 --> 00:05:12.927
In fact, the word slavery never appears

00:05:12.927 --> 00:05:15.500
in the original constitution.

00:05:15.500 --> 00:05:18.297
So, why do they say three
fifths of all other persons

00:05:18.297 --> 00:05:22.100
or migration of such
persons as states think it

00:05:22.100 --> 00:05:23.330
proper to admit?

00:05:23.330 --> 00:05:24.883
And honestly, I think the answer to this

00:05:24.883 --> 00:05:28.310
is that the framers
were ashamed of slavery.

00:05:28.310 --> 00:05:31.355
They were ashamed that this institution

00:05:31.355 --> 00:05:34.550
existed in a democratic society.

00:05:34.550 --> 00:05:36.620
They knew that the eyes of the world,

00:05:36.620 --> 00:05:39.325
the eyes of history, would
look at this document

00:05:39.325 --> 00:05:41.967
and this institution

00:05:41.967 --> 00:05:46.660
completely sullied the idea
of a democratic government.

00:05:46.660 --> 00:05:48.748
So, as it says here,

00:05:48.748 --> 00:05:53.748
their agreement was that for
every five enslaved people

00:05:55.120 --> 00:05:56.254
who lived in a state,

00:05:56.254 --> 00:05:59.022
three of them would be counted

00:05:59.022 --> 00:06:02.220
for the purposes of population.

00:06:02.220 --> 00:06:05.422
This is a huge victory for slaveholders,

00:06:05.422 --> 00:06:08.760
getting more power in Congress

00:06:08.760 --> 00:06:11.290
for having people who can't vote,

00:06:11.290 --> 00:06:12.627
who can't be citizens.

00:06:12.627 --> 00:06:16.290
Why did the delegates of other states

00:06:16.290 --> 00:06:17.600
allow this to happen?

00:06:17.600 --> 00:06:19.930
And I think the simple answer is that

00:06:19.930 --> 00:06:22.890
the constitution would
not have been ratified

00:06:22.890 --> 00:06:26.100
were it not for this
compromise, among others.

00:06:26.100 --> 00:06:29.174
The states of the south were too important

00:06:29.174 --> 00:06:33.930
to getting that nine out
of 13 necessary votes

00:06:33.930 --> 00:06:36.500
to replace the articles of confederation

00:06:36.500 --> 00:06:38.240
with this new constitution.

00:06:38.240 --> 00:06:40.930
So, they made a compromise to make sure

00:06:40.930 --> 00:06:44.040
that the constitution was
ratified and improved.

00:06:44.040 --> 00:06:47.048
But that compromise would
have tremendous consequences

00:06:47.048 --> 00:06:50.960
for the generations of enslaved people

00:06:50.960 --> 00:06:53.060
who would live under that system.

00:06:53.060 --> 00:06:56.660
And for the nation when the
Civil War broke out in 1865.


Constitutional compromises: The Electoral College

WEBVTT
Kind: captions
Language: en

00:00:00.360 --> 00:00:01.670
- [Instructor] In this series of videos

00:00:01.670 --> 00:00:02.880
about the Constitution,

00:00:02.880 --> 00:00:05.230
we've been discussing all the elements

00:00:05.230 --> 00:00:10.230
of balance and compromise that
appear in the Constitution,

00:00:10.400 --> 00:00:13.520
the balance between large
states and small states

00:00:13.520 --> 00:00:16.240
and between the different
branches of government.

00:00:16.240 --> 00:00:17.860
But in this video I want to talk

00:00:17.860 --> 00:00:20.090
about one particular compromise

00:00:20.090 --> 00:00:22.260
made at the Constitutional Convention

00:00:22.260 --> 00:00:25.670
over how the president of
the United States is elected,

00:00:25.670 --> 00:00:28.360
and that is the electoral college.

00:00:28.360 --> 00:00:33.250
I think these compromises
reveal some real conflict

00:00:33.250 --> 00:00:37.050
among the framers over how they think

00:00:37.050 --> 00:00:39.940
the American Revolution did.

00:00:39.940 --> 00:00:43.430
Did they think that the Revolution

00:00:43.430 --> 00:00:45.950
went too far,

00:00:45.950 --> 00:00:48.370
had created too much equality

00:00:48.370 --> 00:00:51.720
and too much liberty for
people who weren't ready

00:00:51.720 --> 00:00:52.770
to deal with it?

00:00:52.770 --> 00:00:56.280
Or did it not go far enough?

00:00:56.280 --> 00:00:58.130
So let's talk about this idea

00:00:58.130 --> 00:01:00.920
that perhaps the Revolution went too far,

00:01:00.920 --> 00:01:04.780
that the average American Joe,

00:01:04.780 --> 00:01:06.610
or Jedediah, I guess,

00:01:06.610 --> 00:01:09.250
could be the Revolutionary version of Joe,

00:01:09.250 --> 00:01:14.120
had too much a sense
of his own importance,

00:01:14.120 --> 00:01:17.660
was going to tear down
the social structures

00:01:17.660 --> 00:01:20.730
that had seemed natural during
the American Revolution,

00:01:20.730 --> 00:01:24.450
the wealthy elites, the middling farmers,

00:01:24.450 --> 00:01:27.040
the rough and rowdy workers.

00:01:27.040 --> 00:01:28.620
You'll remember that one of the incidents

00:01:28.620 --> 00:01:31.560
that led to the decision to revise

00:01:31.560 --> 00:01:35.150
the Articles of Confederation
was Shays's Rebellion

00:01:35.150 --> 00:01:40.150
in which a group of unruly farmers,

00:01:40.690 --> 00:01:42.390
Revolutionary War veterans,

00:01:42.390 --> 00:01:46.560
had marched against the
governor of Massachusetts.

00:01:46.560 --> 00:01:50.290
So the people were used to rebelling,

00:01:50.290 --> 00:01:54.310
and they first had rebelled
against Great Britain.

00:01:54.310 --> 00:01:56.230
But now that war was over,

00:01:56.230 --> 00:02:00.080
and they started rebelling
against state governments.

00:02:00.080 --> 00:02:04.010
So there's a real sense
throughout the Constitution

00:02:04.010 --> 00:02:09.010
that the founders were
attempting to balance democracy,

00:02:11.040 --> 00:02:13.390
a representative government,

00:02:13.390 --> 00:02:17.160
with what they saw as too much democracy

00:02:17.160 --> 00:02:21.050
or mobocracy in their words,

00:02:21.050 --> 00:02:23.450
that unruly mobs

00:02:23.450 --> 00:02:27.060
who perhaps lacked the virtue

00:02:28.020 --> 00:02:32.970
of elite, educated citizens

00:02:32.970 --> 00:02:35.700
would foolishly tear down government

00:02:35.700 --> 00:02:38.820
that they weren't prepared to be part of.

00:02:38.820 --> 00:02:42.143
Now, you see that in
things like the Senate.

00:02:43.500 --> 00:02:48.170
The members of the Senate
were appointed, not elected,

00:02:48.170 --> 00:02:50.570
up until the 20th century.

00:02:50.570 --> 00:02:53.010
The idea that there had to be one part

00:02:53.010 --> 00:02:56.460
of the legislative
branch that was selected

00:02:56.460 --> 00:02:58.440
by the better sort of men,

00:02:58.440 --> 00:03:01.260
the sort of people who really knew

00:03:01.260 --> 00:03:03.050
what good leadership looked like,

00:03:03.050 --> 00:03:06.480
not by a mob that might be swayed

00:03:06.480 --> 00:03:09.430
by any fancy talking politician.

00:03:09.430 --> 00:03:14.430
The founders didn't want all
white men to be able to vote.

00:03:14.770 --> 00:03:19.770
They wanted voting to be
reserved to the elite,

00:03:19.860 --> 00:03:22.703
the propertied, the educated,

00:03:23.630 --> 00:03:27.770
those who were prepared
to be virtuous citizens.

00:03:27.770 --> 00:03:29.900
It wouldn't be until the 1820s

00:03:29.900 --> 00:03:34.390
that all white men
could vote in elections,

00:03:34.390 --> 00:03:36.600
regardless of how much
property they owned.

00:03:36.600 --> 00:03:38.550
Of course, it wouldn't be until the late

00:03:38.550 --> 00:03:42.030
19th and 20th centuries
that women and minorities

00:03:42.030 --> 00:03:43.170
would get the right to vote.

00:03:43.170 --> 00:03:46.550
So they had a very dim idea

00:03:46.550 --> 00:03:49.270
of the average citizen's ability

00:03:49.270 --> 00:03:52.500
to engage productively in democracy.

00:03:52.500 --> 00:03:56.070
And another way that they
show this in the Constitution

00:03:56.070 --> 00:04:01.020
is in the process of
electing the president.

00:04:01.020 --> 00:04:04.880
Article II establishes
the executive branch,

00:04:04.880 --> 00:04:09.480
and it also discusses how
presidential elections shall work.

00:04:09.480 --> 00:04:13.720
And it's a kinda complex
process, the electoral college.

00:04:13.720 --> 00:04:18.720
But the simple version is that
instead of having citizens

00:04:20.290 --> 00:04:23.010
vote directly for the president,

00:04:23.010 --> 00:04:25.890
the citizens would vote in each state,

00:04:25.890 --> 00:04:29.570
and then that state would have electors

00:04:29.570 --> 00:04:33.160
equal to the number of
senators and representatives.

00:04:33.160 --> 00:04:38.160
And those electors would then
cast votes for the president,

00:04:38.390 --> 00:04:41.130
and whoever got the most electoral votes

00:04:41.130 --> 00:04:42.340
should be president.

00:04:42.340 --> 00:04:44.220
And we still have this system today.

00:04:44.220 --> 00:04:49.220
This is a map of the current
number of electoral votes

00:04:49.390 --> 00:04:51.320
that each state has.

00:04:51.320 --> 00:04:54.140
And really, what the
founders intended here

00:04:54.140 --> 00:04:56.340
was to have a safeguard

00:04:58.410 --> 00:05:00.480
of the office of the president,

00:05:00.480 --> 00:05:03.120
believing that it would be possible

00:05:03.120 --> 00:05:06.650
for a mob to be swayed,

00:05:06.650 --> 00:05:09.370
even the better sorts of citizens,

00:05:09.370 --> 00:05:11.700
into voting for a politician

00:05:11.700 --> 00:05:13.680
who wouldn't be good for the office.

00:05:13.680 --> 00:05:18.130
And so they moved away
from direct democracy

00:05:18.130 --> 00:05:23.130
into a slightly more
complicated indirect system

00:05:23.300 --> 00:05:26.700
just to put an extra layer of safety

00:05:26.700 --> 00:05:31.153
in between the office of
president and the unruly masses.


The Emirate of Nejd and the Kingdom of Saudi Arabia

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:02.100
- [Instructor] Where we
left off in the last video,

00:00:02.100 --> 00:00:05.270
the first Saudi state,
the Emirate of Diriyah,

00:00:05.270 --> 00:00:08.270
was ended in the Ottoman-Wahhabi War.

00:00:08.270 --> 00:00:10.020
It was ended by the Ottomans,

00:00:10.020 --> 00:00:12.420
but it was by actual Egyptian forces

00:00:12.420 --> 00:00:15.520
that retook control of Mecca and Medina

00:00:15.520 --> 00:00:18.250
and then laid siege to
the town of Diriyah,

00:00:18.250 --> 00:00:21.360
which was the center of
the Emirate of Diriyah.

00:00:21.360 --> 00:00:23.920
The last emir, who was
also considered imam,

00:00:23.920 --> 00:00:25.230
so a religious leader,

00:00:25.230 --> 00:00:27.770
of the Emirate of Diriyah was Abdullah,

00:00:27.770 --> 00:00:31.240
who was beheaded by the Ottoman Turks.

00:00:31.240 --> 00:00:33.860
Now, that wasn't the end
of the House of Saud.

00:00:33.860 --> 00:00:35.270
As we have alluded to,

00:00:35.270 --> 00:00:38.400
there's going to be two more Saudi states.

00:00:38.400 --> 00:00:42.720
A cousin of Abdullah's
father, Turki bin Abdullah,

00:00:42.720 --> 00:00:44.230
who had gone into hiding after

00:00:44.230 --> 00:00:47.720
the Egyptian forces had
taken control of Diriyah,

00:00:47.720 --> 00:00:51.240
was able to come back
to power in the 1820s,

00:00:51.240 --> 00:00:55.290
and he establishes the second Saudi state,

00:00:55.290 --> 00:00:57.200
the Emirate of Nejd.

00:00:57.200 --> 00:00:58.990
And as we talked about
in the previous video,

00:00:58.990 --> 00:01:02.220
the Nejd refers to that
central Arabian region,

00:01:02.220 --> 00:01:05.830
which was really the stronghold
of the House of Saud.

00:01:05.830 --> 00:01:08.440
And we can see here
that the Emirate of Nejd

00:01:08.440 --> 00:01:11.660
eventually expands to much
of the Arabian Peninsula,

00:01:11.660 --> 00:01:14.310
although it never took
control of Mecca and Medina,

00:01:14.310 --> 00:01:16.760
which were back under Ottoman control.

00:01:16.760 --> 00:01:19.550
Now, this emirate will
last until the last decade

00:01:19.550 --> 00:01:24.550
of the 19th century, as
we get into 1890 and 1891,

00:01:24.720 --> 00:01:29.393
when a rival tribe, the Rashidun,
are able to take Riyadh.

00:01:30.460 --> 00:01:34.000
At that time, Turki bin
Abdullah's grandson,

00:01:34.000 --> 00:01:38.380
Abdul Rahman bin Faisal,
is now emir and imam,

00:01:38.380 --> 00:01:41.870
so both the head of state
and as imam religious leader.

00:01:41.870 --> 00:01:45.400
When the Rashidun take over,
he and his family go into exile

00:01:45.400 --> 00:01:47.160
with Bedouin tribes in the south

00:01:47.160 --> 00:01:49.990
and eventually make their way to Kuwait.

00:01:49.990 --> 00:01:54.040
Abdul Rahman himself eventually
focuses more on religion,

00:01:54.040 --> 00:01:57.980
but his son, Abdul Aziz,
will end up becoming

00:01:57.980 --> 00:02:02.150
one of the most significant
figures in all of Saudi history.

00:02:02.150 --> 00:02:04.200
In 1902,

00:02:04.200 --> 00:02:07.593
he is able to take back Riyadh.

00:02:08.680 --> 00:02:12.490
He and roughly 40 men are able
to scale the protective walls

00:02:12.490 --> 00:02:15.470
of Riyadh by climbing
bent-over palm trees.

00:02:15.470 --> 00:02:17.440
They're able to take the town.

00:02:17.440 --> 00:02:19.000
And then from there,

00:02:19.000 --> 00:02:23.100
they establish what ends up
becoming the third Saudi state.

00:02:23.100 --> 00:02:26.250
He soon takes back the Nejd region

00:02:26.250 --> 00:02:27.640
and then, in World War I,

00:02:27.640 --> 00:02:31.520
takes the side of Great Britain
against the Ottoman Empire.

00:02:31.520 --> 00:02:34.300
After World War I, he's
able to take the Hejaz,

00:02:34.300 --> 00:02:36.300
this western region of
the Arabian Peninsula,

00:02:36.300 --> 00:02:38.770
including Mecca and Medina.

00:02:38.770 --> 00:02:42.630
And in 1932, it's officially
proclaimed a kingdom,

00:02:42.630 --> 00:02:44.880
the Kingdom of Saudi Arabia,

00:02:44.880 --> 00:02:48.910
which included both
the Hejaz and the Nejd.

00:02:48.910 --> 00:02:52.860
And all of this was
done by King Abdulaziz.

00:02:52.860 --> 00:02:54.450
And as you can see here,

00:02:54.450 --> 00:02:57.470
his reign covered half
of the 20th century.

00:02:57.470 --> 00:02:59.660
And over the next 70 years,

00:02:59.660 --> 00:03:03.240
every king of Saudi Arabia
has been one of his sons,

00:03:03.240 --> 00:03:05.730
and the current Crown
Prince of Saudi Arabia,

00:03:05.730 --> 00:03:08.600
Mohammed bin Salman, is his grandson.

00:03:08.600 --> 00:03:11.030
And King Abdulaziz and Saudi Arabia

00:03:11.030 --> 00:03:15.560
become major actors on the
global stage when, in 1938,

00:03:15.560 --> 00:03:17.750
major oil reserves are discovered

00:03:17.750 --> 00:03:20.000
in Saudi Arabia near the Persian Gulf.

00:03:20.000 --> 00:03:22.210
And these end up being some of the largest

00:03:22.210 --> 00:03:24.860
oil reserves in the world.

00:03:24.860 --> 00:03:27.850
So this became a strong
incentive for major world powers

00:03:27.850 --> 00:03:30.000
to become allied with Saudi Arabia,

00:03:30.000 --> 00:03:32.120
especially the United States.

00:03:32.120 --> 00:03:34.600
This right over here is a picture of FDR,

00:03:34.600 --> 00:03:35.790
the American president,

00:03:35.790 --> 00:03:38.660
meeting King Abdulaziz in 1945,

00:03:38.660 --> 00:03:40.700
near the end of World War II.

00:03:40.700 --> 00:03:43.650
And you could appreciate
the strategic importance

00:03:43.650 --> 00:03:45.450
of these oil reserves.

00:03:45.450 --> 00:03:47.810
They're important for growing economies,

00:03:47.810 --> 00:03:50.910
and they are also important
from a military point of view.

00:03:50.910 --> 00:03:53.840
World War I and even more so World War II

00:03:53.840 --> 00:03:56.710
are defined by mechanized militaries

00:03:56.710 --> 00:04:00.320
that need oil in order to operate.

00:04:00.320 --> 00:04:03.260
And just to get an
appreciation of Saudi Arabia's

00:04:03.260 --> 00:04:06.910
continued strategic importance
from an energy point of view,

00:04:06.910 --> 00:04:10.360
this right over here shows
the production of petroleum

00:04:10.360 --> 00:04:12.960
and natural gas for the
United States, Russia,

00:04:12.960 --> 00:04:15.600
and Saudi Arabia over
the last several years.

00:04:15.600 --> 00:04:18.710
And these tend to be
the top three producers

00:04:18.710 --> 00:04:20.530
of petroleum and natural gas.

00:04:20.530 --> 00:04:22.340
But if you focus on petroleum,

00:04:22.340 --> 00:04:24.860
you see that, for most
of this time period,

00:04:24.860 --> 00:04:27.430
Saudi Arabia was the top producer,

00:04:27.430 --> 00:04:30.460
and it's actually true of
most of the post-war era.

00:04:30.460 --> 00:04:34.210
Recently, the United States
has passed up Saudi Arabia,

00:04:34.210 --> 00:04:37.160
but even then Saudi Arabia is incredibly

00:04:37.160 --> 00:04:40.470
strategically important
because it uses far less

00:04:40.470 --> 00:04:42.950
of its petroleum than
the United States does.

00:04:42.950 --> 00:04:44.530
So Saudi Arabia is still the world's

00:04:44.530 --> 00:04:47.110
largest exporter of petroleum,

00:04:47.110 --> 00:04:50.450
and the reserves that
Saudi Arabia has access to

00:04:50.450 --> 00:04:52.560
are far cheaper to access than, say,

00:04:52.560 --> 00:04:55.010
the reserves in the United States.

00:04:55.010 --> 00:04:57.270
So they continue to be very strategic

00:04:57.270 --> 00:05:00.463
and very important actors
on the global stage.


Emirate of Diriyah as the first Saudi State

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.640
- [Man] The history of the Saudi
state and the Saudi dynasty

00:00:03.640 --> 00:00:08.459
really becomes interesting in
1744, in the town of Diriyah,

00:00:08.459 --> 00:00:11.408
which is very close to
the modern city of Riyadh.

00:00:11.408 --> 00:00:14.874
In the region of the Arabian
peninsula known as the Nejd,

00:00:14.874 --> 00:00:17.232
and the Nejd is an important term to know

00:00:17.232 --> 00:00:20.414
if you're studying Saudi Arabia
because this central region

00:00:20.414 --> 00:00:22.647
of the Arabian peninsula
is really the stronghold

00:00:22.647 --> 00:00:24.397
of the House of Saud.

00:00:25.248 --> 00:00:29.713
But Diriyah was a successful
town in the early 18th century,

00:00:29.713 --> 00:00:32.360
but the Emir of Diriyah,
Muhammad bin Saud,

00:00:32.360 --> 00:00:34.909
forges an alliance with
a religious leader,

00:00:34.909 --> 00:00:37.080
Muhammad bin Abdul-Wahhab,

00:00:37.080 --> 00:00:39.743
and that alliance is a very strong one.

00:00:39.743 --> 00:00:43.998
Muhammad bin Saud's son marries
Muhammad bin Abdul-Wahhab's

00:00:43.998 --> 00:00:46.715
daughter to really seal that alliance.

00:00:46.715 --> 00:00:50.344
And the Wahhabis could be
viewed as a fundamentalist sect

00:00:50.344 --> 00:00:52.057
of Sunni Islam.

00:00:52.057 --> 00:00:54.226
And with that purist religious fervor,

00:00:54.226 --> 00:00:56.798
this alliance is very successful.

00:00:56.798 --> 00:00:59.619
Over the next several decades, you have

00:00:59.619 --> 00:01:02.893
the Emirate of Diriyah continuing to grow,

00:01:02.893 --> 00:01:06.969
first conquering much of
the Nejd and Eastern Arabia,

00:01:06.969 --> 00:01:10.324
and then as we go into
the early 19th centuries,

00:01:10.324 --> 00:01:14.271
we go into the early 1800's,
conquers Western Saudi Arabia

00:01:14.271 --> 00:01:16.748
and much of the territory
you would now associate

00:01:16.748 --> 00:01:18.549
with the modern Saudi state.

00:01:18.549 --> 00:01:21.579
Now of note, is that they
conquered the Western portion

00:01:21.579 --> 00:01:25.173
of the Arabian peninsula,
known as the Hejaz.

00:01:25.173 --> 00:01:27.514
And the reason why this is significant,

00:01:27.514 --> 00:01:28.899
you have to remember the context.

00:01:28.899 --> 00:01:31.454
At this time, the Ottoman Empire,

00:01:31.454 --> 00:01:34.612
with Constantinople as
its capital, was nominally

00:01:34.612 --> 00:01:37.363
in control of most of the
Middle East, including Egypt

00:01:37.363 --> 00:01:39.290
and the Arabian peninsula.

00:01:39.290 --> 00:01:41.903
But all of a sudden, you have this emirate

00:01:41.903 --> 00:01:44.717
that starts off in Diriyah,
and now is able to conquer

00:01:44.717 --> 00:01:46.977
a big chunk of the Arabian peninsula,

00:01:46.977 --> 00:01:50.099
including two of the holiest
places in all of Islam.

00:01:50.099 --> 00:01:52.756
On top of that, the Emir of Diriyah

00:01:52.756 --> 00:01:55.467
and the Wahhabi religious leaders say that

00:01:55.467 --> 00:01:58.655
the ruler of the Ottoman
Empire, the Sultan,

00:01:58.655 --> 00:02:02.027
Mustafa the Fourth, is not
the valid leader of Islam,

00:02:02.027 --> 00:02:04.977
that he is not the true caliph of Islam.

00:02:04.977 --> 00:02:08.822
On top of that, in 1802 an
estimated 12,000 Wahhabis

00:02:08.822 --> 00:02:11.592
go to Karbala and sack the city.

00:02:11.592 --> 00:02:15.494
They destroy the shrine of
Husayn, who's a revered figure

00:02:15.494 --> 00:02:18.704
amongst Shia muslims, and
they kill an estimated

00:02:18.704 --> 00:02:21.984
up to four or five thousand people,

00:02:21.984 --> 00:02:24.864
and Karbala is also under Ottoman control,

00:02:24.864 --> 00:02:27.323
so the Ottomans are not happy about this.

00:02:27.323 --> 00:02:30.552
You have a fundamentalist
sect of Islam in the Wahhabis

00:02:30.552 --> 00:02:33.468
controlling most of the
Arabian peninsula, including

00:02:33.468 --> 00:02:35.747
the holiest sites in all of Islam.

00:02:35.747 --> 00:02:39.169
They're saying that the
Ottoman rulers, the Sultan,

00:02:39.169 --> 00:02:42.028
is not the true caliph of
Islam, and on top of that,

00:02:42.028 --> 00:02:44.822
they're causing internal strife
within the Ottoman Empire

00:02:44.822 --> 00:02:47.284
with the Shias by sacking Karbala.

00:02:47.284 --> 00:02:50.493
To make things even more
complicated for the Ottomans,

00:02:50.493 --> 00:02:53.773
in Egypt, you have another
leader who's trying to exercise

00:02:53.773 --> 00:02:56.190
his autonomy in Muhammad Ali.

00:02:57.463 --> 00:02:58.713
And so, the

00:02:58.713 --> 00:03:00.389
Ottomans have a plan.

00:03:00.389 --> 00:03:04.016
Why don't we get Muhammad Ali
to send some of his troops

00:03:04.016 --> 00:03:06.839
to focus on the Emirate of Diriyah?

00:03:06.839 --> 00:03:09.051
And so they order Muhammad Ali to take on

00:03:09.051 --> 00:03:11.772
the Emirate of Diriyah
and this is what becomes

00:03:11.772 --> 00:03:14.154
the Ottoman-Wahhabi War.

00:03:14.154 --> 00:03:16.997
Now by the time the war is in full swing,

00:03:16.997 --> 00:03:21.101
the Sultan in Constantinople
is Mahmud the Second.

00:03:21.101 --> 00:03:24.114
Muhammad Ali sends his sons,
who are eventually successful,

00:03:24.114 --> 00:03:27.178
in taking back Mecca and
Medina, and then they lay siege

00:03:27.178 --> 00:03:30.256
to the town of Diriyah, and
they are eventually able

00:03:30.256 --> 00:03:31.839
to take it in 1818.

00:03:32.771 --> 00:03:35.840
They send the leaders of the
House of Saud, and the Wahhabis

00:03:35.840 --> 00:03:39.580
to Constantinople, including
the current emir at the time,

00:03:39.580 --> 00:03:43.187
Abdullah bin Saud, and
just to get a sense of

00:03:43.187 --> 00:03:47.342
the family tree up to this
point, this is Abdullah,

00:03:47.342 --> 00:03:50.425
who is the last emir of
the Emirate of Diriyah,

00:03:50.425 --> 00:03:52.074
who gets captured by the Egyptian forces

00:03:52.074 --> 00:03:53.647
on behalf of the Ottomans.

00:03:53.647 --> 00:03:57.798
You can see he's the great
grandson of Muhammad bin Saud.

00:03:57.798 --> 00:04:01.832
Bin Saud means Muhammad, son
of Saud, so the whole dynasty

00:04:01.832 --> 00:04:05.173
is named after Saud, who is
the previous Emir of Diriyah,

00:04:05.173 --> 00:04:07.826
but Muhammad bin Saud is
the one who really formed

00:04:07.826 --> 00:04:10.484
that alliance with the
Wahhabis, which enabled

00:04:10.484 --> 00:04:13.296
the Emerate of Diriyah
to become a major power

00:04:13.296 --> 00:04:16.481
in the Arabian peninsula,
eventually taking over most

00:04:16.481 --> 00:04:17.618
of the peninsula.

00:04:17.618 --> 00:04:20.280
Abdullah was sentenced to
death in Constantinople

00:04:20.280 --> 00:04:23.728
but before he was beheaded,
he was forced to listen

00:04:23.728 --> 00:04:26.688
to lute music, because
according to the Wahhabis,

00:04:26.688 --> 00:04:29.597
that was considered sacrilege,
while the Ottomans had

00:04:29.597 --> 00:04:32.893
no problem with it, so they
were really trying to rub it in.

00:04:32.893 --> 00:04:35.523
That before you die, you're
going to have to do something

00:04:35.523 --> 00:04:38.124
that you, the Wahhabis view as sacrilege.

00:04:38.124 --> 00:04:40.702
But I will leave you
there, as we will see,

00:04:40.702 --> 00:04:44.219
the Emirate of Diriyah is
the first of three major

00:04:44.219 --> 00:04:46.472
Saudi states that will each control

00:04:46.472 --> 00:04:48.939
most of the Arabian peninsula.

00:04:48.939 --> 00:04:51.886
In the next video, we're
going to see the Saudi state

00:04:51.886 --> 00:04:55.200
come back with the Emirate
of Nejd, and in blue here,

00:04:55.200 --> 00:04:57.570
you see the rulers of the Emirate of Nejd.

00:04:57.570 --> 00:04:59.944
And then, after that, we will talk about

00:04:59.944 --> 00:05:02.771
the modern Kingdom of Saudi Arabia,

00:05:02.771 --> 00:05:05.186
whose rulers you see here in green,

00:05:05.186 --> 00:05:07.321
and we'll go into a lot
more depth than that.

00:05:07.321 --> 00:05:10.163
But you can see, that
they all are descendant

00:05:10.163 --> 00:05:12.080
from Muhammad bin Saud.


Writing proportional equations

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:02.000
- [Instructor] We're told that Justin runs

00:00:02.000 --> 00:00:07.000
at a constant rate, traveling
17 kilometers in two hours.

00:00:07.960 --> 00:00:11.180
Write an equation that
shows the relationship

00:00:11.180 --> 00:00:14.670
between the distance he
runs, d, in kilometers

00:00:14.670 --> 00:00:18.210
and the time he spends
running, h, in hours.

00:00:18.210 --> 00:00:20.920
So, pause this video and see
if you can work through that

00:00:20.920 --> 00:00:23.600
on your own before we do it together.

00:00:23.600 --> 00:00:25.770
All right, now there's several ways

00:00:25.770 --> 00:00:27.970
to approach this question.

00:00:27.970 --> 00:00:30.840
One way is to say, look, he's
running at a constant rate,

00:00:30.840 --> 00:00:34.240
so his distance is going to
be equal to some constant,

00:00:34.240 --> 00:00:36.370
let's just call that lowercase k,

00:00:36.370 --> 00:00:39.320
times the amount of
time he spends running,

00:00:39.320 --> 00:00:42.390
and the way that we can
figure out what k is

00:00:42.390 --> 00:00:45.140
is by using the information
that they gave us.

00:00:45.140 --> 00:00:47.530
They tell us right over here

00:00:47.530 --> 00:00:50.963
that when our distance is 17 kilometers,

00:00:52.140 --> 00:00:54.840
so when our distance is 17 kilometers,

00:00:54.840 --> 00:00:59.120
that's a situation where he
has been running for two hours.

00:00:59.120 --> 00:01:04.120
So, that is going to be
equal to k times two hours.

00:01:05.670 --> 00:01:07.840
So, what is k going to be?

00:01:07.840 --> 00:01:09.480
Pause the video again and
see if you can figure it out,

00:01:09.480 --> 00:01:10.790
if you didn't figure it out the first time

00:01:10.790 --> 00:01:12.340
I asked you to pause the video.

00:01:13.300 --> 00:01:15.320
All right, well there's a
bunch of ways to solve for k,

00:01:15.320 --> 00:01:16.340
but one way is to say,

00:01:16.340 --> 00:01:18.970
let's just divide both sides by two hours.

00:01:18.970 --> 00:01:22.010
So, if you divide both sides by two hours,

00:01:22.010 --> 00:01:25.920
you are going to get that k

00:01:25.920 --> 00:01:30.890
is going to be equal 17 over
two kilometers per hour,

00:01:30.890 --> 00:01:32.833
which is, 17 over two is 8.5,

00:01:33.730 --> 00:01:38.480
8.5 kilometers per hour.

00:01:38.480 --> 00:01:40.790
And so, if we go back to
the original question,

00:01:40.790 --> 00:01:44.410
which asks us to write an equation

00:01:44.410 --> 00:01:47.200
that shows the relationship
between d and h,

00:01:47.200 --> 00:01:50.850
we can say that d is equal to,

00:01:50.850 --> 00:01:53.160
we now know our proportionality constant,

00:01:53.160 --> 00:01:58.020
it is 8.5 times h.

00:01:58.020 --> 00:02:01.210
8.5 times h, and we're done.

00:02:01.210 --> 00:02:02.440
If we wanted to write their units,

00:02:02.440 --> 00:02:03.730
we could write d is equal to

00:02:03.730 --> 00:02:06.950
8.5 kilometers per hour times h,

00:02:06.950 --> 00:02:09.123
which is given in hours.


Constant of proportionality from tables

WEBVTT
Kind: captions
Language: en

00:00:00.320 --> 00:00:01.590
- [Instructor] We are asked, Which table

00:00:01.590 --> 00:00:05.763
has a constant of proportionality
between y and x of 0.6?

00:00:07.010 --> 00:00:10.440
Pause this video and see
if you can figure that out.

00:00:10.440 --> 00:00:12.050
All right, so just as a reminder,

00:00:12.050 --> 00:00:15.290
the constant of proportionality
between y and x,

00:00:15.290 --> 00:00:16.260
one way to think about it

00:00:16.260 --> 00:00:20.500
is that y is equal to
some constant times x.

00:00:20.500 --> 00:00:22.970
Y is proportional to x.

00:00:22.970 --> 00:00:24.470
And this constant right over here

00:00:24.470 --> 00:00:27.230
is our constant of proportionality.

00:00:27.230 --> 00:00:29.280
So if that's going to be 0.6,

00:00:29.280 --> 00:00:30.600
so in our tables,

00:00:30.600 --> 00:00:31.950
or in the table that has

00:00:31.950 --> 00:00:34.690
a constant of proportionality of 0.6,

00:00:34.690 --> 00:00:38.380
y should be equal to 0.6 times x

00:00:38.380 --> 00:00:40.830
for every x,y pair.

00:00:40.830 --> 00:00:43.000
So let's look at these choices.

00:00:43.000 --> 00:00:47.130
So is seven 0.6 times four?

00:00:47.130 --> 00:00:49.290
Well, no, seven is larger than four.

00:00:49.290 --> 00:00:51.833
0.6 times four would actually be 2.4,

00:00:52.890 --> 00:00:55.100
so this one is not gonna be,

00:00:55.100 --> 00:00:56.210
is definitely not going to have

00:00:56.210 --> 00:00:58.740
a constant of proportionality of 0.6.

00:00:58.740 --> 00:00:59.900
And in fact, this table,

00:00:59.900 --> 00:01:02.800
this isn't even a
proportional relationship.

00:01:02.800 --> 00:01:04.070
For this first one,

00:01:04.070 --> 00:01:06.400
I would have to multiply by 7/4.

00:01:06.400 --> 00:01:09.340
And then here I'm going
to be multiplying by 10/6,

00:01:09.340 --> 00:01:12.020
which is equivalent to 5/3.

00:01:12.020 --> 00:01:14.910
And here I'm multiplying by 13/8,

00:01:14.910 --> 00:01:17.940
so I'm not multiplying by
the same constant every time.

00:01:17.940 --> 00:01:21.540
So this isn't even a
proportional relationship.

00:01:21.540 --> 00:01:23.430
Now let's look at choice B.

00:01:23.430 --> 00:01:25.980
Well, to go from four to 2.4, that is.

00:01:25.980 --> 00:01:27.463
You would multiply by 0.6.

00:01:28.710 --> 00:01:30.120
But that's not enough for us to say

00:01:30.120 --> 00:01:32.840
that this is truly a
proportional relationship.

00:01:32.840 --> 00:01:34.820
It would have to be 0.6 in every scenario.

00:01:34.820 --> 00:01:35.653
So let's see.

00:01:35.653 --> 00:01:39.260
Nine times 0.6, yeah, that is 5.4.

00:01:39.260 --> 00:01:41.060
Nine times six is 54.

00:01:41.060 --> 00:01:43.180
But now this is nine times 6/10.

00:01:43.180 --> 00:01:46.190
It's 54 divided by 10, which is 5.4.

00:01:46.190 --> 00:01:50.270
And now let's see, 14 times six is 84.

00:01:50.270 --> 00:01:53.620
So 14 times 6/10 would indeed be 8.4.

00:01:54.680 --> 00:01:56.990
So this looks like our choice.

00:01:56.990 --> 00:01:59.550
And we can verify that
this would not be the case.

00:01:59.550 --> 00:02:01.110
Let's see, three,

00:02:01.110 --> 00:02:04.250
to get to two we would
be multiplying by 2/3.

00:02:04.250 --> 00:02:07.170
And then here, once again,
we're multiplying by 2/3.

00:02:07.170 --> 00:02:09.883
And then here, once again,
we're multiplying by 2/3.

00:02:10.920 --> 00:02:14.690
So this is actually describing
a proportional relationship,

00:02:14.690 --> 00:02:17.980
but our constant of
proportionality here is 2/3,

00:02:17.980 --> 00:02:20.200
which, if you tried to
express it as a decimal,

00:02:20.200 --> 00:02:22.500
it would be 0.6 repeating.

00:02:22.500 --> 00:02:26.290
2/3 is equal to 0.6 repeating.

00:02:26.290 --> 00:02:29.420
And so it is proportional
but does not have

00:02:29.420 --> 00:02:31.900
this constant of proportionality.

00:02:31.900 --> 00:02:33.343
So we like our choice B.


Introduction to proportional relationships

WEBVTT
Kind: captions
Language: en

00:00:00.130 --> 00:00:01.620
- [Instructor] In this video, we are going

00:00:01.620 --> 00:00:04.350
to talk about proportional relationships,

00:00:04.350 --> 00:00:05.840
and these are relationships

00:00:05.840 --> 00:00:09.010
between two variables where the ratio

00:00:09.010 --> 00:00:11.840
between the variables is equivalent.

00:00:11.840 --> 00:00:14.510
Now that sounds complex
or a little bit fancy.

00:00:14.510 --> 00:00:15.630
It'll hopefully seem a little bit

00:00:15.630 --> 00:00:18.660
more straightforward once
we look at some examples.

00:00:18.660 --> 00:00:20.190
So let's say I'm looking at a recipe

00:00:20.190 --> 00:00:21.610
for some type of baked goods,

00:00:21.610 --> 00:00:23.250
maybe it's some type of pancakes,

00:00:23.250 --> 00:00:25.030
I've been making a lot of those lately,

00:00:25.030 --> 00:00:28.330
and we know that for a
certain number of eggs

00:00:28.330 --> 00:00:30.090
how many cups of milk we need.

00:00:30.090 --> 00:00:34.490
So we have number of eggs,

00:00:34.490 --> 00:00:39.490
and then we're also going
to have cups of milk.

00:00:39.810 --> 00:00:43.230
And in this recipe, we know

00:00:43.230 --> 00:00:46.140
that if we're going to use one egg,

00:00:46.140 --> 00:00:48.390
then we would use two cups of milk,

00:00:48.390 --> 00:00:49.870
and if we use three eggs,

00:00:49.870 --> 00:00:51.520
then we're gonna use six cups of milk,

00:00:51.520 --> 00:00:55.140
and if we use, let's say, 12 eggs,

00:00:55.140 --> 00:00:58.270
then we're going to use 24 cups of milk.

00:00:58.270 --> 00:01:00.500
So is this a proportional relationship

00:01:00.500 --> 00:01:02.710
where the two variables
are the cups of milk

00:01:02.710 --> 00:01:04.450
and the number of eggs?

00:01:04.450 --> 00:01:06.060
Well, to test that we just have

00:01:06.060 --> 00:01:07.240
to think about the ratio

00:01:07.240 --> 00:01:08.460
between these two variables.

00:01:08.460 --> 00:01:09.380
And you can say that the ratio

00:01:09.380 --> 00:01:10.730
of the number of eggs to the cups of milk,

00:01:10.730 --> 00:01:12.010
or the ratio of the cups of milk

00:01:12.010 --> 00:01:13.070
to the number of eggs.

00:01:13.070 --> 00:01:14.510
But you just need to ensure that they

00:01:14.510 --> 00:01:17.400
are always equivalent in these scenarios.

00:01:17.400 --> 00:01:19.230
So let me make another column here,

00:01:19.230 --> 00:01:21.330
and I'm gonna think about the ratio

00:01:21.330 --> 00:01:26.330
of the eggs to the cups of milk.

00:01:28.650 --> 00:01:30.180
Well, in this first scenario

00:01:30.180 --> 00:01:33.160
one egg for two cups of milk.

00:01:33.160 --> 00:01:36.380
This second scenario is three to six.

00:01:36.380 --> 00:01:39.450
This third scenario is 12 to 24.

00:01:39.450 --> 00:01:42.450
Are these equivalent ratios?

00:01:42.450 --> 00:01:44.010
Well, to go from one to three

00:01:44.010 --> 00:01:46.000
you multiply by three,

00:01:46.000 --> 00:01:47.970
and also, to go from two to six,

00:01:47.970 --> 00:01:49.100
you multiply by three.

00:01:49.100 --> 00:01:51.990
So you multiplied both
the variables by three.

00:01:51.990 --> 00:01:53.320
Similarly, if you multiply

00:01:53.320 --> 00:01:55.310
the number of eggs by four,

00:01:55.310 --> 00:01:56.530
then you multiply the number

00:01:56.530 --> 00:01:59.780
of cups of milk by four as well.

00:01:59.780 --> 00:02:02.540
So these indeed are all equivalent ratios,

00:02:02.540 --> 00:02:05.860
one to two, three to six, 12 to 24.

00:02:05.860 --> 00:02:07.940
In every scenario you have twice as much

00:02:07.940 --> 00:02:11.310
cups of milk as you have number of eggs.

00:02:11.310 --> 00:02:13.350
So this would be proportional.

00:02:13.350 --> 00:02:15.140
So check.

00:02:15.140 --> 00:02:16.200
Now what would be an example

00:02:16.200 --> 00:02:18.600
of a non-proportional relationship?

00:02:18.600 --> 00:02:20.990
We'll stay in this baked
goods frame of mind.

00:02:20.990 --> 00:02:23.930
Let's say you're going to a cake store

00:02:23.930 --> 00:02:25.270
and you're curious about how much

00:02:25.270 --> 00:02:26.540
it would cost to buy a cake

00:02:26.540 --> 00:02:28.460
for different numbers of people.

00:02:28.460 --> 00:02:32.420
So let's say number of servings,

00:02:32.420 --> 00:02:34.130
number of servings in one column,

00:02:34.130 --> 00:02:38.470
and then the cost of the cake.

00:02:38.470 --> 00:02:40.973
And let me set up two
columns right over here.

00:02:42.080 --> 00:02:44.500
And so let's say if you have 10 servings,

00:02:44.500 --> 00:02:46.990
the cake costs $20.

00:02:46.990 --> 00:02:51.120
If you have 20 servings,
the cake costs $30.

00:02:51.120 --> 00:02:56.010
And if you have 40 servings,
the cake costs $40.

00:02:56.010 --> 00:02:58.510
Pause this video and see
if you can figure out

00:02:58.510 --> 00:03:00.970
whether this is a
proportional relationship.

00:03:00.970 --> 00:03:01.910
If it is, why?

00:03:01.910 --> 00:03:03.523
If it isn't, why not?

00:03:04.480 --> 00:03:05.360
Alright, well let's just think

00:03:05.360 --> 00:03:07.285
about the ratios again.

00:03:07.285 --> 00:03:08.550
And here our two variables are

00:03:08.550 --> 00:03:10.920
the number of servings
and the cost of cake.

00:03:10.920 --> 00:03:14.280
So if we look at the
ratio of the servings,

00:03:14.280 --> 00:03:18.283
servings, to cost,

00:03:19.450 --> 00:03:23.570
in this first situation it is 10 to 20,

00:03:23.570 --> 00:03:27.090
and then it is 20 to 30,

00:03:27.090 --> 00:03:30.680
and then it is 40 to 40.

00:03:30.680 --> 00:03:33.520
And so to see if these
are equivalent ratios,

00:03:33.520 --> 00:03:35.350
when we go from 10 to 20 on the number

00:03:35.350 --> 00:03:37.790
of servings, we're multiplying by two.

00:03:37.790 --> 00:03:39.770
But when we go from 20 to 30

00:03:39.770 --> 00:03:41.210
on the cost of the cake,

00:03:41.210 --> 00:03:42.740
we aren't multiplying by two,

00:03:42.740 --> 00:03:46.310
we're multiplying by
1.5, or one and a half.

00:03:46.310 --> 00:03:48.940
And similarly, when we go from 20 to 40,

00:03:48.940 --> 00:03:51.120
we are multiplying by two again,

00:03:51.120 --> 00:03:52.600
but to go from 30 to 40

00:03:52.600 --> 00:03:54.530
we aren't multiplying by two,

00:03:54.530 --> 00:03:56.580
we're multiplying by one and one third.

00:03:56.580 --> 00:03:59.070
By one and one third.

00:03:59.070 --> 00:04:01.100
When we multiply our
servings by a given amount,

00:04:01.100 --> 00:04:03.020
we're not multiplying our cost of cake

00:04:03.020 --> 00:04:04.140
by the same amount.

00:04:04.140 --> 00:04:07.543
This tells us that this
is not proportional.

00:04:08.870 --> 00:04:10.860
One way to think about
proportional relationships,

00:04:10.860 --> 00:04:12.660
we already said, that the ratio

00:04:12.660 --> 00:04:14.980
between the variables will be equivalent.

00:04:14.980 --> 00:04:16.570
Another way to think about it is

00:04:16.570 --> 00:04:20.200
one variable will always be some constant

00:04:20.200 --> 00:04:22.340
times the first variable.

00:04:22.340 --> 00:04:24.840
So in our first example right over here

00:04:24.840 --> 00:04:27.330
we said the cups of milk is always

00:04:27.330 --> 00:04:29.120
two times the number of eggs.

00:04:29.120 --> 00:04:30.510
We can write that down.

00:04:30.510 --> 00:04:33.620
So cups of milk

00:04:35.010 --> 00:04:36.620
is always going to be equal

00:04:36.620 --> 00:04:41.053
to two times the number of eggs.

00:04:42.330 --> 00:04:44.410
And this number right over here,

00:04:44.410 --> 00:04:47.350
we call that the constant
of proportionality.

00:04:47.350 --> 00:04:48.680
And you wouldn't be able to set up

00:04:48.680 --> 00:04:51.170
an equation like this in this scenario.

00:04:51.170 --> 00:04:53.070
It would have to be more complicated.

00:04:53.070 --> 00:04:55.020
And so a proportional relationship,

00:04:55.020 --> 00:04:56.930
the ratios are equivalent

00:04:56.930 --> 00:04:58.140
between the two variables

00:04:58.140 --> 00:04:59.340
and you can set it up

00:04:59.340 --> 00:05:00.790
with an equation like this

00:05:00.790 --> 00:05:03.570
where you have a constant

00:05:03.570 --> 00:05:06.793
of proportionality.