Estimating limit numerically

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:02.830
- [Instructor] Consider the
table with function values

00:00:02.830 --> 00:00:07.380
for f of x is equal to x
squared over one minus cosine x

00:00:07.380 --> 00:00:10.660
at positive x-values near zero.

00:00:10.660 --> 00:00:13.210
Notice that there is one
missing value in the table.

00:00:13.210 --> 00:00:14.850
This is the missing one right here.

00:00:14.850 --> 00:00:19.140
Use a calculator to evaluate
f of x at x equals 0.1

00:00:19.140 --> 00:00:20.860
and enter this number in the table

00:00:20.860 --> 00:00:23.120
rounded to the nearest thousandth.

00:00:23.120 --> 00:00:26.560
From the table, what
does the one-sided limit,

00:00:26.560 --> 00:00:28.280
the limit as x approaches zero

00:00:28.280 --> 00:00:29.660
from the positive direction

00:00:31.792 --> 00:00:35.340
of x squared over one minus
cosine of x appear to be?

00:00:35.340 --> 00:00:36.173
So let's see what they did.

00:00:36.173 --> 00:00:39.970
They evaluated when x equals
one, f of x equals 2.175.

00:00:39.970 --> 00:00:41.840
When x gets even a little
bit closer to zero,

00:00:41.840 --> 00:00:42.900
and once again, we're approaching zero

00:00:42.900 --> 00:00:44.290
from values larger than zero.

00:00:44.290 --> 00:00:47.020
That's what this little
superscript positive tell us.

00:00:47.020 --> 00:00:50.900
We're at 0.5, and we're at 2.042.

00:00:50.900 --> 00:00:54.600
Then when we get even closer to zero, 0.2,

00:00:54.600 --> 00:00:57.440
f of x is 2.007.

00:00:57.440 --> 00:00:59.750
And so I'm guessing when
I'm getting even closer,

00:00:59.750 --> 00:01:02.420
it's gonna be even closer
to two right over here.

00:01:02.420 --> 00:01:03.860
But let's verify that.

00:01:03.860 --> 00:01:05.263
Get my calculator out.

00:01:06.160 --> 00:01:11.160
So I wanna evaluate x squared
over one minus cosine of x

00:01:11.170 --> 00:01:15.520
when x is equal to 0.1.

00:01:15.520 --> 00:01:17.320
So the first thing I wanna actually,

00:01:19.434 --> 00:01:21.830
let me verify that I'm in radian mode

00:01:21.830 --> 00:01:24.060
'cause otherwise I might
get a strange answer.

00:01:24.060 --> 00:01:26.810
So I am in radian mode.

00:01:26.810 --> 00:01:28.580
And so let me evaluate it.

00:01:28.580 --> 00:01:30.190
So I'm going to have

00:01:32.396 --> 00:01:35.610
0.1 squared over,

00:01:37.460 --> 00:01:40.190
let me do divide it by

00:01:40.190 --> 00:01:44.190
one minus cosine of 0.1.

00:01:46.020 --> 00:01:50.550
And this gets me 2.0016.

00:01:50.550 --> 00:01:51.830
And let's see, they want us to round

00:01:51.830 --> 00:01:53.710
to the nearest thousandth.

00:01:53.710 --> 00:01:55.743
So that'll be 2.002.

00:01:57.270 --> 00:01:58.210
Type that in.

00:01:58.210 --> 00:02:00.510
2.002.

00:02:00.510 --> 00:02:05.510
And so it looks like the
limit is approaching two.

00:02:05.880 --> 00:02:07.840
It's not approaching 2.005.

00:02:07.840 --> 00:02:12.597
We just crossed 2.005 from 2.007 to 2.002.

00:02:14.190 --> 00:02:15.710
So let's check our answer.

00:02:15.710 --> 00:02:17.060
And we got it right.

00:02:17.060 --> 00:02:19.190
And I always find it fun
to visualize these things,

00:02:19.190 --> 00:02:21.020
and that's what a graphing
calculator is good for.

00:02:21.020 --> 00:02:23.060
It can actually graph things.

00:02:23.060 --> 00:02:26.170
So let's graph this right over here.

00:02:26.170 --> 00:02:27.840
So go into graph mode.

00:02:27.840 --> 00:02:30.760
Let me redefine my function here.

00:02:30.760 --> 00:02:31.593
So let's see.

00:02:31.593 --> 00:02:36.310
It's going to be x squared
divided by one minus cosine of x.

00:02:41.420 --> 00:02:43.070
And then let me make sure that the range

00:02:43.070 --> 00:02:44.920
of my graph is right, so I'm zoomed in

00:02:44.920 --> 00:02:47.860
at the right part that I care about.

00:02:47.860 --> 00:02:49.720
So let me go to the range.

00:02:49.720 --> 00:02:50.553
And let's see.

00:02:50.553 --> 00:02:55.260
I care about approaching zero
from the positive direction,

00:02:56.320 --> 00:02:59.840
but as long as I see values around zero,

00:02:59.840 --> 00:03:01.740
I should be fine.

00:03:01.740 --> 00:03:03.690
But I could actually zoom
in a little bit more,

00:03:03.690 --> 00:03:05.640
so I can make them my minimum x-value.

00:03:05.640 --> 00:03:07.740
I don't know, let me make it negative one.

00:03:09.520 --> 00:03:10.870
Let me make my maximum x-value.

00:03:10.870 --> 00:03:12.290
The maximum x-value here is one,

00:03:12.290 --> 00:03:13.450
but just to get some space here

00:03:13.450 --> 00:03:14.400
I'll make this 1.5.

00:03:16.420 --> 00:03:18.210
So the x-scale is one.

00:03:18.210 --> 00:03:21.000
Y-minimum, it seems like
we're approaching two.

00:03:21.000 --> 00:03:23.520
So the y-max can be
much smaller, let's see.

00:03:23.520 --> 00:03:25.090
We'll make y-max three.

00:03:25.090 --> 00:03:26.763
And now let's graph this thing.

00:03:28.100 --> 00:03:30.900
So let's see what it's doing.

00:03:30.900 --> 00:03:32.420
It looks like...

00:03:33.260 --> 00:03:35.520
And you see here whether,

00:03:35.520 --> 00:03:36.830
and actually it looks like
whether you're approaching

00:03:36.830 --> 00:03:39.710
from the positive direction or
from the negative direction,

00:03:39.710 --> 00:03:41.040
it looks like you're approaching,

00:03:41.040 --> 00:03:43.470
the value of the function approaches two.

00:03:43.470 --> 00:03:46.380
But this problem, we're
only caring about as we have

00:03:46.380 --> 00:03:48.220
x-values that approaching zero

00:03:48.220 --> 00:03:50.010
from values larger than zero.

00:03:50.010 --> 00:03:51.280
So this is the limit.

00:03:51.280 --> 00:03:53.790
This is the one-sided
limit that we care about

00:03:53.790 --> 00:03:56.473
with the two shows up
right over here as well.


Solving equations and inequalities through substitution example 2

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:04.290
- [Instructor] Mario started
the day with five fire flowers.

00:00:04.290 --> 00:00:07.400
He used some, and now only has three left.

00:00:07.400 --> 00:00:10.520
If U is the number of flowers he's used,

00:00:10.520 --> 00:00:12.740
we can write the following equation.

00:00:12.740 --> 00:00:15.580
How many fire flowers did Mario use?

00:00:15.580 --> 00:00:16.930
So he started with five,

00:00:16.930 --> 00:00:18.040
he uses some of them.

00:00:18.040 --> 00:00:19.850
So we subtract the number he used

00:00:19.850 --> 00:00:21.960
and he ends up with three.

00:00:21.960 --> 00:00:23.860
So there's a bunch of
ways that you could do it.

00:00:23.860 --> 00:00:24.760
One way is you literally

00:00:24.760 --> 00:00:25.900
could probably do this in your head.

00:00:25.900 --> 00:00:29.210
Hey I took away some number
from five and I got three.

00:00:29.210 --> 00:00:30.360
And you probably know,

00:00:30.360 --> 00:00:33.100
hey well if I take two away
from five I'm gonna get three.

00:00:33.100 --> 00:00:35.500
So he used two flowers.

00:00:35.500 --> 00:00:37.230
The other way that you
could think about doing it

00:00:37.230 --> 00:00:39.180
is a little, hey we got
only four options here.

00:00:39.180 --> 00:00:40.780
Let me substitute each of them in here.

00:00:40.780 --> 00:00:42.900
Maybe I was able to do
this one in my head,

00:00:42.900 --> 00:00:44.270
but if I had something more complicated

00:00:44.270 --> 00:00:45.722
let me try out these choices.

00:00:45.722 --> 00:00:47.950
So if he only took zero away,

00:00:47.950 --> 00:00:48.960
it'd five minus zero,

00:00:48.960 --> 00:00:50.470
he'd still of course be left with five.

00:00:50.470 --> 00:00:51.490
If he took one away,

00:00:51.490 --> 00:00:53.980
then of course he'd only
have, he'd have four left.

00:00:53.980 --> 00:00:55.900
If you took three away,
he'd have two left.

00:00:55.900 --> 00:00:56.950
But if he took two away,

00:00:56.950 --> 00:00:59.550
five minus two is indeed three.

00:00:59.550 --> 00:01:00.550
And the third way,

00:01:00.550 --> 00:01:02.600
you'll learn this as we go into more,

00:01:02.600 --> 00:01:06.090
more of how, of the how of algebra

00:01:06.090 --> 00:01:07.670
and also the why,

00:01:07.670 --> 00:01:09.360
is that you could systematically solve it,

00:01:09.360 --> 00:01:11.770
but we'll go into that in future videos.

00:01:11.770 --> 00:01:14.633
So he used two fire flowers.


Simple model to understand r and g relationship

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:01.640
- [Instructor] What I
wanna do in this video

00:00:01.640 --> 00:00:04.500
is to create a simple
spreadsheet to help us understand

00:00:04.500 --> 00:00:06.810
why if R greater than G,

00:00:06.810 --> 00:00:10.210
why that might lead to more
and more of national income

00:00:10.210 --> 00:00:14.400
going to the owners of
capital as opposed to labor.

00:00:14.400 --> 00:00:18.400
So let's just say R is 3%,

00:00:18.400 --> 00:00:20.060
we can change that assumption later.

00:00:20.060 --> 00:00:22.210
So that's the return on
capital that we're assuming.

00:00:22.210 --> 00:00:23.470
We're assuming it's just going to be fixed

00:00:23.470 --> 00:00:24.480
at that constant rate

00:00:24.480 --> 00:00:27.190
and let's say that economic growth is 2%.

00:00:27.190 --> 00:00:30.190
So we're assuming it's an area
where R is greater than G.

00:00:30.190 --> 00:00:31.760
Now in this column let me write the years.

00:00:31.760 --> 00:00:34.840
There's Year One, Year
Two, and then we would go.

00:00:34.840 --> 00:00:38.710
Let's see, maybe we can go up to Year,

00:00:38.710 --> 00:00:41.640
let's go to Year 15 just for good measure.

00:00:41.640 --> 00:00:46.510
Now we could think about
what our national capital is

00:00:46.510 --> 00:00:48.670
or I could say Total Capital.

00:00:48.670 --> 00:00:51.410
So the capital in our economy.

00:00:51.410 --> 00:00:54.450
Now we'd also want to think
about the national income.

00:00:54.450 --> 00:00:58.360
So we could view this as
the output of the economy.

00:00:58.360 --> 00:01:01.020
National income.

00:01:01.020 --> 00:01:03.530
And now we could think about
the split of the income

00:01:03.530 --> 00:01:05.290
between capital and labor.

00:01:05.290 --> 00:01:10.290
So income to capital and income to labor.

00:01:11.370 --> 00:01:14.260
And now let's just think
about percentage of total.

00:01:14.260 --> 00:01:18.923
So capital as percent of total.

00:01:20.330 --> 00:01:22.350
So let's come up with some
assumptions right over here.

00:01:22.350 --> 00:01:24.880
So let's say that our aggregate capital

00:01:24.880 --> 00:01:27.170
and I'm just gonna throw
out a random number here.

00:01:27.170 --> 00:01:29.720
Let's say it's 4,000

00:01:29.720 --> 00:01:32.160
and if we're talking
about millions of dollars,

00:01:32.160 --> 00:01:33.780
then this would be four trillion dollars,

00:01:33.780 --> 00:01:36.180
but I'm being currency
agnostic right over here.

00:01:36.180 --> 00:01:37.660
So let's just say it's 4,000.

00:01:37.660 --> 00:01:40.390
It could be, this could,
if it was in millions

00:01:40.390 --> 00:01:41.350
this would be four trillion,

00:01:41.350 --> 00:01:43.200
but let's just say 4,000 for now.

00:01:43.200 --> 00:01:45.980
And let's say our national income is 1,000

00:01:45.980 --> 00:01:49.069
and we've seen charts already
that, at least for the U.S.,

00:01:49.069 --> 00:01:52.810
capital, the national
capital as a percentage of

00:01:52.810 --> 00:01:55.330
national income has
been about 400 or 500%.

00:01:55.330 --> 00:01:57.100
It's kind of oscillated in that range.

00:01:57.100 --> 00:01:59.160
So it's been about four or
five times national income.

00:01:59.160 --> 00:02:03.080
So this is kind of a
not unreasonable ratio.

00:02:03.080 --> 00:02:06.330
And now what's the income
to the owners of capital?

00:02:06.330 --> 00:02:08.420
Well we're saying the
return on capital is 3%.

00:02:08.420 --> 00:02:11.390
So this is going to be equal to

00:02:11.390 --> 00:02:15.400
this 3%, I'm gonna press F4
to put those dollar signs

00:02:15.400 --> 00:02:16.610
and we'll see why that's valuable.

00:02:16.610 --> 00:02:19.190
It'll make sure that we
stay referenced to that cell

00:02:19.190 --> 00:02:22.270
as we drag this down later on.

00:02:22.270 --> 00:02:27.270
It's going to be that times,
times the amount of capital

00:02:28.140 --> 00:02:30.420
that we have in the economy.

00:02:30.420 --> 00:02:32.760
So it's going to be 120

00:02:32.760 --> 00:02:35.620
and notice, I had the
dollar signs on the B1

00:02:35.620 --> 00:02:37.660
because I always wanna refer to this,

00:02:37.660 --> 00:02:39.640
but I didn't put the
dollar signs on the B5

00:02:39.640 --> 00:02:43.330
because as I drag this down
I always want this cell,

00:02:43.330 --> 00:02:45.130
say that when I drag it down here,

00:02:45.130 --> 00:02:46.810
I want it to refer to that same 3%.

00:02:46.810 --> 00:02:49.040
So that's why I kind of anchored it there

00:02:49.040 --> 00:02:50.220
with the dollar signs,

00:02:50.220 --> 00:02:52.930
but I want it to be times
the capital in that row.

00:02:52.930 --> 00:02:55.360
So we'll see how that happens in a second.

00:02:55.360 --> 00:02:57.060
Now what's the income to labor?

00:02:57.060 --> 00:02:58.187
Well it's gonna be what's left over.

00:02:58.187 --> 00:03:02.880
National income minus
the income to capital

00:03:02.880 --> 00:03:04.820
and then capital as a percentage of total?

00:03:04.820 --> 00:03:07.524
The income to capital is a
percentage of total income,

00:03:07.524 --> 00:03:10.210
well that's just going to be equal to,

00:03:10.210 --> 00:03:13.990
I can select income to capital, D5,

00:03:13.990 --> 00:03:18.210
divided by, divided by national income

00:03:18.210 --> 00:03:20.090
and so it's 12%.

00:03:20.090 --> 00:03:22.294
And we're also going to
assume that every year

00:03:22.294 --> 00:03:25.580
that income to capital all
gets reinvested as capital.

00:03:25.580 --> 00:03:27.880
So it doesn't get consumed in some way.

00:03:27.880 --> 00:03:30.740
So, then Year Two the capital
that we start off with

00:03:30.740 --> 00:03:32.720
is going to be the capital from last year

00:03:32.720 --> 00:03:35.060
plus the new income to capital.

00:03:35.060 --> 00:03:37.610
That capital, that income
to capital's going to get

00:03:37.610 --> 00:03:39.200
reinvested as capital

00:03:39.200 --> 00:03:41.400
is just going to be my assumption there.

00:03:41.400 --> 00:03:44.470
And national income,
well we know its growth.

00:03:44.470 --> 00:03:47.490
It's going to be the previous
year's national income

00:03:47.490 --> 00:03:50.300
plus, I guess we could say times,

00:03:50.300 --> 00:03:55.300
times one plus this number,
plus our economic growth.

00:03:56.910 --> 00:03:58.360
So there and I'll press F4

00:03:58.360 --> 00:04:01.568
because I keep, I wanna
stay referencing that.

00:04:01.568 --> 00:04:05.160
And so notice, we grew by 2%

00:04:05.160 --> 00:04:07.610
and then these two over here.

00:04:07.610 --> 00:04:09.900
Actually all three of these over here

00:04:09.900 --> 00:04:12.010
we can just drag down.

00:04:12.010 --> 00:04:14.115
And now hopefully you see
the value of what I did

00:04:14.115 --> 00:04:16.550
when I put the dollar signs here

00:04:16.550 --> 00:04:18.614
because now when I dragged it down,

00:04:18.614 --> 00:04:21.860
this is still referring to B1

00:04:21.860 --> 00:04:22.910
'cause it had the dollar signs there.

00:04:22.910 --> 00:04:25.595
So it's taking the 3% times B6.

00:04:25.595 --> 00:04:27.230
So B6 is this.

00:04:27.230 --> 00:04:28.306
So it's looking at the right,

00:04:28.306 --> 00:04:29.990
at the B column,

00:04:29.990 --> 00:04:32.040
but in its row now.

00:04:32.040 --> 00:04:33.750
That's what's one of
the really useful things

00:04:33.750 --> 00:04:34.710
about spreadsheets

00:04:34.710 --> 00:04:37.420
and actually this
column, let me make this.

00:04:37.420 --> 00:04:39.230
Let me make this a percentage

00:04:39.230 --> 00:04:41.170
just so it becomes a little bit cleaner.

00:04:41.170 --> 00:04:42.710
Okay, there you go.

00:04:42.710 --> 00:04:44.632
And then we can just keep on going.

00:04:44.632 --> 00:04:47.470
So let's just keep on going down

00:04:48.750 --> 00:04:49.960
and what do we get?

00:04:49.960 --> 00:04:50.793
And actually let me get rid of

00:04:50.793 --> 00:04:52.170
some of the decimal places here.

00:04:52.170 --> 00:04:53.720
It's making a little bit messy.

00:04:54.694 --> 00:04:56.070
So, let me.

00:04:56.070 --> 00:04:57.150
So, there you go.

00:04:57.150 --> 00:04:59.500
That actually makes it
a little bit cleaner.

00:04:59.500 --> 00:05:01.713
And so what do we have going on over here?

00:05:01.713 --> 00:05:06.510
Well we see that when R
is a fixed rate of return

00:05:06.510 --> 00:05:08.620
on capital that is greater than growth,

00:05:08.620 --> 00:05:09.940
we see that capital,

00:05:09.940 --> 00:05:12.200
the income going to
capital as a percentage

00:05:12.200 --> 00:05:15.610
of the total national
income is increasing.

00:05:15.610 --> 00:05:19.590
Now, this could be used
as a proxy for inequality

00:05:19.590 --> 00:05:21.020
because as we've said before

00:05:21.020 --> 00:05:23.640
capital does tend to
be concentrated amongst

00:05:23.640 --> 00:05:27.242
the upper percentile, decile, quartile,

00:05:27.242 --> 00:05:29.660
however you want to define it,

00:05:29.660 --> 00:05:31.750
but once again this is just a proxy.

00:05:31.750 --> 00:05:34.110
And the other thing you
have to keep in mind is

00:05:34.110 --> 00:05:37.117
inequality is a natural by-product of

00:05:37.117 --> 00:05:39.352
a capitalistic market economy.

00:05:39.352 --> 00:05:42.450
And even though more
and more of the income

00:05:42.450 --> 00:05:44.840
is going to the owners of capital,

00:05:44.840 --> 00:05:47.250
the labor, the income going to labor,

00:05:47.250 --> 00:05:48.500
is also increasing.

00:05:48.500 --> 00:05:50.480
Now that by itself doesn't necessarily say

00:05:50.480 --> 00:05:51.313
it's a good thing.

00:05:51.313 --> 00:05:53.710
For example, if the
population were increasing

00:05:53.710 --> 00:05:54.561
faster than this

00:05:54.561 --> 00:05:58.190
then the income to labor
divided by the population,

00:05:58.190 --> 00:06:00.450
which would be kind of the
per capita income to labor.

00:06:00.450 --> 00:06:03.560
A kind of a proxy for how
much the individual person

00:06:03.560 --> 00:06:04.618
working is making,

00:06:04.618 --> 00:06:06.987
then if that, even if this is going up

00:06:06.987 --> 00:06:08.920
but the population is
growing faster than that

00:06:08.920 --> 00:06:10.080
that might not be a good thing

00:06:10.080 --> 00:06:12.420
because that means per
capita income might not be

00:06:12.420 --> 00:06:13.253
where it needs to be.

00:06:13.253 --> 00:06:16.250
But if we assume the population is stable

00:06:16.250 --> 00:06:17.763
or it's not growing as fast as this,

00:06:17.763 --> 00:06:20.520
even though we see inequality

00:06:20.520 --> 00:06:22.870
or at least this measure
of inequality going up,

00:06:22.870 --> 00:06:25.840
more and more of the
income is going to capital.

00:06:25.840 --> 00:06:28.910
These people still,
still might be better off

00:06:28.910 --> 00:06:29.771
in this reality.

00:06:29.771 --> 00:06:33.570
But to kind of test the
sensitivity of this model

00:06:33.570 --> 00:06:34.600
that we've created,

00:06:34.600 --> 00:06:35.980
we can actually try different things.

00:06:35.980 --> 00:06:38.130
So if the return on
capital is much larger,

00:06:38.130 --> 00:06:40.497
say it's 5%, we see that the disparity,

00:06:40.497 --> 00:06:44.140
the disparity becomes much
much more significant.

00:06:44.140 --> 00:06:46.530
And let's say if it was 10%,

00:06:46.530 --> 00:06:49.630
now you see a situation that
could get not so pleasant.

00:06:49.630 --> 00:06:51.330
'Cause in this situation,

00:06:51.330 --> 00:06:53.010
and this is a fairly extreme situation

00:06:53.010 --> 00:06:54.440
we have right over here,

00:06:54.440 --> 00:06:56.500
now all of a sudden the income to labor

00:06:56.500 --> 00:06:58.290
and not even on per capita terms,

00:06:58.290 --> 00:06:59.900
is actually decreasing.

00:06:59.900 --> 00:07:02.490
So it really does matter a lot
where these two numbers are,

00:07:02.490 --> 00:07:05.410
but of course, if economic
growth was also pretty robust

00:07:05.410 --> 00:07:06.243
now all of a sudden,

00:07:06.243 --> 00:07:07.280
well this is still decreasing,

00:07:07.280 --> 00:07:09.480
but if economic growth was 9%,

00:07:09.480 --> 00:07:12.020
now all of a sudden this could
be a pretty good scenario

00:07:12.020 --> 00:07:12.889
for everyone involved.

00:07:12.889 --> 00:07:14.047
You do have,

00:07:14.047 --> 00:07:15.890
at least this is kind of a proxy for

00:07:15.890 --> 00:07:18.310
increased income to capital

00:07:18.310 --> 00:07:20.819
which could be a proxy for inequality,

00:07:20.819 --> 00:07:23.290
but maybe the economy
is growing fast enough

00:07:23.290 --> 00:07:25.200
that everyone is benefiting.

00:07:25.200 --> 00:07:27.630
So I encourage you to either
build a model like this

00:07:27.630 --> 00:07:29.590
or I'll give you the link
to this model as well

00:07:29.590 --> 00:07:30.670
and play with these numbers.

00:07:30.670 --> 00:07:32.800
And just try to get a
better understanding for

00:07:32.800 --> 00:07:35.267
if we assumed R and G were constant

00:07:35.267 --> 00:07:37.300
and we made some assumptions about

00:07:37.300 --> 00:07:39.160
starting capital and national income,

00:07:39.160 --> 00:07:41.770
how that ends up breaking down as we go

00:07:41.770 --> 00:07:44.043
further and further into the future.


Statistical and non statistical questions

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:01.820
- [Instructor] What I
want to do in this video

00:00:01.820 --> 00:00:03.429
is think about the types of questions

00:00:03.429 --> 00:00:05.750
that we need statistics to address

00:00:05.750 --> 00:00:06.940
and the types of questions that

00:00:06.940 --> 00:00:08.790
we don't need statistics to address.

00:00:08.790 --> 00:00:11.610
And we could call the ones
where we need statistics

00:00:11.610 --> 00:00:16.610
as statistical questions,
statistical questions.

00:00:17.410 --> 00:00:20.940
And I'll circle the statistical
questions in yellow.

00:00:20.940 --> 00:00:23.480
And I encourage you to pause this video

00:00:23.480 --> 00:00:25.560
and try to figure this out yourself first.

00:00:25.560 --> 00:00:27.810
Look at each of these
questions and think about

00:00:27.810 --> 00:00:29.710
whether you think you
need statistics to answer

00:00:29.710 --> 00:00:31.520
this question or you
don't need statistics.

00:00:31.520 --> 00:00:34.430
Whether these are
statistical questions or not.

00:00:34.430 --> 00:00:36.573
So I'm assuming you've given a pass at it,

00:00:36.573 --> 00:00:38.450
and now we can go through this together.

00:00:38.450 --> 00:00:41.680
So this first question is how old are you.

00:00:41.680 --> 00:00:45.080
So we're talking about how
old is a particular person.

00:00:45.080 --> 00:00:46.990
There is an answer here, and we don't need

00:00:46.990 --> 00:00:49.510
any tools of statistics to answer this.

00:00:49.510 --> 00:00:52.340
So this is not a statistical question.

00:00:52.340 --> 00:00:56.980
How old are the people who have
watched this video in 2013?

00:00:56.980 --> 00:00:58.330
Now this is interesting.

00:00:58.330 --> 00:01:00.110
We're assuming that multiple people

00:01:00.110 --> 00:01:02.943
will have watched this video in 2013,

00:01:04.400 --> 00:01:06.740
and they're not all gonna be the same age.

00:01:06.740 --> 00:01:09.350
There's gonna be some
variability in their age.

00:01:09.350 --> 00:01:11.880
So one person might be 10 years old,

00:01:11.880 --> 00:01:15.310
another person might be 20,
another person might be 15.

00:01:15.310 --> 00:01:17.770
So what answer do you give here?

00:01:17.770 --> 00:01:19.550
Would you give all of the ages?

00:01:19.550 --> 00:01:21.430
But we want to get a sense of in general,

00:01:21.430 --> 00:01:22.690
how old are the people?

00:01:22.690 --> 00:01:24.950
So this is where statistics
might be valuable.

00:01:24.950 --> 00:01:27.100
We might wanna find some
type of central tendency,

00:01:27.100 --> 00:01:29.760
an average, a median age for this.

00:01:29.760 --> 00:01:32.660
So this is absolutely
a statistical question.

00:01:32.660 --> 00:01:35.920
And you might already be
seeing kind of a pattern here.

00:01:35.920 --> 00:01:37.820
The first question, we were asking about

00:01:37.820 --> 00:01:40.810
a particular person, there
was only one answer here.

00:01:40.810 --> 00:01:42.860
There's no variability in the answer.

00:01:42.860 --> 00:01:45.410
The second one, we're asking about a trait

00:01:45.410 --> 00:01:48.030
of a bunch of people and there's
variability in that trait.

00:01:48.030 --> 00:01:49.790
They're not all the same age.

00:01:49.790 --> 00:01:53.110
And so we'll need statistics
to come up with some features

00:01:53.110 --> 00:01:55.930
of the data set to be able
to make some conclusions.

00:01:55.930 --> 00:01:58.660
We might say, on average
the people who have watched

00:01:58.660 --> 00:02:02.860
this video in 2013 are 18
years old or 22 years old,

00:02:02.860 --> 00:02:06.160
or the median is 24 years old.

00:02:06.160 --> 00:02:08.200
Whatever it might be.

00:02:08.200 --> 00:02:12.200
Do dogs run faster than cats?

00:02:12.200 --> 00:02:15.610
So once again, there are
many dogs and many cats.

00:02:15.610 --> 00:02:17.680
And they all run at different speeds.

00:02:17.680 --> 00:02:20.250
Some dogs run faster than some cats,

00:02:20.250 --> 00:02:23.160
and some cats run faster than some dogs.

00:02:23.160 --> 00:02:26.240
So we would need some
statistics to get a sense of,

00:02:26.240 --> 00:02:29.800
in general, or on average,
how fast do dogs run?

00:02:29.800 --> 00:02:32.660
And then maybe on average
how fast do cats run?

00:02:32.660 --> 00:02:34.930
And then we could compare those averages,

00:02:34.930 --> 00:02:37.470
or we could compare the
medians in some way.

00:02:37.470 --> 00:02:39.790
So this is definitely
a statistical question.

00:02:39.790 --> 00:02:42.230
Once again, we're
talking about in general,

00:02:42.230 --> 00:02:43.840
a whole population of dogs,

00:02:43.840 --> 00:02:45.680
the whole species of dogs versus cats,

00:02:45.680 --> 00:02:47.620
and there's variation in how fast

00:02:47.620 --> 00:02:50.470
dogs run and how fast cats run.

00:02:50.470 --> 00:02:52.690
If we were talking about a particular dog

00:02:52.690 --> 00:02:54.410
and a particular cat, well then

00:02:54.410 --> 00:02:55.990
there would just be an answer.

00:02:55.990 --> 00:02:58.480
Does dog A run faster
than cat B, well sure,

00:02:58.480 --> 00:03:00.420
that's not going to be
a statistical question,

00:03:00.420 --> 00:03:02.640
you don't have to use
the tools of statistics.

00:03:02.640 --> 00:03:05.020
And this next question
actually fits that pattern.

00:03:05.020 --> 00:03:06.770
Do wolves weigh, actually no,

00:03:06.770 --> 00:03:08.840
this fits the pattern of the previous one.

00:03:08.840 --> 00:03:10.980
Do wolves weigh more than dogs?

00:03:10.980 --> 00:03:14.790
So, once again, there
are some very light dogs

00:03:14.790 --> 00:03:16.530
and some very heavy
wolves, so those wolves

00:03:16.530 --> 00:03:18.230
definitely weigh more than those dogs.

00:03:18.230 --> 00:03:21.080
But there are some very,
very, very heavy dogs.

00:03:21.080 --> 00:03:22.530
And so what you would want to do here,

00:03:22.530 --> 00:03:25.340
because we have variability
in each of these,

00:03:25.340 --> 00:03:27.270
is you might want to come
with some central tendency.

00:03:27.270 --> 00:03:29.170
On average, what is
the median wolf weight?

00:03:29.170 --> 00:03:31.780
What's the average, the mean wolf weight?

00:03:31.780 --> 00:03:33.880
Compare that to the mean dogs' weight.

00:03:33.880 --> 00:03:36.910
So once again, since we're
speaking in general about wolves

00:03:36.910 --> 00:03:40.103
not a particular wolf,
and in general about dogs,

00:03:41.070 --> 00:03:42.790
and there's variation in the data,

00:03:42.790 --> 00:03:45.300
and we're trying to glean
some numbers from that

00:03:45.300 --> 00:03:49.440
to compare, this is definitely
a statistical question.

00:03:49.440 --> 00:03:51.253
Definitely a statistical question.

00:03:52.090 --> 00:03:54.470
Does your dog weigh more than that wolf?

00:03:54.470 --> 00:03:56.860
And we're assuming that we're
pointing at a particular wolf.

00:03:56.860 --> 00:03:58.060
So now this is the particular.

00:03:58.060 --> 00:04:00.140
We're comparing a dog,

00:04:00.140 --> 00:04:02.030
a particular dog to a particular wolf.

00:04:02.030 --> 00:04:03.700
We could put each of them
on a weighing machine

00:04:03.700 --> 00:04:05.490
and come up with an absolute answer.

00:04:05.490 --> 00:04:07.210
There's no variability
in this dog's weight,

00:04:07.210 --> 00:04:09.140
at least at the moment that we weigh it.

00:04:09.140 --> 00:04:10.450
No variability at the wolf's weight

00:04:10.450 --> 00:04:11.470
at the moment that we weigh it.

00:04:11.470 --> 00:04:13.330
So this is not a statistical question.

00:04:13.330 --> 00:04:15.010
So I'll put an X next to the ones

00:04:15.010 --> 00:04:16.823
that are not statistical questions.

00:04:17.690 --> 00:04:21.850
Does it rain more in
Seattle than Singapore?

00:04:21.850 --> 00:04:24.430
So once again, there's variation here.

00:04:24.430 --> 00:04:25.560
And we'd also probably want to know

00:04:25.560 --> 00:04:27.450
does it rain more in
Seattle than Singapore

00:04:27.450 --> 00:04:30.000
in a given year, over
a decade, or whatever.

00:04:30.000 --> 00:04:32.330
But regardless of those
questions, however we ask it,

00:04:32.330 --> 00:04:35.110
in some years, it might
rain more in Seattle,

00:04:35.110 --> 00:04:37.200
and other years, it might
rain more in Singapore.

00:04:37.200 --> 00:04:39.920
Or if we just pick Seattle, it rains

00:04:39.920 --> 00:04:41.470
a different amount from year to year.

00:04:41.470 --> 00:04:42.303
In Singapore, it rains

00:04:42.303 --> 00:04:43.650
a different amount from year to year.

00:04:43.650 --> 00:04:45.070
So how do we compare?

00:04:45.070 --> 00:04:47.280
Well, that's where the
statistics could be valuable.

00:04:47.280 --> 00:04:49.370
There's variability in the data,

00:04:49.370 --> 00:04:51.750
so we can look at the data set for Seattle

00:04:51.750 --> 00:04:53.350
and come up with some type of an average,

00:04:53.350 --> 00:04:54.670
some type of a central tendency

00:04:54.670 --> 00:04:58.310
and compare that to the average, the mean,

00:04:58.310 --> 00:05:00.350
the mode, whatever you
want, the mode probably

00:05:00.350 --> 00:05:02.350
wouldn't be that useful
here, to Singapore.

00:05:02.350 --> 00:05:06.570
So this is definitely, definitely
a statistical question.

00:05:06.570 --> 00:05:08.540
Definitely statistical.

00:05:08.540 --> 00:05:09.970
What was the difference in rainfall

00:05:09.970 --> 00:05:12.900
between Singapore and Seattle in 2013?

00:05:12.900 --> 00:05:15.020
Well, these two numbers aren't known.

00:05:15.020 --> 00:05:16.190
They can be measured.

00:05:16.190 --> 00:05:18.270
Both the rainfall in
Singapore can be measured,

00:05:18.270 --> 00:05:21.750
the rainfall in Seattle can be measured.

00:05:21.750 --> 00:05:22.680
And assuming that this has

00:05:22.680 --> 00:05:25.080
already happened and we can measure 'em,

00:05:25.080 --> 00:05:27.100
then we can just find the difference.

00:05:27.100 --> 00:05:28.800
So you don't need statistics here.

00:05:30.692 --> 00:05:32.170
You just have to have
both of these measurements

00:05:32.170 --> 00:05:33.620
and subtract the difference.

00:05:33.620 --> 00:05:36.070
So not a statistical question.

00:05:36.070 --> 00:05:38.610
In general, will I use less gas

00:05:38.610 --> 00:05:42.100
driving at 55 miles an hour
than 70 miles per hour?

00:05:42.100 --> 00:05:43.440
So this feels statistical,

00:05:43.440 --> 00:05:46.380
because it probably depends
on the circumstance.

00:05:46.380 --> 00:05:49.330
It might depend on the car,
or even for a given car,

00:05:49.330 --> 00:05:51.230
when you drive at 55 miles per hour,

00:05:51.230 --> 00:05:53.250
there's some variation
in your gas mileage.

00:05:53.250 --> 00:05:55.980
It might be how recent
an oil change happened,

00:05:55.980 --> 00:05:58.010
what the wind conditions are like,

00:05:58.010 --> 00:05:59.883
what the road conditions are like.

00:06:02.370 --> 00:06:04.770
You know, exactly how
you're driving the car,

00:06:04.770 --> 00:06:06.690
are you turning, are you
going in a straight line.

00:06:06.690 --> 00:06:09.010
And same thing for 70 miles an hour.

00:06:09.010 --> 00:06:10.470
So when we're saying in general,

00:06:10.470 --> 00:06:12.510
there's variation in
what the gas mileage is

00:06:12.510 --> 00:06:16.310
at 55 miles an hour and
at 70 miles an hour.

00:06:16.310 --> 00:06:17.660
So what you probably want to do is say,

00:06:17.660 --> 00:06:19.050
well, what's my average mileage

00:06:19.050 --> 00:06:20.690
when I drive at 55 miles an hour,

00:06:20.690 --> 00:06:23.890
and compare that to the average
mileage when I drive at 70.

00:06:23.890 --> 00:06:26.290
So because we have this
variability in each of those cases,

00:06:26.290 --> 00:06:30.483
this is definitely a statistical question.

00:06:31.510 --> 00:06:35.050
Do English professors get paid
less than math professors?

00:06:35.050 --> 00:06:38.300
So once again, all English professors

00:06:38.300 --> 00:06:39.530
don't get paid the same amount,

00:06:39.530 --> 00:06:42.100
and all math professors don't
get paid the same amount.

00:06:42.100 --> 00:06:43.930
Some English professors
might do quite well,

00:06:43.930 --> 00:06:44.980
some might make very little.

00:06:44.980 --> 00:06:46.610
Same thing for math professors.

00:06:46.610 --> 00:06:49.070
So we'd probably want to
find some type of an average

00:06:49.070 --> 00:06:52.660
to represent the central
tendency for each of these.

00:06:52.660 --> 00:06:56.520
So once again, this is
a statistical question.

00:06:56.520 --> 00:07:00.180
This is a statistical question.

00:07:00.180 --> 00:07:03.290
Does the most highly paid
English professor at Harvard

00:07:03.290 --> 00:07:05.380
get paid more than the most highly paid

00:07:05.380 --> 00:07:07.860
math professor at MIT?

00:07:07.860 --> 00:07:10.520
Well, now we're talking about
two particular individuals.

00:07:10.520 --> 00:07:12.170
You could go look at their tax forms,

00:07:12.170 --> 00:07:15.810
see how much each of them get paid.

00:07:15.810 --> 00:07:17.100
I guess especially if we assume

00:07:17.100 --> 00:07:19.070
that this is in a particular year.

00:07:19.070 --> 00:07:21.100
Let's say, and let's
just make it that way.

00:07:21.100 --> 00:07:25.380
Say in 2013, just so that we
can remove some variability

00:07:25.380 --> 00:07:26.920
that they might make from year to year,

00:07:26.920 --> 00:07:28.450
make it a little bit more concrete.

00:07:28.450 --> 00:07:30.230
So if this was, does the most highly paid

00:07:30.230 --> 00:07:31.530
English professor at Harvard

00:07:31.530 --> 00:07:33.960
get paid more than the most highly paid

00:07:33.960 --> 00:07:36.510
math professor at MIT in 2013,

00:07:36.510 --> 00:07:38.850
then you have an absolute
number for each of these,

00:07:38.850 --> 00:07:40.160
each of these people, and then

00:07:40.160 --> 00:07:41.600
you could just compare them directly.

00:07:41.600 --> 00:07:43.710
And so, when we're talking
about a particular year,

00:07:43.710 --> 00:07:45.570
particular people, then this is

00:07:45.570 --> 00:07:48.210
no longer a statistical question,

00:07:48.210 --> 00:07:49.963
or it isn't a statistical question.


Reading tables 1

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:01.870
- [Instructor] The table below shows

00:00:01.870 --> 00:00:04.090
solar panel installations by state

00:00:04.090 --> 00:00:06.400
during the last fiscal year.

00:00:06.400 --> 00:00:08.050
How many total solar panels

00:00:08.050 --> 00:00:12.670
were installed last year in Wyoming?

00:00:12.670 --> 00:00:16.140
So we look at the state, so
this right over here is Wyoming

00:00:16.140 --> 00:00:19.250
and this whole table is
about solar installations

00:00:19.250 --> 00:00:21.230
by states during the last year,

00:00:21.230 --> 00:00:25.200
so this is the Wyoming
during the last year.

00:00:25.200 --> 00:00:27.410
This is the first quarter,
so the first three months,

00:00:27.410 --> 00:00:29.690
the second three months,
the third quarter,

00:00:29.690 --> 00:00:31.300
the fourth quarter and
this right over here

00:00:31.300 --> 00:00:33.180
is the total number of solar panels

00:00:33.180 --> 00:00:35.280
that were installed in Wyoming last year,

00:00:35.280 --> 00:00:36.810
which is what I think
they're asking us for,

00:00:36.810 --> 00:00:39.120
so let's just type that in, 104.

00:00:39.120 --> 00:00:39.953
We got it right.

00:00:39.953 --> 00:00:41.290
Let's do another one.

00:00:41.290 --> 00:00:43.530
The table below indicates
the points scored

00:00:43.530 --> 00:00:46.540
by four players in a
charity basketball game.

00:00:46.540 --> 00:00:48.960
How many points did Ben have in total?

00:00:48.960 --> 00:00:50.970
So here is Ben right over here.

00:00:50.970 --> 00:00:52.310
It's broken down by quarter,

00:00:52.310 --> 00:00:55.610
but we care about his total
number of points, which is 31.

00:00:55.610 --> 00:00:58.243
So Ben had a total of 31 points.

00:01:00.330 --> 00:01:02.760
So this indicates the points
scored by four players

00:01:02.760 --> 00:01:04.560
in a charity basketball game again.

00:01:04.560 --> 00:01:07.650
How many points did Vanessa
score in the fourth quarter?

00:01:07.650 --> 00:01:09.690
So let's look at, this is Vanessa,

00:01:09.690 --> 00:01:12.500
and we care about the fourth quarter,

00:01:12.500 --> 00:01:15.910
so she scored four points
in the fourth quarter.

00:01:15.910 --> 00:01:18.380
Put four right over there.

00:01:18.380 --> 00:01:19.320
Alright.

00:01:19.320 --> 00:01:22.450
So once again, we're talking
about solar panel installations

00:01:22.450 --> 00:01:24.850
by state during the last fiscal year.

00:01:24.850 --> 00:01:26.580
How many solar panels were installed

00:01:26.580 --> 00:01:29.970
in Alabama in the second quarter, in Q2?

00:01:29.970 --> 00:01:33.190
So Alabama in the second quarter,

00:01:33.190 --> 00:01:35.344
so that's this right over here,

00:01:35.344 --> 00:01:38.690
28 solar panels were installed.

00:01:38.690 --> 00:01:40.223
So 28.

00:01:42.390 --> 00:01:44.630
So this is once again in a charity game.

00:01:44.630 --> 00:01:47.000
How many points did Kevin
score in the third quarter?

00:01:47.000 --> 00:01:51.967
So Kevin in the third quarter,
he scored nine, nine points.

00:01:53.040 --> 00:01:54.730
And one more just for fun.

00:01:54.730 --> 00:01:57.310
So how many points did Gabriela
score in the third quarter.

00:01:57.310 --> 00:02:00.140
Well, same idea, Gabriela, third quarter,

00:02:00.140 --> 00:02:02.433
she scored five points.


Reading tables 2

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:02.910
- [Instructor] The table below
shows the number of chickens

00:00:02.910 --> 00:00:05.460
sold by four friends from January to March

00:00:05.460 --> 00:00:07.750
except one entry is missing.

00:00:07.750 --> 00:00:09.760
What number should go in the empty cell?

00:00:09.760 --> 00:00:12.120
So let's see, this is the
empty cell right over here.

00:00:12.120 --> 00:00:14.780
They haven't filled out how many,

00:00:14.780 --> 00:00:15.910
let's see, this is chickens sold,

00:00:15.910 --> 00:00:20.580
so we don't know how many
chickens Michael sold in March.

00:00:20.580 --> 00:00:24.100
Well, we know how many he
sold in January and February

00:00:24.100 --> 00:00:27.260
and we also know the total
he sold in the three months.

00:00:27.260 --> 00:00:29.370
So he sold a combined four plus six,

00:00:29.370 --> 00:00:31.790
he sold 10 in January and February.

00:00:31.790 --> 00:00:33.550
He sold 10 in January and February

00:00:33.550 --> 00:00:36.680
and we know he sold a total of 11.

00:00:36.680 --> 00:00:40.520
So this 11 is really the sum
of these three right over here

00:00:40.520 --> 00:00:42.960
so this right over here in
March has got to be one.

00:00:42.960 --> 00:00:45.610
He has got to have sold one chicken

00:00:45.610 --> 00:00:49.060
because we see four
plus six plus one is 11.

00:00:49.060 --> 00:00:50.710
So let's check our answer.

00:00:50.710 --> 00:00:52.530
There we go, let's try this one.

00:00:52.530 --> 00:00:55.970
So here's the missing
cell and so once again,

00:00:55.970 --> 00:00:57.160
we just need to look at the total.

00:00:57.160 --> 00:00:59.130
The total is 107.

00:00:59.130 --> 00:01:02.720
So these four numbers right
over here and one is missing

00:01:02.720 --> 00:01:05.270
you need to add up to be 107

00:01:05.270 --> 00:01:08.330
for the solar panel
installations in Wyoming.

00:01:08.330 --> 00:01:12.210
So let's see, 32 plus 27 is 59.

00:01:12.210 --> 00:01:13.740
59.

00:01:13.740 --> 00:01:18.740
59 plus 26 is, let's see, 79, it is 85.

00:01:21.470 --> 00:01:23.940
So these three add up to 85

00:01:23.940 --> 00:01:25.990
but they're missing this
one right over here.

00:01:25.990 --> 00:01:30.380
So this is going to have to
be whatever 107 is minus 85.

00:01:30.380 --> 00:01:32.040
So that is 22.

00:01:32.040 --> 00:01:36.340
22 solar panels in Q3
because if 22 is here,

00:01:36.340 --> 00:01:39.883
22 plus this, this and this
should be equal to 107.

00:01:41.920 --> 00:01:44.760
So we're missing how many
points Gabriela scores

00:01:44.760 --> 00:01:46.980
in the first quarter right over here.

00:01:46.980 --> 00:01:50.020
Once again, we know that she
scores 35 points in the game.

00:01:50.020 --> 00:01:52.851
In the other three quarters,
she scores a combined,

00:01:52.851 --> 00:01:55.570
let's see, 10 plus 10
plus eight is 28 points.

00:01:55.570 --> 00:01:58.060
So if she's 28 in the other three quarters

00:01:58.060 --> 00:02:01.720
and she gets to 35, 35 minus 28 is,

00:02:01.720 --> 00:02:05.520
she must have scored seven
points in the first quarter.

00:02:05.520 --> 00:02:06.760
Seven points.

00:02:06.760 --> 00:02:08.370
Let's do one more.

00:02:08.370 --> 00:02:11.220
So in Alabama, how many
solar panel installations

00:02:11.220 --> 00:02:12.670
in the fourth quarter?

00:02:12.670 --> 00:02:17.670
See, 33 plus 26 is 59 plus 21 is 80.

00:02:20.410 --> 00:02:22.570
So to get from 80 to 101,

00:02:22.570 --> 00:02:25.563
we need to get another 21
sold in the fourth quarter.

00:02:26.430 --> 00:02:27.473
We got it right.


Measuring segments

WEBVTT
Kind: captions
Language: en

00:00:00.820 --> 00:00:03.790
- [Instructor] What is BD, so
when they're just saying BD,

00:00:03.790 --> 00:00:06.770
they're saying literally
the length of segment BD.

00:00:06.770 --> 00:00:11.090
So they're saying the length
from B, point B to point D.

00:00:11.090 --> 00:00:14.520
And B is sitting here at
negative two, D here is at five.

00:00:14.520 --> 00:00:15.930
So you're looking at really the distance

00:00:15.930 --> 00:00:18.970
between negative two and five.

00:00:18.970 --> 00:00:21.450
So you literally could just
say, well, that's gonna be five,

00:00:21.450 --> 00:00:24.640
that's kind of our endpoint,
minus negative two,

00:00:24.640 --> 00:00:26.400
so five minus negative two

00:00:26.400 --> 00:00:29.460
is the same thing as
five plus two, or seven.

00:00:29.460 --> 00:00:31.440
Another way, you could
literally just count it out.

00:00:31.440 --> 00:00:35.430
Count out the distance, you
go one, two, to get to zero,

00:00:35.430 --> 00:00:39.940
three, four, five, six, seven,
to get all the way to five.

00:00:39.940 --> 00:00:41.370
Or you could say, look, I got to get,

00:00:41.370 --> 00:00:43.520
I got to go two to the right just to get,

00:00:43.520 --> 00:00:45.600
I got to go two to the
right just to get to zero,

00:00:45.600 --> 00:00:48.460
and I got to go five more
to the right to get to D,

00:00:48.460 --> 00:00:51.010
so that's going to be seven.

00:00:51.010 --> 00:00:54.270
Let's do a couple more, what is BD again?

00:00:54.270 --> 00:00:55.900
Well, here we're going from one to four.

00:00:55.900 --> 00:00:58.563
The distance between
one and four is three.

00:01:01.370 --> 00:01:04.290
And then let's do one more, what is AD?

00:01:04.290 --> 00:01:09.040
So A is a negative five,
D is all the way at four.

00:01:09.040 --> 00:01:12.610
So we're going from negative
five to zero, which is five,

00:01:12.610 --> 00:01:16.690
and then we're gonna go four
more, five plus four is nine.

00:01:16.690 --> 00:01:19.040
Or you could say, hey look,
we start at negative five,

00:01:19.040 --> 00:01:22.550
we end up at four, four
minus negative five

00:01:22.550 --> 00:01:26.683
is the same thing as four
plus five, which is nine.