Dependent and independent variables exercise: graphing the equation

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.827
On your math quiz, you earn
5 points for each question

00:00:03.827 --> 00:00:04.910
that you answer correctly.

00:00:04.910 --> 00:00:08.189
In the table below, x represents
the number of questions

00:00:08.189 --> 00:00:11.337
that you answer correctly, and
y represents the total number

00:00:11.337 --> 00:00:12.920
of points that you
score on your quiz.

00:00:12.920 --> 00:00:13.527
Fair enough?

00:00:13.527 --> 00:00:15.360
The relationship between
these two variables

00:00:15.360 --> 00:00:17.193
can be expressed by the
following equation--

00:00:17.193 --> 00:00:18.540
y is equal to 5x.

00:00:18.540 --> 00:00:21.360
Graph the equation below.

00:00:21.360 --> 00:00:23.360
So you could look at a
couple of your points.

00:00:23.360 --> 00:00:25.568
You could say, well, look,
if I got 0 questions right

00:00:25.568 --> 00:00:27.260
I'm going to have 0 points.

00:00:27.260 --> 00:00:29.010
So you could literally
graph that point--

00:00:29.010 --> 00:00:32.009
if I got 0 questions
right, I get 0 points.

00:00:32.009 --> 00:00:33.550
And then if I get
one question right,

00:00:33.550 --> 00:00:35.430
and the table tells us
that, or we could logically

00:00:35.430 --> 00:00:37.430
think about it, every
question I would get right

00:00:37.430 --> 00:00:39.180
I'm going to get 5 more points.

00:00:39.180 --> 00:00:43.300
So if I get one question right,
I'm going to get 5 more points,

00:00:43.300 --> 00:00:45.170
and we saw that in
our table as well.

00:00:45.170 --> 00:00:48.132
We saw that right over here--
one question, 5 points.

00:00:48.132 --> 00:00:50.090
We could also plot it
two questions, 10 points,

00:00:50.090 --> 00:00:50.631
any of these.

00:00:50.631 --> 00:00:53.730
But you only have to do two
points to define a line.

00:00:53.730 --> 00:00:58.260
So it looks like we
are actually done here.


5 Triangle angle measure

WEBVTT
Kind: captions
Language: en

00:00:00.453 --> 00:00:04.910
In the figure above, triangle
ABC is inscribed in the

00:00:04.910 --> 00:00:08.919
circle with center O and diameter AC.

00:00:08.919 --> 00:00:11.298
So AC is the diameter right over there.

00:00:11.298 --> 00:00:16.298
If AB is equal to AO,
so this is equal to that

00:00:18.751 --> 00:00:21.050
right over there, AO is the radius so AB

00:00:21.050 --> 00:00:23.092
is the same length of
the radius, what is the

00:00:23.092 --> 00:00:27.888
degree measure of angle ABO?

00:00:27.888 --> 00:00:30.175
So the main realization here is that

00:00:30.175 --> 00:00:33.309
this length right over here,
AO, the length of segment AO,

00:00:33.309 --> 00:00:36.095
that's a radius so this is
a radius right over here.

00:00:36.095 --> 00:00:37.790
It goes from the center of the circle

00:00:37.790 --> 00:00:42.387
to the circle itself, to point A.

00:00:42.387 --> 00:00:45.926
Well this is also a radius of a circle.

00:00:47.202 --> 00:00:48.976
This is going to have the same length.

00:00:48.976 --> 00:00:50.973
They already told us that
AB has the same length

00:00:50.973 --> 00:00:55.253
as AO which, because AO
is a radius and OB is also

00:00:55.253 --> 00:00:58.816
a radius, it's going to have
the same length as OB or BO.

00:00:58.816 --> 00:01:00.477
And so then what do we have?

00:01:00.477 --> 00:01:02.369
We have an equilateral triangle.

00:01:02.369 --> 00:01:04.551
What do we know about
an equilateral triangle?

00:01:04.551 --> 00:01:07.570
All of the sides have
the same length, but also

00:01:07.570 --> 00:01:09.593
all of these interior angles are

00:01:09.593 --> 00:01:11.083
going to have the same measures.

00:01:11.083 --> 00:01:13.846
So that angle is equal to
that is equal to that angle

00:01:13.846 --> 00:01:16.051
and all three of them,
because they have to add up

00:01:16.051 --> 00:01:18.883
to 180 degrees, and you
have three equal angles

00:01:18.883 --> 00:01:20.672
adding up to 180 degrees, each of these

00:01:20.672 --> 00:01:25.215
have to be 60 degree angles.

00:01:26.477 --> 00:01:29.875
So angle ABO, which is the
one they're asking about,

00:01:29.875 --> 00:01:33.810
is going to be a 60 degree angle.


Ways to represent data

WEBVTT
Kind: captions
Language: en

00:00:00.278 --> 00:00:01.359
- [Voiceover] What I
want to do in this video

00:00:01.359 --> 00:00:03.135
is think about all of
all the different ways

00:00:03.135 --> 00:00:06.118
that we can represent data.

00:00:06.118 --> 00:00:08.394
So right over here, we have a list

00:00:08.394 --> 00:00:10.935
of, and I'm just using this as one form

00:00:10.935 --> 00:00:13.503
of data, a list of students' scores on,

00:00:13.503 --> 00:00:16.545
say, the last test, so
Amy got 90 percent right,

00:00:16.545 --> 00:00:18.061
Bill got 95 percent right,

00:00:18.061 --> 00:00:19.401
Cam got 100 percent right,

00:00:19.401 --> 00:00:21.623
Efra also got 100 percent right,

00:00:21.623 --> 00:00:23.761
and Farah got 80 percent right.

00:00:23.761 --> 00:00:25.541
This is one way to show data.

00:00:25.541 --> 00:00:28.206
Remember, data is just
recorded information,

00:00:28.206 --> 00:00:30.005
and it could be numeric like this,

00:00:30.005 --> 00:00:32.026
it could be quantitative,
so you're recording

00:00:32.026 --> 00:00:34.545
actual numbers, or it could even be things

00:00:34.545 --> 00:00:36.658
you could record data on
how do they like the test,

00:00:36.658 --> 00:00:38.701
and they could have scored it based on,

00:00:38.701 --> 00:00:40.199
I really liked it, I kind of liked it,

00:00:40.199 --> 00:00:42.049
I didn't like it, or they might have

00:00:42.049 --> 00:00:44.317
rated it on a scale of zero to five,

00:00:44.317 --> 00:00:45.821
which would have been numbers,

00:00:45.821 --> 00:00:48.089
but it's numbers that are
measuring peoples' opinions,

00:00:48.089 --> 00:00:50.537
as opposed to, here, we have numbers

00:00:50.537 --> 00:00:52.742
that are measuring their actual scores.

00:00:52.742 --> 00:00:54.344
So there's all different types of data,

00:00:54.344 --> 00:00:56.085
and I don't want to get into all of that,

00:00:56.085 --> 00:00:57.119
but let's just start thinking about

00:00:57.119 --> 00:00:59.591
different ways to represent this data.

00:00:59.591 --> 00:01:00.961
So this is one way, you could view this

00:01:00.961 --> 00:01:05.228
as a table where you have the name,

00:01:05.228 --> 00:01:07.443
and then you have the score.

00:01:07.443 --> 00:01:09.451
So you have your name column,

00:01:09.451 --> 00:01:10.844
and then you have your score column.

00:01:10.844 --> 00:01:12.725
And I can construct it as a table,

00:01:12.725 --> 00:01:15.949
so clearly, it looks like a table.

00:01:15.949 --> 00:01:17.882
Like that, that's one way,

00:01:17.882 --> 00:01:21.331
one very common way

00:01:21.331 --> 00:01:25.414
of representing data,

00:01:25.414 --> 00:01:27.388
just like that.

00:01:27.388 --> 00:01:29.814
That's actually how most
traditional databases

00:01:29.814 --> 00:01:32.368
record data, in tables like this.

00:01:32.368 --> 00:01:34.610
But you could also do it in other ways.

00:01:34.610 --> 00:01:37.702
So you could record it as a...

00:01:38.149 --> 00:01:40.407
Often times called a bar graph,

00:01:40.407 --> 00:01:43.006
or sometimes, a histogram,

00:01:43.006 --> 00:01:48.006
so you could put score on
the vertical axis here,

00:01:48.384 --> 00:01:51.957
and then you could have
your names over here.

00:01:51.957 --> 00:01:53.224
And let's see the scores,

00:01:53.224 --> 00:01:56.938
let's see, maybe we'll make this a 50.

00:01:56.938 --> 00:01:59.180
Actually, let me just mark them off.

00:01:59.180 --> 00:02:03.358
So this is 10, 20, 30, that's too big.

00:02:03.358 --> 00:02:06.922
10, 20, 30, 40, 50,

00:02:06.922 --> 00:02:10.928
60, 70, 80, 90, so that's...

00:02:10.928 --> 00:02:13.474
And then 100, so that's 100.

00:02:13.474 --> 00:02:15.284
One, two, three, four, five,

00:02:15.284 --> 00:02:17.652
that would be 50 right over there,

00:02:17.652 --> 00:02:19.371
and then you can go person by person.

00:02:19.371 --> 00:02:24.270
So Amy got a 90 on the exam,

00:02:24.270 --> 00:02:27.207
so the bar will go up to 90.

00:02:27.207 --> 00:02:31.294
So that is Amy, and then you have Bill,

00:02:31.294 --> 00:02:33.533
got a 95, so it's going

00:02:33.533 --> 00:02:35.695
to be between 90 and 100, so it's going

00:02:35.695 --> 00:02:37.608
to be right over there.

00:02:37.608 --> 00:02:40.834
Bill got a 95,

00:02:40.834 --> 00:02:43.457
and so it'll look like this.

00:02:43.457 --> 00:02:46.558
Bill, so that is Bill.

00:02:46.558 --> 00:02:49.840
And then you have Cam,
who got 100, on the exam.

00:02:49.840 --> 00:02:53.194
So, make sure, you see
I'm hand drawing it,

00:02:53.194 --> 00:02:56.144
so it's not as precise as if I were

00:02:56.144 --> 00:02:58.286
to do it on a computer.

00:02:58.286 --> 00:03:01.746
So this right over there,
that is Cam's score.

00:03:01.746 --> 00:03:03.859
Efra got the same score as Cam,

00:03:03.859 --> 00:03:06.063
so her score is going to be,

00:03:06.063 --> 00:03:09.245
let me do that in Efra's color,

00:03:09.245 --> 00:03:11.497
and that's Efra's score right there.

00:03:11.497 --> 00:03:13.901
She also got 100.

00:03:13.901 --> 00:03:17.568
So, Efra...

00:03:18.384 --> 00:03:20.941
Efra, and then finally, Farah got an 80,

00:03:20.941 --> 00:03:25.941
so 60, 70, 80, so Farah got an 80.

00:03:26.107 --> 00:03:28.758
So this is Farah's score right over here.

00:03:28.758 --> 00:03:33.007
So this is another way
of representing the data,

00:03:33.007 --> 00:03:34.853
and here we see it in visual form,

00:03:34.853 --> 00:03:36.574
but it has the same information.

00:03:36.574 --> 00:03:38.616
You can look up someone's name,

00:03:38.616 --> 00:03:39.777
and then figure out their score.

00:03:39.777 --> 00:03:43.515
Amy scored a 90, Bill scored a 95,

00:03:43.515 --> 00:03:46.313
Cam scored 100, Efra also scored 100,

00:03:46.313 --> 00:03:48.449
Farah scored an 80.

00:03:48.449 --> 00:03:49.807
And there's even other ways you can

00:03:49.807 --> 00:03:52.175
have some of this information.

00:03:52.175 --> 00:03:54.055
In fact sometimes, you might
not even know their names,

00:03:54.055 --> 00:03:55.665
and so then it would be less information

00:03:55.665 --> 00:03:57.407
but (mumbles) a list of scores.

00:03:57.407 --> 00:03:59.056
The professor might say,
"Hey, here are the five scores

00:03:59.056 --> 00:04:01.750
"that people got on the exam."

00:04:01.750 --> 00:04:06.750
And they were listed 90, 95,

00:04:07.075 --> 00:04:09.394
100...

00:04:10.470 --> 00:04:14.861
100, and 80, now,

00:04:14.861 --> 00:04:18.024
if it was listed, this
was all the data you got,

00:04:18.024 --> 00:04:19.846
this is less information than the data

00:04:19.846 --> 00:04:23.190
that's in this bar
graph, or this histogram,

00:04:23.190 --> 00:04:25.838
or the data that's given in this table

00:04:25.838 --> 00:04:27.754
right over here, because here, not only

00:04:27.754 --> 00:04:30.877
do we know the scores, but
we know who got what score.

00:04:30.877 --> 00:04:34.012
Here, we only know the list of scores,

00:04:34.012 --> 00:04:34.894
but there's even other ways,

00:04:34.894 --> 00:04:37.994
and this is not an exhaustive video of all

00:04:37.994 --> 00:04:40.548
of the different ways
you can represent data.

00:04:40.548 --> 00:04:42.301
You can also represent data by looking

00:04:42.301 --> 00:04:44.936
at the frequency of scores.

00:04:44.936 --> 00:04:48.592
So, the frequency of
scores right over here,

00:04:48.592 --> 00:04:49.998
so instead of writing the people,

00:04:49.998 --> 00:04:51.518
you could write the scores.

00:04:51.518 --> 00:04:55.280
So let's see, you could say this is 80,

00:04:55.280 --> 00:04:59.993
85, 90, 95,

00:04:59.993 --> 00:05:02.918
and 100, and then you could record

00:05:02.918 --> 00:05:04.950
the frequency that
people got these scores.

00:05:04.950 --> 00:05:07.690
So how many times do we
have a score of an 80?

00:05:07.690 --> 00:05:09.443
Well, Farah is the only
person with a score

00:05:09.443 --> 00:05:12.240
of 80, so you put one data point there.

00:05:12.240 --> 00:05:15.664
No one got an 85, one person got a 90,

00:05:15.664 --> 00:05:18.022
so you put a data point there.

00:05:18.022 --> 00:05:20.448
One person got a 95, so you can put

00:05:20.448 --> 00:05:22.365
that data point right over there.

00:05:22.365 --> 00:05:24.361
And then two people got 100.

00:05:24.361 --> 00:05:28.563
So this is one and two.

00:05:28.563 --> 00:05:30.747
Let's see the other 100 is in this color,

00:05:30.747 --> 00:05:31.697
so I'll just do it in the color.

00:05:31.697 --> 00:05:34.204
You wouldn't necessarily have
to color code it like this.

00:05:34.204 --> 00:05:36.307
So this is another way to represent,

00:05:36.307 --> 00:05:39.325
and this axis, you could
just view it as the number.

00:05:39.325 --> 00:05:41.558
So this tells you how many 80s there were,

00:05:41.558 --> 00:05:43.683
how many 90s there are,

00:05:43.683 --> 00:05:45.727
how many 95s, and how many 100s.

00:05:45.727 --> 00:05:47.781
So this right over here,

00:05:47.781 --> 00:05:50.522
has the same data as this list of numbers.

00:05:50.522 --> 00:05:52.819
It's just another way of looking at it.

00:05:52.819 --> 00:05:54.143
And once you have your data arranged

00:05:54.143 --> 00:05:56.070
in any of these ways,

00:05:56.070 --> 00:05:58.984
we can start to ask interesting questions.

00:05:58.984 --> 00:06:01.364
We can ask ourselves things like,

00:06:01.364 --> 00:06:04.104
well, what is the range of data?

00:06:04.104 --> 00:06:08.027
What is the range in the data?

00:06:08.027 --> 00:06:09.548
And the range is just the spread

00:06:09.548 --> 00:06:12.844
between the lowest point
and the highest point.

00:06:12.844 --> 00:06:14.099
So the range in this data,

00:06:14.099 --> 00:06:15.561
is going to be the difference between

00:06:15.561 --> 00:06:17.465
the highest score, and the highest scores

00:06:17.465 --> 00:06:19.718
are 100, and the lowest score,

00:06:19.718 --> 00:06:21.842
an 80, so the range is
going to be the difference

00:06:21.842 --> 00:06:25.383
between the max, minus the min.

00:06:25.383 --> 00:06:27.867
The maximum score minus the minimum score.

00:06:27.867 --> 00:06:30.398
So it's going to be 100 minus 80

00:06:30.398 --> 00:06:33.184
is equal to 20, so that
gives you a sense of things,

00:06:33.184 --> 00:06:35.261
it kind of gives you a sense of spread.

00:06:35.261 --> 00:06:36.760
You could also ask yourself, well,

00:06:36.760 --> 00:06:40.776
how many people scored below 100?

00:06:40.776 --> 00:06:42.982
And these are just interesting questions.

00:06:42.982 --> 00:06:45.861
Below 100, and you can actually answer

00:06:45.861 --> 00:06:47.405
that question, well, actually,

00:06:47.405 --> 00:06:48.718
you could have answered
either of these questions

00:06:48.718 --> 00:06:51.401
with any of these different
ways of looking at the data.

00:06:51.401 --> 00:06:53.455
If you say, how many
people scored below 100?

00:06:53.455 --> 00:06:55.306
Well, one, two, three?

00:06:55.306 --> 00:06:56.745
How many people scored below 100?

00:06:56.745 --> 00:06:58.046
Well, 100 is up here, so it's giong

00:06:58.046 --> 00:07:00.341
to be one, two, three.

00:07:00.341 --> 00:07:02.464
How many people scored below 100?

00:07:02.464 --> 00:07:04.513
One, two, three.

00:07:04.513 --> 00:07:06.590
How many people scored below 100?

00:07:06.590 --> 00:07:09.714
One, two, three.

00:07:09.714 --> 00:07:11.188
And so, any way you look at it,

00:07:11.188 --> 00:07:13.696
you would have gotten three.

00:07:13.696 --> 00:07:15.065
And you could also ask yourself,

00:07:15.065 --> 00:07:17.515
what is the most frequent score?

00:07:17.515 --> 00:07:20.672
So, most frequent.

00:07:22.727 --> 00:07:24.527
And once again, you could
answer that question

00:07:24.527 --> 00:07:27.569
with any of these ways of
representing this data.

00:07:27.569 --> 00:07:28.928
You could look at our original table

00:07:28.928 --> 00:07:30.309
and you say, look, there's only one 90,

00:07:30.309 --> 00:07:32.828
one 95, one 80, there's two 100s.

00:07:32.828 --> 00:07:35.916
So you'd say look, the
most frequent score is 100.

00:07:35.916 --> 00:07:37.053
You'd see that over here too,

00:07:37.053 --> 00:07:38.945
you actually have two
100s, there's only one

00:07:38.945 --> 00:07:40.710
of each of the other scores.

00:07:40.710 --> 00:07:43.067
Here you also see the 200s,

00:07:43.067 --> 00:07:44.529
and here is probably the clearest

00:07:44.529 --> 00:07:46.445
if you're looking at frequency.

00:07:46.445 --> 00:07:49.919
Sometimes this might be
called a frequency plot.

00:07:53.704 --> 00:07:55.657
It's often called a frequency plot.

00:07:55.657 --> 00:07:57.499
And you see here, the most frequent one

00:07:57.499 --> 00:07:58.828
is the one that has the most dots on it

00:07:58.828 --> 00:08:00.929
which is 100.

00:08:00.929 --> 00:08:02.950
So anyway, that's just a very, high level

00:08:02.950 --> 00:08:05.110
overview of how you can look at data

00:08:05.110 --> 00:08:07.210
in different ways to represent data,

00:08:07.210 --> 00:08:09.091
but the one thing I really
want you to get from this,

00:08:09.091 --> 00:08:10.913
is that these are all different ways

00:08:10.913 --> 00:08:12.968
of representing the same data.

00:08:12.968 --> 00:08:15.291
And we could probably invent other ways

00:08:15.291 --> 00:08:17.229
of doing it as well.


15 Possible values for expression

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:01.833
- [Voiceover] If p and n are integers

00:00:01.833 --> 00:00:05.366
such that p is greater than
n, is greater than zero.

00:00:05.366 --> 00:00:08.140
So this tells us since p
is strictly greater than n

00:00:08.140 --> 00:00:09.863
that means that they're
two different integers.

00:00:09.863 --> 00:00:12.233
p is larger of these
two different integers

00:00:12.233 --> 00:00:13.325
and that they're both positive.

00:00:13.325 --> 00:00:15.367
They're both greater than zero

00:00:15.367 --> 00:00:17.433
and they tell us that p
squared minus n squared

00:00:17.433 --> 00:00:18.740
is equal to 12.

00:00:18.740 --> 00:00:22.566
p squared minus n squared is equal to 12.

00:00:22.566 --> 00:00:26.366
Which of the following can
be the value of p minus,

00:00:26.366 --> 00:00:29.566
the value of p minus n?

00:00:29.566 --> 00:00:32.233
So one thing that might
jump immediately out of use

00:00:32.233 --> 00:00:34.366
and you could factor p
squared minus n squared,

00:00:34.366 --> 00:00:35.900
that this is a difference of squares.

00:00:35.900 --> 00:00:38.100
And why it especially jumps out this,

00:00:38.100 --> 00:00:40.858
when you factor it, one of
the factors is p minus n.

00:00:40.858 --> 00:00:45.666
So this could be factored as p plus n,

00:00:46.366 --> 00:00:47.879
times p minus n.

00:00:47.879 --> 00:00:50.673
This is just factoring
a difference of squares,

00:00:50.673 --> 00:00:52.776
is equal to 12.

00:00:52.776 --> 00:00:55.600
Is equal to 12.

00:00:56.167 --> 00:00:57.166
Now, what we essentially have

00:00:57.166 --> 00:01:00.166
is a product of two
expressions is equal to 12

00:01:00.166 --> 00:01:01.233
but each of expressions,

00:01:01.233 --> 00:01:03.300
this expression right over here, p plus n,

00:01:03.300 --> 00:01:05.458
this is the sum of two positive integers.

00:01:05.458 --> 00:01:07.304
This is a difference of
two positive integers

00:01:07.304 --> 00:01:09.776
so both of these are going to be positive.

00:01:09.776 --> 00:01:11.500
Both of these are positive integers.

00:01:11.500 --> 00:01:13.376
The reason why we know p
minus n is a positive integer

00:01:13.376 --> 00:01:15.842
is they tell us that p is greater than n.

00:01:15.842 --> 00:01:17.900
So this is going to be a positive value.

00:01:17.900 --> 00:01:19.966
So let's think about all
the different combinations

00:01:19.966 --> 00:01:22.900
of positive integers that
we could take their product

00:01:22.900 --> 00:01:24.900
and get to 12.

00:01:24.900 --> 00:01:27.596
So, it could be and this of course,

00:01:27.596 --> 00:01:30.325
this of course is going
to be larger of the two.

00:01:30.325 --> 00:01:32.500
This is going to be the larger of the two

00:01:32.500 --> 00:01:34.633
and this is going to be the smaller.

00:01:34.633 --> 00:01:37.366
Smaller of the two.

00:01:37.366 --> 00:01:38.300
So let's see.

00:01:38.300 --> 00:01:42.950
You could have 12 times one.

00:01:42.950 --> 00:01:46.166
12 times one is definitely equal to 12.

00:01:46.166 --> 00:01:48.792
You could have, let's see.

00:01:48.792 --> 00:01:53.792
You could have six times
two which is equal to 12.

00:01:54.766 --> 00:01:58.966
You could have four times three

00:01:58.966 --> 00:02:00.966
which is also equal to 12.

00:02:00.966 --> 00:02:02.607
So these are essentially
all of the contenders

00:02:02.607 --> 00:02:06.167
for p plus n and p minus n.

00:02:06.167 --> 00:02:10.242
So let's think about how
we can rule some of these,

00:02:10.242 --> 00:02:12.303
some of these scenarios.

00:02:12.303 --> 00:02:15.366
Some of these scenarios out.

00:02:15.366 --> 00:02:16.361
So let's take the first scenario.

00:02:16.361 --> 00:02:18.402
Let's say that p plus n.

00:02:18.402 --> 00:02:20.509
p, let me write it in green.

00:02:20.509 --> 00:02:22.433
So the first scenario.

00:02:22.433 --> 00:02:26.146
Let's say that p plus n is equal to 12

00:02:26.146 --> 00:02:28.997
and p minus n is equal to one.

00:02:28.997 --> 00:02:32.633
p minus n is equal to one.

00:02:32.633 --> 00:02:35.033
Well, we could solve for p here.

00:02:35.033 --> 00:02:37.023
We could add the left hand side.

00:02:37.023 --> 00:02:39.149
It's the left hand side
as the n is canceled out.

00:02:39.149 --> 00:02:41.366
You're left with two p.

00:02:41.366 --> 00:02:42.695
And you add the right hand side.

00:02:42.695 --> 00:02:45.012
You get 13, p.

00:02:45.012 --> 00:02:46.366
You divide both sides by two,

00:02:46.366 --> 00:02:49.533
you get p is equal to 13 over two.

00:02:49.873 --> 00:02:51.273
Now, this is not cool

00:02:51.273 --> 00:02:54.234
because this contradicts
one of our first assumptions

00:02:54.234 --> 00:02:56.633
that p needs to be an integer.

00:02:56.633 --> 00:02:59.700
This right over here is not an integer.

00:02:59.700 --> 00:03:02.233
So, p plus n could not be 12

00:03:02.233 --> 00:03:04.376
and p minus n cannot be one.

00:03:04.376 --> 00:03:07.642
So, p minus n cannot be one.

00:03:07.642 --> 00:03:10.118
So actually we can rule
this out right over here.

00:03:10.118 --> 00:03:12.500
p minus n cannot be one.

00:03:12.500 --> 00:03:14.233
Now let's think about this scenario

00:03:14.233 --> 00:03:16.700
where p plus n is equal to six.

00:03:16.700 --> 00:03:19.381
p plus n is equal to six.

00:03:19.381 --> 00:03:21.500
That's the larger of the two factors.

00:03:21.500 --> 00:03:26.500
And p minus n is equal to two.

00:03:26.766 --> 00:03:28.730
Well, same idea.

00:03:28.730 --> 00:03:30.330
We can add the left hand side.

00:03:30.330 --> 00:03:31.842
We go off with two p

00:03:31.842 --> 00:03:33.233
and then the right hand side,

00:03:33.233 --> 00:03:35.166
we get six plus two is equal to eight.

00:03:35.166 --> 00:03:36.530
Divide both sides by two,

00:03:36.530 --> 00:03:39.366
you get p is equal to four.

00:03:39.366 --> 00:03:42.730
And if p is equal to four then you see,

00:03:42.730 --> 00:03:45.361
well, see four plus what is equal to six?

00:03:45.361 --> 00:03:47.227
Well, four plus two is equal to six

00:03:47.227 --> 00:03:49.263
and is equal to two.

00:03:49.263 --> 00:03:50.100
And that works out here.

00:03:50.100 --> 00:03:52.366
Four minus two is equal to two.

00:03:52.366 --> 00:03:53.966
So, this is completely consistent.

00:03:53.966 --> 00:03:55.612
Both of these are positive integers.

00:03:55.612 --> 00:03:57.100
They're different positive integers,

00:03:57.100 --> 00:03:58.700
p is greater than n.

00:03:58.700 --> 00:04:01.263
So it's completely possible that p minus n

00:04:01.263 --> 00:04:02.761
is equal to two.

00:04:02.761 --> 00:04:07.635
So, this is completely a
possible value of p minus n.

00:04:07.635 --> 00:04:09.382
And then let's look at this last.

00:04:09.382 --> 00:04:10.827
So this one works out.

00:04:10.827 --> 00:04:13.433
Let's look at this last scenario.

00:04:13.433 --> 00:04:17.433
So we have p plus n is equal to four

00:04:17.447 --> 00:04:22.447
and p minus n is equal to three.

00:04:23.300 --> 00:04:25.196
Well, same idea.

00:04:25.196 --> 00:04:29.889
The left hand side you get
two p is equal to seven.

00:04:29.889 --> 00:04:32.366
Well once again, we
divide both sides by two.

00:04:32.366 --> 00:04:34.100
p is equal to 7/2.

00:04:34.100 --> 00:04:35.300
This is not an integer.

00:04:35.300 --> 00:04:37.094
Just like here, this cannot be the case

00:04:37.094 --> 00:04:39.166
because p needs to be an integer.

00:04:39.166 --> 00:04:41.156
In this case, p is being forced.

00:04:41.156 --> 00:04:45.566
If p plus n was four
and p minus n is three

00:04:45.566 --> 00:04:47.725
then you are not going to have,

00:04:47.725 --> 00:04:50.100
then p is not going to be an integer.

00:04:50.100 --> 00:04:54.633
So this is also not a possibility.

00:04:54.633 --> 00:04:56.300
Now, we could have ruled this out

00:04:56.300 --> 00:04:58.166
even without doing that exercise

00:04:58.166 --> 00:05:00.776
because if p minus n was four,

00:05:00.776 --> 00:05:02.635
if p minus n was four,

00:05:02.635 --> 00:05:05.033
then p plus n would have to be three.

00:05:05.033 --> 00:05:08.700
But p plus n is going to be
the larger of the two values.

00:05:08.700 --> 00:05:13.700
So, p minus n even if this
were a possible scenario,

00:05:14.279 --> 00:05:17.500
even if we didn't end up with
a non-integer right over here,

00:05:17.500 --> 00:05:19.967
we could have gotten p minus
n to three but not to four.

00:05:19.967 --> 00:05:23.109
p minus n is going to be the
lesser of the two factors

00:05:23.109 --> 00:05:25.946
between if four and three
were even contenders.

00:05:25.946 --> 00:05:27.833
But we know here that they aren't.

00:05:27.833 --> 00:05:29.633
But anyway, we could rule that one.

00:05:29.633 --> 00:05:31.909
We can rule that one out as well.

00:05:31.909 --> 00:05:34.300
So the only possible value of p minus n

00:05:34.300 --> 00:05:37.166
or which of the following
can be a value of p minus n?

00:05:37.166 --> 00:05:38.466
Two.


Reflecting line across another line example

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:04.100
We're asked to reflect
this line across the line y

00:00:04.100 --> 00:00:05.707
is equal to negative 4.

00:00:05.707 --> 00:00:07.290
So you have this
line right over here,

00:00:07.290 --> 00:00:09.960
and they actually give us all
of these transformation tools

00:00:09.960 --> 00:00:11.585
right over here where
we can Translate,

00:00:11.585 --> 00:00:15.460
Rotate, Reflect, Dilate, or
Undo what we were about to do.

00:00:15.460 --> 00:00:19.000
So they want us to
reflect this line across y

00:00:19.000 --> 00:00:20.560
is equal to negative 4.

00:00:20.560 --> 00:00:23.980
So let's get the
Reflection tool out.

00:00:23.980 --> 00:00:30.250
So I clicked reflect and
let's make this line be y

00:00:30.250 --> 00:00:31.880
equals negative 4.

00:00:31.880 --> 00:00:33.160
So let's see.

00:00:33.160 --> 00:00:38.200
That would look something
y equals negative 4.

00:00:38.200 --> 00:00:41.020
So we're going to reflect
this line across y

00:00:41.020 --> 00:00:42.550
equals negative 4.

00:00:42.550 --> 00:00:45.370
So let's reflect it,
and there you have it.

00:00:45.370 --> 00:00:48.200
We reflected it across
y equals negative 4.

00:00:48.200 --> 00:00:50.490
And then they ask us to
answer some questions.

00:00:50.490 --> 00:00:55.390
When a line is reflected,
the image created is a line.

00:00:55.390 --> 00:00:58.350
We know this is true because
a reflected image, well,

00:00:58.350 --> 00:01:00.350
the original image
had zero endpoints.

00:01:00.350 --> 00:01:03.111
These are just two points on
the line, they're not endpoints,

00:01:03.111 --> 00:01:04.819
they're not defining
the end of the line.

00:01:04.819 --> 00:01:07.110
If they were both endpoints
it would be a line segment,

00:01:07.110 --> 00:01:09.510
but this line keeps
going in both directions.

00:01:09.510 --> 00:01:11.780
So we know this is true
because a reflected image has

00:01:11.780 --> 00:01:15.630
zero endpoints, and
we got it right.


How to match function input to output given the graph (example)

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.740
The graph of the function g is shown below

00:00:03.810 --> 00:00:06.546
What is the input value for which g of x

00:00:06.694 --> 00:00:08.801
is equal to negative 2?

00:00:08.806 --> 00:00:11.836
So what they do here is along the x axis

00:00:11.856 --> 00:00:14.536
these are the inputs and then the graph

00:00:14.623 --> 00:00:16.263
shows us what's the output

00:00:16.294 --> 00:00:17.804
so when x is equal to 7,

00:00:17.903 --> 00:00:19.973
G of 7 we see here is 1.

00:00:20.003 --> 00:00:22.263
If x equals 9,

00:00:22.297 --> 00:00:24.259
g of 9 here is 2.

00:00:24.259 --> 00:00:25.739
If x equals 6,

00:00:25.739 --> 00:00:30.259
g of 6 is equal to the y coordinate at

00:00:30.259 --> 00:00:31.760
this point is equal to 0.

00:00:31.760 --> 00:00:35.480
so what is the input value for which g is

00:00:35.490 --> 00:00:37.783
equal to negative 2?

00:00:37.873 --> 00:00:40.293
so this graph over here, this is y equals

00:00:40.293 --> 00:00:41.363
g of x.

00:00:41.363 --> 00:00:43.768
So g of x equaling to negative 2 means y

00:00:43.771 --> 00:00:45.241
of x is equal to negative 2

00:00:45.251 --> 00:00:47.776
And so when does y equal negative 2?

00:00:47.776 --> 00:00:49.786
(muttering) when does equal negative 2?

00:00:49.786 --> 00:00:51.854
(talking) it looks like that happens right

00:00:51.854 --> 00:00:52.774
at this point.

00:00:52.778 --> 00:00:55.528
And that happens when you input negative 9

00:00:55.548 --> 00:00:56.519
into g.

00:00:56.519 --> 00:00:59.715
g of negative 9 is negative 2

00:00:59.715 --> 00:01:02.762
so this is going to be negative 9.

00:01:02.772 --> 00:01:05.802
And we're done.


Constructing a box and whisker plot

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.770
The owner of a restaurant
wants to find out more about

00:00:03.770 --> 00:00:06.020
where his patrons
are coming from.

00:00:06.020 --> 00:00:08.140
One day, he decided
to gather data

00:00:08.140 --> 00:00:10.650
about the distance
in miles that people

00:00:10.650 --> 00:00:12.930
commuted to get
to his restaurant.

00:00:12.930 --> 00:00:15.750
People reported the
following distances traveled.

00:00:15.750 --> 00:00:17.940
So here are all the
distances traveled.

00:00:17.940 --> 00:00:20.250
He wants to create
a graph that helps

00:00:20.250 --> 00:00:23.170
him understand the
spread of the distances--

00:00:23.170 --> 00:00:26.210
this is a key word--
the spread of distances

00:00:26.210 --> 00:00:32.520
and the median distance
that people traveled

00:00:32.520 --> 00:00:33.570
or that people travel.

00:00:33.570 --> 00:00:36.700
What kind of graph
should he create?

00:00:36.700 --> 00:00:39.542
So the answer of what kind
of graph he should create,

00:00:39.542 --> 00:00:41.500
that might be a little
bit more straightforward

00:00:41.500 --> 00:00:45.440
than the actual creation of the
graph, which we will also do.

00:00:45.440 --> 00:00:49.670
But he's trying to visualize
the spread of information.

00:00:49.670 --> 00:00:52.360
And at the same time,
he wants the median.

00:00:52.360 --> 00:00:55.950
So what a graph captures
both of that information?

00:00:55.950 --> 00:00:58.090
Well, a box and whisker plot.

00:00:58.090 --> 00:01:01.520
So let's actually try to
draw a box and whisker plot.

00:01:01.520 --> 00:01:03.980
And to do that, we need to
come up with the median.

00:01:03.980 --> 00:01:06.880
And we'll also see the median
of the two halves of the data

00:01:06.880 --> 00:01:07.380
as well.

00:01:07.380 --> 00:01:09.796
And whenever we're trying to
take the median of something,

00:01:09.796 --> 00:01:11.680
it's really helpful
to order our data.

00:01:11.680 --> 00:01:16.220
So let's start off by
attempting to order our data.

00:01:16.220 --> 00:01:18.940
So what is the
smallest number here?

00:01:18.940 --> 00:01:19.760
Well, let's see.

00:01:19.760 --> 00:01:20.650
There's one 2.

00:01:20.650 --> 00:01:22.240
So let me mark it off.

00:01:22.240 --> 00:01:25.560
And then we have another two.

00:01:25.560 --> 00:01:27.250
So we've got all the 2's.

00:01:27.250 --> 00:01:30.180
And then we have this 3.

00:01:30.180 --> 00:01:32.430
Then we have this 3.

00:01:32.430 --> 00:01:34.230
I think we've got all the 3's.

00:01:34.230 --> 00:01:36.890
Then we have that 4.

00:01:36.890 --> 00:01:40.690
Then we have this 4.

00:01:40.690 --> 00:01:42.090
Do we have any 5's?

00:01:42.090 --> 00:01:42.590
No.

00:01:42.590 --> 00:01:43.350
Do we have any 6's?

00:01:43.350 --> 00:01:43.850
Yep.

00:01:43.850 --> 00:01:45.490
We have that 6.

00:01:45.490 --> 00:01:47.670
And that looks like the only 6.

00:01:47.670 --> 00:01:49.291
Any 7's?

00:01:49.291 --> 00:01:49.790
Yep.

00:01:49.790 --> 00:01:51.790
We have this 7 right over here.

00:01:51.790 --> 00:01:54.460
And I just realized
that I missed this 1.

00:01:54.460 --> 00:01:57.070
So let me put the 1 at
the beginning of our set.

00:01:57.070 --> 00:01:58.494
So I got that 1
right over there.

00:01:58.494 --> 00:01:59.660
Actually, there was two 1's.

00:01:59.660 --> 00:02:00.576
I missed both of them.

00:02:00.576 --> 00:02:04.340
So both of those 1's
are right over there.

00:02:04.340 --> 00:02:07.260
So I have the 1's,
2's, 3's, 4's, no 5's.

00:02:07.260 --> 00:02:08.789
This is one 6.

00:02:08.789 --> 00:02:10.039
There was one 7.

00:02:10.039 --> 00:02:13.230
There's one 8 right over here.

00:02:13.230 --> 00:02:15.090
And then, let's see, any 9's?

00:02:15.090 --> 00:02:15.920
No 9's.

00:02:15.920 --> 00:02:16.560
Any 10s?

00:02:16.560 --> 00:02:17.060
Yep.

00:02:17.060 --> 00:02:18.660
There's a 10.

00:02:18.660 --> 00:02:19.700
Any 11s?

00:02:19.700 --> 00:02:22.010
We have an 11 right over there.

00:02:22.010 --> 00:02:23.060
Any 12s?

00:02:23.060 --> 00:02:23.620
Nope.

00:02:23.620 --> 00:02:27.210
13, 14?

00:02:27.210 --> 00:02:30.660
Then we have a 15.

00:02:30.660 --> 00:02:34.830
And then we have a
20 and then a 22.

00:02:34.830 --> 00:02:36.230
So we've ordered all our data.

00:02:36.230 --> 00:02:38.360
Now it should be relatively
straightforward to find

00:02:38.360 --> 00:02:41.162
the middle of our
data, the median.

00:02:41.162 --> 00:02:42.620
So how many data
points do we have?

00:02:42.620 --> 00:02:50.402
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17.

00:02:50.402 --> 00:02:51.860
So the middle number
is going to be

00:02:51.860 --> 00:02:53.940
a number that has 8
numbers larger than it

00:02:53.940 --> 00:02:55.520
and 8 numbers smaller than it.

00:02:55.520 --> 00:02:56.520
So let's think about it.

00:02:56.520 --> 00:03:00.470
1, 2, 3, 4, 5, 6, 7, 8.

00:03:00.470 --> 00:03:04.090
So the number 6 here is
larger than 8 of the values.

00:03:04.090 --> 00:03:05.990
And if I did the
calculations right,

00:03:05.990 --> 00:03:08.230
it should be smaller
than 8 of the values.

00:03:08.230 --> 00:03:12.020
1, 2, 3, 4, 5, 6, 7, 8.

00:03:12.020 --> 00:03:17.390
So it is, indeed, the median.

00:03:17.390 --> 00:03:21.340
Now, when we're trying to
construct a box and whisker

00:03:21.340 --> 00:03:24.530
plot, the convention is,
OK, we have our median.

00:03:24.530 --> 00:03:27.230
And it's essentially dividing
our data into two sets.

00:03:27.230 --> 00:03:30.560
Now, let's take the median
of each of those sets.

00:03:30.560 --> 00:03:32.810
And the convention is to
take our median out and have

00:03:32.810 --> 00:03:34.240
the sets that are left over.

00:03:34.240 --> 00:03:35.520
Sometimes people leave it in.

00:03:35.520 --> 00:03:37.910
But the standard convention,
take this median out.

00:03:37.910 --> 00:03:39.450
And now, look
separately at this set

00:03:39.450 --> 00:03:41.700
and look separately at this set.

00:03:41.700 --> 00:03:44.680
So if we look at this first
bottom half of our numbers

00:03:44.680 --> 00:03:49.060
essentially, what's the
median of these numbers?

00:03:49.060 --> 00:03:54.530
Well, we have 1, 2, 3, 4,
5, 6, 7, 8 data points.

00:03:54.530 --> 00:03:57.000
So we're actually going to
have two middle numbers.

00:03:57.000 --> 00:04:00.725
So the two middle numbers
are this 2 and this 3,

00:04:00.725 --> 00:04:02.460
three numbers less
than these two,

00:04:02.460 --> 00:04:03.785
three numbers greater than it.

00:04:03.785 --> 00:04:05.410
And so when we're
looking for a median,

00:04:05.410 --> 00:04:06.580
you have two middle numbers.

00:04:06.580 --> 00:04:08.260
We take the mean of
these two numbers.

00:04:08.260 --> 00:04:12.910
So halfway in between
two and three is 2.5.

00:04:12.910 --> 00:04:16.720
Or you can say 2 plus 3
is 5 divided by 2 is 2.5.

00:04:16.720 --> 00:04:22.440
So here we have a median
of this bottom half of 2.5.

00:04:22.440 --> 00:04:25.160
And then the middle
of the top half,

00:04:25.160 --> 00:04:26.970
once again, we
have 8 data points.

00:04:26.970 --> 00:04:30.030
So our middle two
numbers are going

00:04:30.030 --> 00:04:33.550
to be this 11 and this 14.

00:04:33.550 --> 00:04:35.940
And so if we want to take the
mean of these two numbers,

00:04:35.940 --> 00:04:38.730
11 plus 14 is 25.

00:04:38.730 --> 00:04:42.610
Halfway in between
the two is 12.5.

00:04:42.610 --> 00:04:47.300
So 12.5 is exactly
halfway between 11 and 14.

00:04:47.300 --> 00:04:49.480
And now, we've figured
out all of the information

00:04:49.480 --> 00:04:51.620
we need to actually
plot or actually

00:04:51.620 --> 00:04:55.000
create or actually draw
our box and whisker plot.

00:04:55.000 --> 00:05:03.080
So let me draw a number line,
so my best attempt at a number

00:05:03.080 --> 00:05:04.900
line.

00:05:04.900 --> 00:05:07.280
So that's my number line.

00:05:07.280 --> 00:05:09.930
And let's say that this
right over here is a 0.

00:05:09.930 --> 00:05:13.610
I need to make sure I get all
the way up to 22 or beyond 22.

00:05:13.610 --> 00:05:15.150
So let's say that's 0.

00:05:15.150 --> 00:05:17.190
Let's say this is 5.

00:05:17.190 --> 00:05:18.240
This is 10.

00:05:18.240 --> 00:05:21.060
That could be 15.

00:05:21.060 --> 00:05:23.080
And that could be 20.

00:05:23.080 --> 00:05:25.460
This could be 25.

00:05:25.460 --> 00:05:29.650
We could keep
going-- 30, maybe 35.

00:05:29.650 --> 00:05:32.630
So the first thing we might
want to think about-- there's

00:05:32.630 --> 00:05:34.070
several ways to draw it.

00:05:34.070 --> 00:05:37.280
We want to think about
the box part of the box

00:05:37.280 --> 00:05:39.260
and whisker
essentially represents

00:05:39.260 --> 00:05:41.150
the middle half of our data.

00:05:41.150 --> 00:05:45.720
So it's essentially trying to
represent this data right over

00:05:45.720 --> 00:05:51.660
here, so the data between the
medians of the two halves.

00:05:51.660 --> 00:05:53.720
So this is a part
that we would attempt

00:05:53.720 --> 00:05:55.360
to represent with the box.

00:05:55.360 --> 00:06:00.070
So we would start right
over here at this 2.5.

00:06:00.070 --> 00:06:02.244
This is essentially
separating the first quartile

00:06:02.244 --> 00:06:04.660
from the second quartile, the
first quarter of our numbers

00:06:04.660 --> 00:06:06.580
from the second
quarter of our numbers.

00:06:06.580 --> 00:06:07.913
So let's put it right over here.

00:06:07.913 --> 00:06:10.260
So this is 2.5.

00:06:10.260 --> 00:06:12.710
2.5 is halfway between 0 and 5.

00:06:12.710 --> 00:06:15.100
So that's 2.5.

00:06:15.100 --> 00:06:17.390
And then up here, we have 12.5.

00:06:17.390 --> 00:06:21.650
And 12.5 is right
over-- let's see.

00:06:21.650 --> 00:06:24.580
This is 10.

00:06:24.580 --> 00:06:29.490
So this right over here would be
halfway between, well, halfway

00:06:29.490 --> 00:06:31.890
between 10 and 15 is 12.5.

00:06:31.890 --> 00:06:32.640
So let me do this.

00:06:32.640 --> 00:06:37.720
So this is 12.5 right over here.

00:06:37.720 --> 00:06:40.110
So that separates
the third quartile

00:06:40.110 --> 00:06:41.460
from the fourth quartile.

00:06:41.460 --> 00:06:43.530
And then our boxes,
everything in between,

00:06:43.530 --> 00:06:45.696
so this is literally the
middle half of our numbers.

00:06:47.960 --> 00:06:50.110
And we'd want to show
where the actual median is.

00:06:50.110 --> 00:06:51.735
And that was actually
one of the things

00:06:51.735 --> 00:06:54.064
we wanted to be able
to think about when

00:06:54.064 --> 00:06:55.480
the owner of the
restaurant wanted

00:06:55.480 --> 00:06:58.420
to think about how far
people are traveling from.

00:06:58.420 --> 00:07:00.130
So the median is 6.

00:07:00.130 --> 00:07:01.800
So we can plot it
right over here.

00:07:01.800 --> 00:07:05.880
So this right here is about six.

00:07:05.880 --> 00:07:08.120
Let me do that same pink color.

00:07:08.120 --> 00:07:11.620
So this right over here is 6.

00:07:11.620 --> 00:07:14.700
And then the whiskers of
the box and whisker plot

00:07:14.700 --> 00:07:17.480
essentially show us
the range of our data.

00:07:17.480 --> 00:07:21.412
And I can do this in a different
color that I haven't used yet.

00:07:21.412 --> 00:07:22.370
I'll do this in orange.

00:07:22.370 --> 00:07:24.036
So essentially, if
we want to see, look,

00:07:24.036 --> 00:07:25.640
the numbers go all
the way up to 22.

00:07:25.640 --> 00:07:27.265
So they go all the
way up to-- so let's

00:07:27.265 --> 00:07:29.520
say that this is
22 right over here.

00:07:29.520 --> 00:07:31.850
Our numbers go all
the way up to 22.

00:07:36.850 --> 00:07:39.490
And they go as low as 1.

00:07:39.490 --> 00:07:42.950
So 1 is right about here.

00:07:42.950 --> 00:07:43.840
Let me label that.

00:07:43.840 --> 00:07:45.040
So that's 1.

00:07:45.040 --> 00:07:47.586
And they go as low as 1.

00:07:47.586 --> 00:07:48.460
So there you have it.

00:07:48.460 --> 00:07:50.150
We have our box
and whisker plot.

00:07:50.150 --> 00:07:52.150
And you can see if you
have a plot like this,

00:07:52.150 --> 00:07:53.610
just visually, you
can immediately

00:07:53.610 --> 00:07:55.110
see, OK, what is the median?

00:07:55.110 --> 00:07:57.502
It's the middle of
the box, essentially.

00:07:57.502 --> 00:07:58.710
It shows you the middle half.

00:07:58.710 --> 00:08:00.293
So it shows you how
far they're spread

00:08:00.293 --> 00:08:02.450
or where the meat
of the spread is.

00:08:02.450 --> 00:08:05.320
And then it shows, well, beyond
that, we have the range that

00:08:05.320 --> 00:08:10.020
goes well beyond that or how
far the total spread of our data

00:08:10.020 --> 00:08:10.520
is.

00:08:10.520 --> 00:08:13.790
So this gives a pretty good
sense of both the median

00:08:13.790 --> 00:08:17.056
and the spread of our data.


Inverse of a 2x2 matrix

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:04.390
Let's attempt to take the
inverse of this 2 by 2 matrix.

00:00:04.390 --> 00:00:06.170
And you'll see the
2 by 2 matrices

00:00:06.170 --> 00:00:08.480
are about the only
size of matrices

00:00:08.480 --> 00:00:10.670
that it's somewhat pleasant
to take the inverse of.

00:00:10.670 --> 00:00:13.810
Anything larger than that,
it becomes very unpleasant.

00:00:13.810 --> 00:00:16.030
So the inverse of
a 2 by 2 matrix

00:00:16.030 --> 00:00:21.870
is going to be equal to 1 over
the determinant of the matrix

00:00:21.870 --> 00:00:24.990
times the adjugate
of the matrix, which

00:00:24.990 --> 00:00:26.460
sounds like a very fancy word.

00:00:26.460 --> 00:00:30.887
But we'll see for by a 2 by 2
matrix, it's not too involved.

00:00:30.887 --> 00:00:33.470
So first let's think about what
the determinant of this matrix

00:00:33.470 --> 00:00:34.350
is.

00:00:34.350 --> 00:00:35.990
Well, we've seen this before.

00:00:35.990 --> 00:00:38.650
We just look along
the two diagonals.

00:00:38.650 --> 00:00:43.880
It's 3 times 2 minus
negative 7 times 5.

00:00:43.880 --> 00:00:56.210
So this is going to be equal to
1 over 3 times 2 minus negative

00:00:56.210 --> 00:00:58.290
7 times 5.

00:01:03.580 --> 00:01:06.250
And then the adjugate
of A-- and here

00:01:06.250 --> 00:01:08.874
I'm really just teaching
you the mechanics of it.

00:01:08.874 --> 00:01:11.290
And it's a little unfortunate
that in a typical algebra II

00:01:11.290 --> 00:01:13.790
class you kind of just go
into the mechanics of it.

00:01:13.790 --> 00:01:17.410
But at least this will get
us to where we need to go.

00:01:17.410 --> 00:01:22.200
So the adjugate of
A, you literally just

00:01:22.200 --> 00:01:26.800
need to swap the two
elements on this diagonal.

00:01:26.800 --> 00:01:30.340
So put the 2 where the 3 is
and the 3 where the 2 is.

00:01:30.340 --> 00:01:34.260
So this element right
over here, this 3

00:01:34.260 --> 00:01:36.590
will go right over there.

00:01:36.590 --> 00:01:41.710
This 2 will go right over here.

00:01:41.710 --> 00:01:45.580
And then these two elements, you
just take the negative of them.

00:01:45.580 --> 00:01:51.241
So the negative-- let
me do a new color.

00:01:51.241 --> 00:01:52.740
Actually, I'm running
out of colors.

00:01:52.740 --> 00:01:55.680
The negative of
that is negative 5.

00:01:55.680 --> 00:01:59.480
The negative of
that is positive 7.

00:01:59.480 --> 00:02:01.300
So we are left
with-- this is going

00:02:01.300 --> 00:02:05.670
to be equal to 1
over-- 3 times 2 is 6.

00:02:05.670 --> 00:02:08.612
Negative 7 times
5 is negative 35.

00:02:08.612 --> 00:02:10.320
But then we have this
positive over here.

00:02:10.320 --> 00:02:13.500
So this whole thing
becomes plus 35.

00:02:13.500 --> 00:02:15.890
So 6 plus 35 is 41.

00:02:15.890 --> 00:02:18.520
So the determinant
of our matrix is 41.

00:02:18.520 --> 00:02:20.360
We're going to take 1
over the determinant

00:02:20.360 --> 00:02:26.030
and multiply it times our
adjugate, times 2, negative 5,

00:02:26.030 --> 00:02:28.000
7, and 3.

00:02:28.000 --> 00:02:33.710
So we get -- so this is the
drum roll part -- 2 over 41,

00:02:33.710 --> 00:02:35.290
negative 5 over 41.

00:02:35.290 --> 00:02:39.040
I'm just multiplying each of
these elements times 1 over 41.

00:02:39.040 --> 00:02:43.980
7 over 41, and 3 over 41.

00:02:43.980 --> 00:02:46.589
And we are done.


Curly arrow conventions in organic chemistry

WEBVTT
Kind: captions
Language: en

00:00:00.233 --> 00:00:02.166
Sal: What I want to do in
this video is talk a little

00:00:02.166 --> 00:00:04.700
bit about the curly
arrow conventions used in

00:00:04.700 --> 00:00:07.300
organic chemistry and the
slight variations I use

00:00:07.300 --> 00:00:10.400
in many of the videos
here on Khan Academy.

00:00:10.433 --> 00:00:13.233
There's two types of
curly arrows you will see.

00:00:13.233 --> 00:00:17.333
You will see a curly full arrow like this,

00:00:17.433 --> 00:00:19.966
a curly full arrow like this.

00:00:19.966 --> 00:00:21.500
And I make sure to draw it curly,

00:00:21.500 --> 00:00:23.566
you will always see the curly like this.

00:00:23.566 --> 00:00:27.800
And you will see a curly half
arrow that looks like this,

00:00:27.966 --> 00:00:31.546
curly half arrow or fish hook arrow.

00:00:31.546 --> 00:00:34.700
The convention is a full
arrow or a typical arrow

00:00:34.700 --> 00:00:36.433
that you're used to
seeing, this is talking

00:00:36.433 --> 00:00:39.466
about the movement of
pairs, of electron pairs.

00:00:39.500 --> 00:00:46.500
Movement of pairs is the convention.

00:00:46.500 --> 00:00:48.900
I'll show you in a second
that I do a slight variation

00:00:48.900 --> 00:00:50.833
of that, and I do that
because it helps me account

00:00:50.833 --> 00:00:53.766
for electrons, and it
helps me at least visualize

00:00:53.766 --> 00:00:56.700
or conceptualize how
things are, or essentially

00:00:56.700 --> 00:00:59.633
how things are happening,
a little bit better.

00:00:59.633 --> 00:01:01.633
The general convention is
that this is movement of

00:01:01.633 --> 00:01:06.766
pairs and this is movement
of electron by itself.

00:01:06.766 --> 00:01:12.800
Electron, electron not part,

00:01:14.581 --> 00:01:17.933
electron by itself, maybe
I'll write it this way.

00:01:18.700 --> 00:01:21.966
By itself.

00:01:21.966 --> 00:01:24.633
The full arrow is what you're going to

00:01:24.633 --> 00:01:27.634
see through most of organic chemistry.

00:01:27.634 --> 00:01:28.633
This is the one that you're going to

00:01:28.633 --> 00:01:31.766
see most typically, the movement of pairs.

00:01:31.766 --> 00:01:34.566
The movement of electrons
by itself, this is going to

00:01:34.566 --> 00:01:37.633
show up more in free radical
reactions, which we do do,

00:01:37.633 --> 00:01:41.233
but this is later on, and
most of organic chemistry

00:01:41.233 --> 00:01:44.200
is going to be dealing
with the movement of pairs.

00:01:44.300 --> 00:01:47.766
What I've drawn over here is a curly

00:01:47.766 --> 00:01:50.233
arrow showing the same thing happening.

00:01:50.233 --> 00:01:52.433
I'm showing you the slight
variation that I do.

00:01:52.433 --> 00:01:55.233
I do it because it helps me, once again,

00:01:55.233 --> 00:01:57.300
account for the electrons, and it helps

00:01:57.300 --> 00:02:00.000
me conceptualize what is going on.

00:02:00.033 --> 00:02:05.233
The typical way that this type
of mechanism will be shown,

00:02:05.233 --> 00:02:09.300
we'll say you have this
electron pair on this oxygen,

00:02:09.300 --> 00:02:12.100
and this electron pair,
sometimes we will say,

00:02:12.100 --> 00:02:14.366
and you will learn about this
reaction in not too long,

00:02:14.366 --> 00:02:16.966
is going to the carbon,
or I guess you could say

00:02:16.966 --> 00:02:19.433
it's attacking the carbon right over here.

00:02:19.433 --> 00:02:23.100
The reason why this I find
a little bit less intuitive

00:02:23.100 --> 00:02:26.366
is that the whole pair is
not going to the carbon,

00:02:26.366 --> 00:02:29.766
that the oxygen is still
going to maintain half

00:02:29.766 --> 00:02:32.356
of this pair and it's
going to form a bond.

00:02:32.356 --> 00:02:34.900
Essentially one end of this
pair is going to end up at

00:02:34.900 --> 00:02:37.433
the carbon, one end of this
pair is going to end up

00:02:37.433 --> 00:02:41.100
at the oxygen, and they
are going to form a bond.

00:02:41.100 --> 00:02:43.900
The way I draw it, still
drawing the full arrow.

00:02:43.900 --> 00:02:47.100
Here I'm still talking
about pairs but I'm talking

00:02:47.100 --> 00:02:50.300
about the movement of an
electron as part of a pair.

00:02:50.300 --> 00:02:52.833
That's kind of the slight non-conventional

00:02:52.833 --> 00:02:55.300
thing that I do with the full arrow.

00:02:55.300 --> 00:03:08.733
Movement, movement of electron,
electron as part of pair.

00:03:09.166 --> 00:03:14.100
I'll often times draw the back
of the arrow from that electron,

00:03:14.100 --> 00:03:17.966
but It's important to recognize
that electron is not moving

00:03:17.966 --> 00:03:21.900
by itself, it's just ending
up on one side of a bond,

00:03:21.900 --> 00:03:24.273
it is moving as part of a pair.

00:03:24.273 --> 00:03:26.367
Another way to think of
it is this electron is

00:03:26.367 --> 00:03:28.900
going to be on the other side of the bond.

00:03:28.900 --> 00:03:30.966
I also want to be clear again.

00:03:30.966 --> 00:03:33.766
When I talk about electrons
on either side of bonds,

00:03:33.766 --> 00:03:35.233
I like to think about that because it

00:03:35.233 --> 00:03:38.033
helps me do it for accounting purposes.

00:03:38.033 --> 00:03:41.100
We know that these covalent
bonds, this one electron just

00:03:41.100 --> 00:03:42.900
doesn't sit on one side of a
bond and the other electron

00:03:42.900 --> 00:03:45.700
doesn't just sit on the
other side of the bond.

00:03:45.700 --> 00:03:47.300
In fact everything we do
in organic chemistry isn't

00:03:47.300 --> 00:03:50.633
anywhere near as clean as
the way we draw it, but I do

00:03:50.633 --> 00:03:54.700
this to remind myself that
there are two electrons here,

00:03:54.700 --> 00:03:57.366
and when you have a bond
there is some probability

00:03:57.366 --> 00:03:59.566
that one of the electrons
is closer to the hydrogen

00:03:59.566 --> 00:04:01.966
and there's some probability
that that electron is

00:04:01.966 --> 00:04:04.033
closer to the carbon, and
so you can kind of imagine

00:04:04.033 --> 00:04:07.700
that there are electrons on
either sides of the bond.

00:04:07.700 --> 00:04:09.566
The actual reality is that
there's a blur over them and

00:04:09.566 --> 00:04:11.433
depending on which molecule
is more electronegative

00:04:11.433 --> 00:04:14.767
the probability blur is a
little bit more weighted on

00:04:14.767 --> 00:04:17.833
one side or another, but
of course we like to clean

00:04:17.833 --> 00:04:20.733
things up with these
formalisms right over here.

00:04:20.767 --> 00:04:23.566
This is kind of the example
when you have this attacking

00:04:23.566 --> 00:04:26.166
pair, why I like to
think of the full arrow

00:04:26.166 --> 00:04:28.933
as the movement of an
electron as part of a pair.

00:04:28.967 --> 00:04:32.233
And this breaking bond over
here is another example.

00:04:32.233 --> 00:04:34.766
In the typical convention
you have this bond here.

00:04:34.766 --> 00:04:37.100
Remember a bond is made up
of two, this covalent bond

00:04:37.100 --> 00:04:39.433
right over here is made
up of two electrons.

00:04:39.433 --> 00:04:42.233
If they wanted to show this
bond breaking and both of

00:04:42.233 --> 00:04:45.766
these electrons going to
this bromine, the convention

00:04:45.766 --> 00:04:49.966
is to go from the middle
of the bond to the bromine.

00:04:49.966 --> 00:04:53.766
That I've never found that
intuitive because here,

00:04:53.766 --> 00:04:57.900
once again, bromine already
essentially had part of

00:04:57.900 --> 00:05:00.233
the bond, it was already
on one end of the bond.

00:05:00.233 --> 00:05:02.700
I like to visualize that it's
getting the other electron

00:05:02.700 --> 00:05:07.001
that it wasn't, it's now
getting both electrons.

00:05:07.033 --> 00:05:10.766
One part of the bond was
already closer to the bromine,

00:05:10.766 --> 00:05:14.800
now it's getting the other,
it's the other part of the bond.

00:05:14.900 --> 00:05:17.233
We're going to use full
arrows for these mechanisms,

00:05:17.233 --> 00:05:19.366
just as we would typically use
full arrows, but I'll often

00:05:19.366 --> 00:05:22.900
conceptualize it as the movement
of an electron as part of

00:05:22.900 --> 00:05:27.166
a pair, as opposed to the
entire pair, but the full arrows

00:05:27.166 --> 00:05:29.933
are still used the way it
would be conventionally used.

00:05:30.033 --> 00:05:33.833
Later on when we do free
radical reactions we're

00:05:33.833 --> 00:05:36.700
going to talk about an
electron moving by itself.

00:05:36.700 --> 00:05:39.833
Notice this electron right
over here, it's moving or

00:05:39.833 --> 00:05:44.053
it's doing something and
it's not part of a pair,

00:05:44.053 --> 00:05:46.500
it's by itself so we use
the fish hook arrows.

00:05:46.500 --> 00:05:49.100
Right over here we see a
bond breaking but instead

00:05:49.100 --> 00:05:52.967
of both electrons going
to one of the atoms or

00:05:52.967 --> 00:05:54.767
another one of the atoms,
as right over here.

00:05:54.767 --> 00:05:57.433
When both electrons went to one
of the atoms we use the full

00:05:57.433 --> 00:06:00.346
arrow, this already you can say
had one and now it's gaining

00:06:00.346 --> 00:06:03.433
another one so use the full
arrow, but here the bond is

00:06:03.433 --> 00:06:07.833
breaking and each electron
is going to a different atom.

00:06:07.833 --> 00:06:10.509
Once again the electron is moving,

00:06:10.509 --> 00:06:12.300
the electron is moving by itself.

00:06:12.300 --> 00:06:15.767
Maybe I'll put this
right, moving by itself,

00:06:15.767 --> 00:06:19.566
and here is a movement of the
electron as part of a pair.

00:06:19.566 --> 00:06:21.700
Hopefully that clarifies it a little bit.

00:06:21.700 --> 00:06:24.100
It's important to keep in
mind a lot of the notation

00:06:24.100 --> 00:06:27.100
I use is a departure from
the traditional organic

00:06:27.100 --> 00:06:30.033
chemistry notation, but I
think at least in my mind it's

00:06:30.033 --> 00:06:32.300
helped me build more of an
intuition of what's going

00:06:32.300 --> 00:06:35.700
on in the mechanisms and
account for the electrons.

00:06:35.700 --> 00:06:40.966
If you're in a course, and
especially depending on how it's

00:06:40.966 --> 00:06:44.566
graded, you might want to
stick to whatever the professor

00:06:44.566 --> 00:06:46.966
uses, which is probably going
to be a little bit closer to

00:06:46.966 --> 00:06:51.966
the using the full arrow as
the whole pair, and going from

00:06:51.966 --> 00:06:54.767
the middle of the bonds, the
middle of the pairs, as opposed

00:06:54.767 --> 00:06:57.734
from one of the electrons
moving as part of the pair.


Evaluating expressions with and without parentheses

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:03.700
What I want to do is think about
whether this expression right

00:00:03.700 --> 00:00:06.520
over here would evaluate
the same way whether or not

00:00:06.520 --> 00:00:08.240
we had parentheses.

00:00:08.240 --> 00:00:09.770
So to think about
that, let's first

00:00:09.770 --> 00:00:11.290
think about how
it would evaluate

00:00:11.290 --> 00:00:13.119
if we add the parentheses.

00:00:13.119 --> 00:00:14.660
So if we add the
parentheses, we want

00:00:14.660 --> 00:00:16.460
to do what's ever in
the parentheses first.

00:00:16.460 --> 00:00:20.140
And so here we have 8 minus
3, which is equal to 5.

00:00:20.140 --> 00:00:28.360
So this simplifies to 5
times 5 times 8 minus 3.

00:00:28.360 --> 00:00:30.140
And now we want to
do the multiplication

00:00:30.140 --> 00:00:31.270
before we do subtraction.

00:00:31.270 --> 00:00:32.900
This goes back to
order of operations.

00:00:32.900 --> 00:00:35.140
You do your multiplication
and division first.

00:00:35.140 --> 00:00:37.160
Well, you do your
parentheses first.

00:00:37.160 --> 00:00:39.760
Then if you have multiplication,
division, addition,

00:00:39.760 --> 00:00:41.730
and subtraction
all in a row, you

00:00:41.730 --> 00:00:45.390
want to do your multiplication
and your division first.

00:00:45.390 --> 00:00:49.520
So here we're going to
multiply 5 times 8 to get 40,

00:00:49.520 --> 00:00:53.920
and then we're going to
subtract 3 to get 37.

00:00:53.920 --> 00:00:57.430
Now, let's think about
what this would evaluate

00:00:57.430 --> 00:01:00.350
to if we did not
have the parentheses.

00:01:00.350 --> 00:01:06.330
So it would be 8 minus
3 times 8 minus 3.

00:01:06.330 --> 00:01:08.260
So we just have to
remind ourselves

00:01:08.260 --> 00:01:09.770
about the order of operations.

00:01:09.770 --> 00:01:13.350
The convention is to do
your multiplication first.

00:01:13.350 --> 00:01:17.050
So you're actually going to
multiply the 3 times the 8

00:01:17.050 --> 00:01:19.780
before you subtract
it from this 8

00:01:19.780 --> 00:01:22.490
and then before you
subtract this 3.

00:01:22.490 --> 00:01:25.260
So we took away the parentheses,
but the order of operations

00:01:25.260 --> 00:01:26.900
say, hey, do this
multiplication first.

00:01:26.900 --> 00:01:29.760
We could even put a parentheses
here to emphasize that.

00:01:29.760 --> 00:01:36.279
So this will become
8 minus 8 minus 24.

00:01:36.279 --> 00:01:37.320
Let me write it this way.

00:01:37.320 --> 00:01:44.920
8 minus 24 minus 3.

00:01:44.920 --> 00:01:48.720
8 minus 24 minus 3.

00:01:48.720 --> 00:01:52.520
Now, 8 minus 24 is negative 16.

00:01:52.520 --> 00:01:57.010
You subtract another 3, you're
going to get to negative 19.

00:01:57.010 --> 00:02:01.380
So clearly, you get very,
very different values

00:02:01.380 --> 00:02:05.142
depending on whether or
not you have parentheses.


How to convert hours to minutes and minutes to seconds

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:02.317
We have multiple
lengths of time here,

00:00:02.317 --> 00:00:03.650
and they're measured in minutes.

00:00:03.650 --> 00:00:05.108
So what we want to
do is say, well,

00:00:05.108 --> 00:00:07.020
what's an equivalent
number of seconds?

00:00:07.020 --> 00:00:09.000
And so the first one is
pretty straightforward.

00:00:09.000 --> 00:00:12.570
1 minute is equal to 60 seconds.

00:00:12.570 --> 00:00:16.910
By definition, a second
is 1/60 of a minute.

00:00:16.910 --> 00:00:18.720
But what about 5 minutes?

00:00:18.720 --> 00:00:20.380
Well, we could try
to visualize that.

00:00:20.380 --> 00:00:24.050
If 1 minute-- so let's say
that is 1 minute-- if that

00:00:24.050 --> 00:00:28.150
is equal to 60 seconds,
then 5 minutes--

00:00:28.150 --> 00:00:32.270
let's just make
five copies of this.

00:00:32.270 --> 00:00:42.950
So that's 1, 2, minutes,
3, minutes, 4 minutes,

00:00:42.950 --> 00:00:45.590
and then 5 minutes.

00:00:45.590 --> 00:00:50.330
So you see that 5 minutes
would be 5 times 60 seconds.

00:00:50.330 --> 00:00:53.650
And 5 times 60 is 300.

00:00:53.650 --> 00:00:55.880
So here you have
so this is going

00:00:55.880 --> 00:01:02.610
to be 5 times 60, which
is equal to 300 seconds.

00:01:02.610 --> 00:01:04.535
So what about 10 minutes?

00:01:04.535 --> 00:01:05.960
Well, same idea.

00:01:05.960 --> 00:01:10.480
10 times 60 is 600 seconds.

00:01:10.480 --> 00:01:12.470
And what about 60 minutes?

00:01:12.470 --> 00:01:18.580
Well, 60 minutes or 60 seconds
per minute times 60 minutes

00:01:18.580 --> 00:01:22.254
is equal to 3,600 seconds.

00:01:22.254 --> 00:01:23.170
So that's interesting.

00:01:23.170 --> 00:01:25.044
Let's take things a
little bit further other.

00:01:25.044 --> 00:01:34.430
Let's convert between
hours, minutes, and seconds.

00:01:34.430 --> 00:01:38.570
So let's start
off with something

00:01:38.570 --> 00:01:40.870
that's a little bit
more straightforward.

00:01:40.870 --> 00:01:43.720
Let's say that we have 1 hour.

00:01:43.720 --> 00:01:49.800
Well, we know by definition 1
hour is equal to 60 minutes.

00:01:49.800 --> 00:01:53.060
And how many seconds
is 60 minutes equal to?

00:01:53.060 --> 00:01:54.650
Well, we just figured that out.

00:01:54.650 --> 00:01:57.470
60 minutes times 60
seconds per minute

00:01:57.470 --> 00:02:04.020
gets us to 3,600 seconds.

00:02:04.020 --> 00:02:05.160
Now, what about 2 hours?

00:02:07.870 --> 00:02:10.050
Well, it would be
the same exact idea.

00:02:10.050 --> 00:02:12.270
60 minutes per hour,
if I have 2 hours,

00:02:12.270 --> 00:02:14.686
it's going to be 2 times 60.

00:02:14.686 --> 00:02:19.870
2 hours times 60 minutes
per hour gets us to 120.

00:02:19.870 --> 00:02:22.287
Well, how many seconds
is that going to be?

00:02:22.287 --> 00:02:24.620
Well, there's a couple of
ways you could think about it.

00:02:24.620 --> 00:02:29.390
You could take 120 minutes
times 60 seconds per minute.

00:02:29.390 --> 00:02:35.310
Or you could say 2 hours
times 3,600 seconds per hour.

00:02:35.310 --> 00:02:36.820
So let's do it that way.

00:02:36.820 --> 00:02:46.660
So two 2 times 3,600 seconds per
hour gets us to 7,200 seconds.

00:02:46.660 --> 00:02:49.880
Now, what if we were
to say 10 hours?

00:02:49.880 --> 00:02:53.600
Well, once again, 10
times 60, 10 hours

00:02:53.600 --> 00:02:57.390
times 60 minutes per hour
would get us to 600 minutes.

00:02:57.390 --> 00:03:05.170
And if we want to think
in terms of seconds,

00:03:05.170 --> 00:03:10.240
10 hours times 3,600 seconds
per hour gets us to-- well,

00:03:10.240 --> 00:03:12.190
it's going to be
3,600 tens, which

00:03:12.190 --> 00:03:19.600
would be 36,000, or 10 times
36 hundred, or 10 times 3,600.


4 Solving for an angle

WEBVTT
Kind: captions
Language: en

00:00:00.371 --> 00:00:01.776
- [Voiceover] In the figure above,

00:00:01.776 --> 00:00:06.776
lines l and k intersect at point Q.

00:00:06.921 --> 00:00:11.183
If m is equal to 40, so
this right over here,

00:00:11.183 --> 00:00:13.482
that right over there is 40 degrees

00:00:13.482 --> 00:00:14.562
because it says m degrees.

00:00:14.562 --> 00:00:18.451
So if m is equal to 40
and p is equal to 25,

00:00:18.451 --> 00:00:22.746
so p, this angle right
over here is 25 degrees.

00:00:22.746 --> 00:00:25.439
What is the value of x?

00:00:25.439 --> 00:00:27.123
I encourage you to pause the video

00:00:27.123 --> 00:00:29.363
and think about this on your own.

00:00:29.363 --> 00:00:32.183
So the big realization right over here

00:00:32.183 --> 00:00:35.608
is to realize that this angle

00:00:35.608 --> 00:00:39.568
and this entire angle are vertical angles.

00:00:39.568 --> 00:00:41.239
They're on the opposite sides

00:00:41.239 --> 00:00:43.584
of this point of interesection at point Q.

00:00:43.584 --> 00:00:45.163
So vertical angles are always

00:00:45.163 --> 00:00:47.380
going to have the same measure.

00:00:47.380 --> 00:00:48.843
So if this is 40 degrees,

00:00:48.843 --> 00:00:52.488
then this whole thing is going to be,

00:00:52.488 --> 00:00:55.158
that whole thing is
going to be 40 degrees.

00:00:55.158 --> 00:00:57.585
And if that whole thing is 40 degrees,

00:00:57.585 --> 00:01:02.145
then x plus 25 is going
to need to be equal

00:01:02.145 --> 00:01:03.424
to the whole 40 degrees.

00:01:03.424 --> 00:01:04.897
So let's write that down.

00:01:04.897 --> 00:01:08.197
So x plus 25

00:01:10.005 --> 00:01:13.222
is going to be equal to 40.

00:01:13.222 --> 00:01:16.080
Subtract 25 from both sides.

00:01:18.791 --> 00:01:23.791
And we are left with x is equal to 15.

00:01:26.710 --> 00:01:30.053
So x, this angle right
over here is 15 degrees

00:01:30.053 --> 00:01:32.398
or x is equal to 15.

00:01:32.398 --> 00:01:34.174
15 plus 25 is 40,

00:01:34.174 --> 00:01:38.147
it's a vertical angle, it
has the exact same measure

00:01:38.147 --> 00:01:40.994
as this angle right over here, 40 degrees.


Constructing quadrilateral based on symmetry

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:04.160
Two of the points that define
a certain quadrilateral

00:00:04.160 --> 00:00:07.250
are 0 comma 9 and 3 comma 4.

00:00:07.250 --> 00:00:10.960
The quadrilateral is left
unchanged by a reflection

00:00:10.960 --> 00:00:15.400
over the line y is
equal to 3 minus x.

00:00:15.400 --> 00:00:18.124
Draw and classify
the quadrilateral.

00:00:18.124 --> 00:00:19.790
Now, I encourage you
to pause this video

00:00:19.790 --> 00:00:21.760
and try to draw and
classify it on your own

00:00:21.760 --> 00:00:24.230
before I'm about to explain it.

00:00:24.230 --> 00:00:26.880
So let's at least plot the
information they give us.

00:00:26.880 --> 00:00:32.400
So the point 0
comma 9, that's one

00:00:32.400 --> 00:00:35.160
of the vertices of
the quadrilateral.

00:00:35.160 --> 00:00:37.360
So 0 comma 9.

00:00:37.360 --> 00:00:39.450
That's that point
right over there.

00:00:39.450 --> 00:00:46.290
And another one of the
vertices is 3 comma 4.

00:00:46.290 --> 00:00:47.850
That's that right over there.

00:00:47.850 --> 00:00:50.020
And then they tell us
that the quadrilateral

00:00:50.020 --> 00:00:53.200
is left unchanged by reflection
over the line y is equal to

00:00:53.200 --> 00:00:55.000
3 minus x.

00:00:55.000 --> 00:00:58.890
So when x is 0, y is 3--
that's our y-intercept--

00:00:58.890 --> 00:01:00.810
and it has a slope
of negative 1.

00:01:00.810 --> 00:01:03.290
You could view
this as 3 minus 1x.

00:01:03.290 --> 00:01:04.640
So it has a slope of negative 1.

00:01:04.640 --> 00:01:06.690
So the line looks like this.

00:01:06.690 --> 00:01:11.250
So every time we increase our
x by 1, we decrease our y by 1.

00:01:11.250 --> 00:01:16.260
So the line looks something like
this. y is equal to 3 minus x.

00:01:16.260 --> 00:01:21.180
Try to draw it relatively,
pretty carefully.

00:01:21.180 --> 00:01:23.210
So that's what it looks like.

00:01:23.210 --> 00:01:25.320
y is equal to 3 minus x.

00:01:25.320 --> 00:01:28.240
So that's my best attempt
at drawing it. y is equal to

00:01:28.240 --> 00:01:30.430
3 minus x.

00:01:30.430 --> 00:01:33.650
So the quadrilateral is
left unchanged by reflection

00:01:33.650 --> 00:01:34.490
over this.

00:01:34.490 --> 00:01:36.999
So that means if I were to
reflect each of these vertices,

00:01:36.999 --> 00:01:38.540
I would, essentially,
end up with one

00:01:38.540 --> 00:01:40.415
of the other vertices
on it, and if those get

00:01:40.415 --> 00:01:42.498
reflected you're going to
end up with one of these

00:01:42.498 --> 00:01:44.600
so the thing is not
going to be different.

00:01:44.600 --> 00:01:46.660
So let's think about
where these other two

00:01:46.660 --> 00:01:49.670
vertices of this
quadrilateral need to be.

00:01:49.670 --> 00:01:53.420
So this point, let's just
reflect it over this line,

00:01:53.420 --> 00:01:56.150
over y is equal to 3 minus x.

00:01:56.150 --> 00:02:00.090
So if we were to try to drop
a perpendicular to this line--

00:02:00.090 --> 00:02:02.860
notice, we have gone
diagonally across one,

00:02:02.860 --> 00:02:04.830
two, three of
these squares so we

00:02:04.830 --> 00:02:06.580
need to go diagonally
across three of them

00:02:06.580 --> 00:02:08.039
on the left-hand side.

00:02:08.039 --> 00:02:13.830
So one, two, three gets
us right over there.

00:02:13.830 --> 00:02:17.010
This is the reflection of
this point across that line.

00:02:17.010 --> 00:02:19.970
Now, let's do the same
thing for this blue point.

00:02:19.970 --> 00:02:23.560
To drop a perpendicular
to this line,

00:02:23.560 --> 00:02:26.410
we have to go diagonally
across two of these squares.

00:02:26.410 --> 00:02:29.690
So let's go diagonally across
two more of these squares just

00:02:29.690 --> 00:02:31.740
like that to get to that
point right over there.

00:02:31.740 --> 00:02:34.370
And now we've defined
our quadrilateral.

00:02:34.370 --> 00:02:35.970
Our quadrilateral
looks like this.

00:02:41.310 --> 00:02:42.940
Both of these lines
are perpendicular

00:02:42.940 --> 00:02:44.600
to that original
line, so they're

00:02:44.600 --> 00:02:46.210
going to have the same slope.

00:02:46.210 --> 00:02:49.090
So that line is parallel
to that line over there.

00:02:49.090 --> 00:02:53.850
And then we have this line
and then we have this line.

00:02:53.850 --> 00:02:55.940
So what type of
quadrilateral is this?

00:02:55.940 --> 00:03:01.260
Well, I have one pair
of parallel sides,

00:03:01.260 --> 00:03:05.200
so this is a trapezoid.


Percentage of a whole number

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:07.280
Let's see if we can figure
out what 30% of 6 is.

00:00:07.280 --> 00:00:09.730
So one way of thinking
about 30%-- this

00:00:09.730 --> 00:00:11.740
literally means 30 per 100.

00:00:11.740 --> 00:00:17.620
So you could view
this as 30/100 times 6

00:00:17.620 --> 00:00:19.700
is the same thing as 30% of 6.

00:00:19.700 --> 00:00:23.320
Or you could view this
as 30 hundredths times 6,

00:00:23.320 --> 00:00:27.115
so 0.30 times 6.

00:00:27.115 --> 00:00:28.490
Now we could solve
both of these,

00:00:28.490 --> 00:00:30.500
and you'll see that we'll
get the same answer.

00:00:30.500 --> 00:00:32.500
If you do this multiplication
right over here,

00:00:32.500 --> 00:00:35.190
30/100-- and you
could view this times

00:00:35.190 --> 00:00:42.027
6/1-- this is equal to 180/100.

00:00:42.027 --> 00:00:42.610
And let's see.

00:00:42.610 --> 00:00:43.520
We can simplify.

00:00:43.520 --> 00:00:46.097
We can divide the numerator
and the denominator by 10.

00:00:46.097 --> 00:00:48.430
And then we can divide the
numerator and the denominator

00:00:48.430 --> 00:00:49.220
by 2.

00:00:49.220 --> 00:00:55.780
And we will get 9/5, which is
the same thing as 1 and 4/5.

00:00:55.780 --> 00:01:00.800
And then if we wanted to write
this as a decimal, 4/5 is 0.8.

00:01:00.800 --> 00:01:02.340
And if you want to
verify that, you

00:01:02.340 --> 00:01:05.304
could verify that
5 goes into 4--

00:01:05.304 --> 00:01:06.720
and there's going
to be a decimal.

00:01:06.720 --> 00:01:08.570
So let's throw some
decimals in there.

00:01:08.570 --> 00:01:10.170
It goes into 4 zero times.

00:01:10.170 --> 00:01:11.711
So we don't have to
worry about that.

00:01:11.711 --> 00:01:13.640
It goes into 40 eight times.

00:01:13.640 --> 00:01:15.590
8 times 5 is 40.

00:01:15.590 --> 00:01:16.270
Subtract.

00:01:16.270 --> 00:01:19.090
You have no remainder, and
you just have 0's left here.

00:01:19.090 --> 00:01:21.192
So 4/5 is 0.8.

00:01:21.192 --> 00:01:22.150
You've got the 1 there.

00:01:22.150 --> 00:01:24.270
This is the same
thing as 1.8, which

00:01:24.270 --> 00:01:26.220
you would have gotten
if you divided 5 into 9.

00:01:26.220 --> 00:01:27.900
You would've gotten 1.8.

00:01:27.900 --> 00:01:31.030
So 30% of 6 is equal to 1.8.

00:01:31.030 --> 00:01:33.290
And we can verify it
doing this way as well.

00:01:33.290 --> 00:01:37.880
So if we were to multiply
0.30 times 6-- let's do that.

00:01:37.880 --> 00:01:43.680
And I could just write that
literally as 0.3 times 6.

00:01:43.680 --> 00:01:45.980
Well, 3 times 6 is 18.

00:01:45.980 --> 00:01:48.285
I have only one digit
behind the decimal

00:01:48.285 --> 00:01:50.410
amongst both of these
numbers that I'm multiplying.

00:01:50.410 --> 00:01:52.410
I only have the 3 to the
right of the decimal.

00:01:52.410 --> 00:01:53.910
So I'm only going
to have one number

00:01:53.910 --> 00:01:55.350
to the right of
the decimal here.

00:01:55.350 --> 00:01:56.890
So I just count one number.

00:01:56.890 --> 00:01:58.700
It's going to be 1.8.

00:01:58.700 --> 00:02:02.910
So either way you think about
it or calculate it, 30% of 6

00:02:02.910 --> 00:02:05.280
is 1.8.


Constructing square inscribed in circle

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.870
Construct a square
inscribed inside the circle.

00:00:04.870 --> 00:00:06.550
And in order to do
this, we just have

00:00:06.550 --> 00:00:08.863
to remember that a square,
what we know of a square

00:00:08.863 --> 00:00:11.120
is all four sides are
congruent and they

00:00:11.120 --> 00:00:12.870
intersect at right angles.

00:00:12.870 --> 00:00:15.830
And we also have to
remember that the two

00:00:15.830 --> 00:00:18.470
diagonals of the
square are going

00:00:18.470 --> 00:00:21.840
to be perpendicular
bisectors of each other.

00:00:21.840 --> 00:00:24.040
So let's see if we can
construct two lines that

00:00:24.040 --> 00:00:26.570
are perpendicular
bisectors of each other.

00:00:26.570 --> 00:00:29.940
And essentially, where those
two lines intersect our bigger

00:00:29.940 --> 00:00:33.730
circle, those are going to be
the vertices of our square.

00:00:33.730 --> 00:00:37.430
So let's throw a straight
edge right over here.

00:00:37.430 --> 00:00:39.340
And let's make a diameter.

00:00:39.340 --> 00:00:43.145
So that's a diameter
right over here.

00:00:43.145 --> 00:00:45.006
It just goes through
the circle, goes

00:00:45.006 --> 00:00:46.380
through the center
of the circle,

00:00:46.380 --> 00:00:48.021
to two sides of the circle.

00:00:48.021 --> 00:00:49.395
And now, let's
think about how we

00:00:49.395 --> 00:00:52.290
can construct a perpendicular
bisector of this.

00:00:52.290 --> 00:00:54.860
And we've done this in
other compass construction

00:00:54.860 --> 00:00:56.550
or construction videos.

00:00:56.550 --> 00:00:59.650
But what we can do is we
can put a circle-- let's

00:00:59.650 --> 00:01:01.340
throw a circle right over here.

00:01:01.340 --> 00:01:04.297
We've got to make its radius
bigger than the center.

00:01:04.297 --> 00:01:06.630
And what we're going to do
is we're going to reuse this.

00:01:06.630 --> 00:01:07.680
We're going to make
another circle that's

00:01:07.680 --> 00:01:08.680
the exact same size.

00:01:08.680 --> 00:01:09.610
Put it there.

00:01:09.610 --> 00:01:12.700
And where they intersect
is going to be exactly

00:01:12.700 --> 00:01:14.450
along-- those two
points of intersection

00:01:14.450 --> 00:01:17.310
are going to be along a
perpendicular bisector.

00:01:17.310 --> 00:01:18.410
So that's one of them.

00:01:18.410 --> 00:01:19.650
Let's do another one.

00:01:19.650 --> 00:01:22.640
I want a circle of the
exact same dimensions.

00:01:22.640 --> 00:01:24.800
So I'll center it
at the same place.

00:01:24.800 --> 00:01:26.090
I'll drag it out there.

00:01:26.090 --> 00:01:27.590
That looks pretty good.

00:01:27.590 --> 00:01:31.890
I'll move it on to this side,
the other side of my diameter.

00:01:31.890 --> 00:01:33.490
So that looks pretty good.

00:01:33.490 --> 00:01:36.050
And notice, if I connect
that point to that point,

00:01:36.050 --> 00:01:38.470
I will have constructed
a perpendicular bisector

00:01:38.470 --> 00:01:40.520
of this original segment.

00:01:40.520 --> 00:01:41.380
So let's do that.

00:01:41.380 --> 00:01:43.150
Let's connect those two points.

00:01:43.150 --> 00:01:45.390
So that point and that point.

00:01:45.390 --> 00:01:49.150
And then, we could just
keep going all the way

00:01:49.150 --> 00:01:51.580
to the end of the circle.

00:01:51.580 --> 00:01:56.730
Go all the way over there.

00:01:56.730 --> 00:01:59.160
That looks pretty good.

00:01:59.160 --> 00:02:01.970
And now, we just have
to connect these four

00:02:01.970 --> 00:02:04.800
points to have a square.

00:02:04.800 --> 00:02:06.020
So let's do that.

00:02:06.020 --> 00:02:11.640
So I'll connect
to that and that.

00:02:11.640 --> 00:02:14.940
And then I will connect, throw
another straight edge there.

00:02:14.940 --> 00:02:20.650
I will connect that with that.

00:02:20.650 --> 00:02:23.530
And then, two more to go.

00:02:23.530 --> 00:02:29.980
I'll connect this with
that, and then one more.

00:02:29.980 --> 00:02:35.470
I can connect this with
that, and there you go.

00:02:35.470 --> 00:02:40.240
I have a shape whose vertices
intersect the circle.

00:02:40.240 --> 00:02:44.060
And its diagonals, this diagonal
and this diagonal, these

00:02:44.060 --> 00:02:46.335
are perpendicular bisectors.


What is capital?

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.460
The title of Thomas Piketty's book

00:00:02.460 --> 00:00:04.890
is Capital in the 21st Century.

00:00:04.890 --> 00:00:07.330
It's probably worth having a conversation

00:00:07.330 --> 00:00:09.810
about what capital is.

00:00:09.810 --> 00:00:13.620
If you're going to produce anything,

00:00:13.620 --> 00:00:15.000
you need some input,

00:00:15.000 --> 00:00:18.480
you need some factors for that production.

00:00:18.480 --> 00:00:19.750
You'd put them together.

00:00:19.750 --> 00:00:24.800
Let's say that you are a farm.

00:00:24.800 --> 00:00:27.000
Your output is food.

00:00:27.000 --> 00:00:28.690
What are your inputs going to be?

00:00:28.690 --> 00:00:30.660
This is a farm right over here.

00:00:30.660 --> 00:00:32.890
Let me a draw a little circle here.

00:00:32.890 --> 00:00:36.840
Farm and your output is food.

00:00:36.840 --> 00:00:39.000
What are your inputs going to be?

00:00:39.000 --> 00:00:42.800
You're going to need some land.

00:00:42.800 --> 00:00:47.000
You're going to need some water.

00:00:47.000 --> 00:00:56.420
You're going to need some equipment.

00:00:56.420 --> 00:01:00.000
And you are going to need some seeds

00:01:00.000 --> 00:01:07.470
and animals so, I'll just put generally here, supplies.

00:01:07.470 --> 00:01:09.490
And of course, you're going to need people to

00:01:09.490 --> 00:01:10.750
put all these things together

00:01:10.750 --> 00:01:14.000
and essentially work the land,

00:01:14.000 --> 00:01:16.630
to plow the soil and plant the seeds

00:01:16.630 --> 00:01:18.000
and harvest the crops

00:01:18.000 --> 00:01:20.480
and manage the whole operations.

00:01:20.480 --> 00:01:25.470
You're going to need labor as well.

00:01:25.470 --> 00:01:29.360
You take all of these factors of production together.

00:01:29.360 --> 00:01:31.610
They essentially represent a farm

00:01:31.610 --> 00:01:34.000
that is going to output food.

00:01:34.000 --> 00:01:36.000
When people talk about capital

00:01:36.000 --> 00:01:38.000
as a factor of production,

00:01:38.000 --> 00:01:40.000
sometimes they talk about it in a fairly narrow way.

00:01:40.000 --> 00:01:42.000
They'll separate things like land and resources

00:01:42.000 --> 00:01:45.000
or something separate and they'll say capital

00:01:45.000 --> 00:01:48.000
is hey, that's your equipment and your supplies.

00:01:48.000 --> 00:01:50.000
When people talk in more general terms,

00:01:50.000 --> 00:01:51.410
especially, in the focus of this book

00:01:51.410 --> 00:01:55.020
when there are people talking about labor versus capital.

00:01:55.020 --> 00:01:57.000
Capital when we just group things

00:01:57.000 --> 00:01:59.200
in only those two categories

00:01:59.200 --> 00:02:03.140
in that context, capital would essentially represent

00:02:03.140 --> 00:02:05.000
pretty much everything else.

00:02:05.000 --> 00:02:08.000
This right over here would be capital

00:02:08.000 --> 00:02:09.820
and one way to think about capital

00:02:09.820 --> 00:02:12.490
is the things that you could maybe you,

00:02:12.490 --> 00:02:14.810
they are assets that you have

00:02:14.810 --> 00:02:18.390
that can be valued that will give you future benefit.

00:02:18.390 --> 00:02:20.790
You can buy and sell those assets

00:02:20.790 --> 00:02:22.450
and we could think about other businesses here.

00:02:22.450 --> 00:02:25.500
Other things that are trying to produce something.

00:02:25.500 --> 00:02:35.880
Let's say you have a mine.

00:02:35.880 --> 00:02:37.580
This is the operation

00:02:37.580 --> 00:02:39.000
and it's going to produce,

00:02:39.000 --> 00:02:40.590
let's just say it's a goldmine.

00:02:40.590 --> 00:02:43.000
It's going to produce gold.

00:02:43.000 --> 00:02:45.520
What are the inputs that you would need there?

00:02:45.520 --> 00:02:47.460
You're clearly gonna need some labor.

00:02:47.460 --> 00:02:49.000
You're gonna need the miners and the people

00:02:49.000 --> 00:02:52.560
who would manage the operation of the goldmine

00:02:52.560 --> 00:02:54.950
but that's not all you need.

00:02:54.950 --> 00:02:58.760
You also need energy

00:02:58.760 --> 00:03:01.530
maybe to operate your equipment

00:03:01.530 --> 00:03:03.490
and I actually, I will put energy in here too.

00:03:03.490 --> 00:03:05.000
Some of which comes from the sun

00:03:05.000 --> 00:03:08.000
but some of which you might have to purchase

00:03:08.000 --> 00:03:09.740
or have somehow

00:03:09.740 --> 00:03:15.000
and then you would also need equipment.

00:03:15.000 --> 00:03:21.000
And you would need supplies.

00:03:21.000 --> 00:03:23.000
I mean we could consider energy as supply

00:03:23.000 --> 00:03:24.650
if you like or you use supply of energy

00:03:24.650 --> 00:03:28.000
but you also might need tires or something like that.

00:03:28.000 --> 00:03:31.690
You might need food in order to

00:03:31.690 --> 00:03:33.000
provide for the labor.

00:03:33.000 --> 00:03:35.720
Who knows what else you might need in your mine?

00:03:35.720 --> 00:03:38.000
And you also would need, of course, land.

00:03:38.000 --> 00:03:40.330
You need access to land to actually mine.

00:03:40.330 --> 00:03:42.500
Here, if we're thinking very broad terms,

00:03:42.500 --> 00:03:50.350
you could view all of this as the capital

00:03:50.350 --> 00:03:51.640
and of course, you have your labor

00:03:51.640 --> 00:03:53.350
and you put your labor and your capital together

00:03:53.350 --> 00:03:56.770
and you produce the gold.

00:03:56.770 --> 00:04:01.000
Another idea that you will hear in life a lot

00:04:01.000 --> 00:04:02.830
but obviously, in a book about capital,

00:04:02.830 --> 00:04:04.000
this will come up a lot,

00:04:04.000 --> 00:04:13.640
it's the idea of return on capital.

00:04:13.640 --> 00:04:17.000
This is just a measure of giving a value of capital

00:04:17.000 --> 00:04:18.000
that you've employed,

00:04:18.000 --> 00:04:21.590
how much income are you getting for that capital.

00:04:21.590 --> 00:04:27.000
For example, let's say this farm.

00:04:27.000 --> 00:04:29.420
The total value of this capital,

00:04:29.420 --> 00:04:31.000
let's just say it is,

00:04:31.000 --> 00:04:33.000
I don't know, I'll make up a number here.

00:04:33.000 --> 00:04:37.960
Let's just say it is $1 million.

00:04:37.960 --> 00:04:41.000
$1 million is the total value of the land

00:04:41.000 --> 00:04:43.000
and maybe you have access to your own lake

00:04:43.000 --> 00:04:45.000
and the equipment and the supplies.

00:04:45.000 --> 00:04:49.000
It's $1 million of capital.

00:04:49.000 --> 00:04:52.070
The value of that capital is $1 million.

00:04:52.070 --> 00:04:58.000
Let's say that the income of the farm

00:04:58.000 --> 00:05:00.250
after you pay the labor.

00:05:00.250 --> 00:05:03.400
Actually, let's just do this in a little broader term.

00:05:03.400 --> 00:05:05.000
Let's say the food's value,

00:05:05.000 --> 00:05:10.350
the food that's produced has a value of $100,000

00:05:10.350 --> 00:05:11.650
in the market.

00:05:11.650 --> 00:05:13.850
Now, out of that $100,000, you obviously had

00:05:13.850 --> 00:05:15.800
to pay your labor.

00:05:15.800 --> 00:05:22.680
Let's say that $50,000 of it, goes to the labor.

00:05:22.680 --> 00:05:25.000
And then we're saying this is in a given year.

00:05:25.000 --> 00:05:27.230
You use the capital of the labor,

00:05:27.230 --> 00:05:29.450
You produce $100,000 worth of food.

00:05:29.450 --> 00:05:32.000
$50,000 goes to the labor was gonna

00:05:32.000 --> 00:05:34.000
be returned on capital.

00:05:34.000 --> 00:05:37.780
You have another 50,000 leftover for the capital

00:05:37.780 --> 00:05:39.370
for the owners of the capital

00:05:39.370 --> 00:05:41.600
whoever owned the farm.

00:05:41.600 --> 00:05:45.860
The owners of the capital would get the other

00:05:45.860 --> 00:05:47.290
$50,000.

00:05:47.290 --> 00:05:51.570
Your return on capital is going to be $50,000

00:05:51.570 --> 00:05:53.470
that's what the owner of the capital gets

00:05:53.470 --> 00:05:55.770
and it's thee return on their investment

00:05:55.770 --> 00:06:01.600
of $1 million.

00:06:01.600 --> 00:06:04.000
And this is going to be the same thing

00:06:04.000 --> 00:06:07.360
as 5 divided by 100

00:06:07.360 --> 00:06:09.640
or it would be 5%.

00:06:09.640 --> 00:06:11.600
You have a 5% return on capital.

00:06:11.600 --> 00:06:14.760
You invest $1 million, you're going to get $50,000.

00:06:14.760 --> 00:06:17.000
It was a 6% return on capital,

00:06:17.000 --> 00:06:18.560
you invest $1 million.

00:06:18.560 --> 00:06:21.650
You get $60,000.

00:06:21.650 --> 00:06:24.000
Now, that we've thought about that a little bit,

00:06:24.000 --> 00:06:27.000
let's actually at a pretty neat chart

00:06:27.000 --> 00:06:29.800
from the book and once again, it's pretty neat.

00:06:29.800 --> 00:06:31.000
If you look right over here

00:06:31.000 --> 00:06:35.530
piketty.psc.ens.frcapital21c.

00:06:35.530 --> 00:06:37.000
He has all the charts from his book,

00:06:37.000 --> 00:06:41.710
which make for interesting analysis at minimum

00:06:41.710 --> 00:06:44.530
and he's gathered all of this information.

00:06:44.530 --> 00:06:45.500
This is pretty interesting.

00:06:45.500 --> 00:06:48.510
There's capital and slavery in the United States.

00:06:48.510 --> 00:06:50.000
And what's interesting about that is

00:06:50.000 --> 00:06:51.800
when you study American history,

00:06:51.800 --> 00:06:53.460
you talk a lot about slavery

00:06:53.460 --> 00:06:54.550
but you don't realize that

00:06:54.550 --> 00:06:58.000
in the time of slavery as abhorrent as it is

00:06:58.000 --> 00:07:02.000
and was, people viewed slaves as capital,

00:07:02.000 --> 00:07:03.000
not as laborer.

00:07:03.000 --> 00:07:05.850
They view them as something that they could buy or sell

00:07:05.850 --> 00:07:09.000
that they own and that would create future income

00:07:09.000 --> 00:07:11.250
for them and this just gives us a sense

00:07:11.250 --> 00:07:13.750
of kind of the breakdown of capital over history.

00:07:13.750 --> 00:07:16.430
You see down, at least in the United States,

00:07:16.430 --> 00:07:18.520
you see agricultural land.

00:07:18.520 --> 00:07:22.000
In 1770, it was a reasonable percentage.

00:07:22.000 --> 00:07:24.000
This is a value capital as a percentage

00:07:24.000 --> 00:07:26.000
of national income.

00:07:26.000 --> 00:07:28.000
The value of the land is a percentage

00:07:28.000 --> 00:07:31.920
of national income was much higher than it is, today

00:07:31.920 --> 00:07:32.860
and we see the other trend

00:07:32.860 --> 00:07:35.140
that the other domestic capital has become

00:07:35.140 --> 00:07:37.000
much more value.

00:07:37.000 --> 00:07:39.000
Well, what is this other domestic capital?

00:07:39.000 --> 00:07:42.970
This could be things like infrastructure.

00:07:42.970 --> 00:07:45.000
It could be technology.

00:07:45.000 --> 00:07:47.000
It could be mechanical technology.

00:07:47.000 --> 00:07:50.700
Things like trains and cars and trucks and buses

00:07:50.700 --> 00:07:55.000
or, and factories or it could be software.

00:07:55.000 --> 00:07:56.150
It could be computers,

00:07:56.150 --> 00:07:58.000
whatever else and as we see as we went through

00:07:58.000 --> 00:07:59.000
the industrial revolution,

00:07:59.000 --> 00:08:02.450
the value of other forms of technology became...

00:08:02.450 --> 00:08:04.480
or other forms of capital other than land

00:08:04.480 --> 00:08:06.750
became more and more important

00:08:06.750 --> 00:08:11.450
for our economy and obviously, slavery ended

00:08:11.450 --> 00:08:16.420
in the mid late 1860's and obviously, that went away

00:08:16.420 --> 00:08:18.460
but it's just an interesting way to think

00:08:18.460 --> 00:08:22.000
about macroeconomic trends in the United States.

00:08:22.000 --> 00:08:25.610
Land in 1770 was a major factor of production

00:08:25.610 --> 00:08:27.160
and still is obviously.

00:08:27.160 --> 00:08:29.440
We still need land in order to produce things,

00:08:29.440 --> 00:08:30.760
especially, agriculture.

00:08:30.760 --> 00:08:32.050
Well, this is agriculture

00:08:32.050 --> 00:08:34.600
and it's still a major part.

00:08:34.600 --> 00:08:36.000
It's not like the land has disappeared.

00:08:36.000 --> 00:08:38.000
It's just that the economy has grown

00:08:38.000 --> 00:08:41.000
to be much more than just agriculture.

00:08:41.000 --> 00:08:43.560
In 1770, agriculture was a big part

00:08:43.560 --> 00:08:44.550
of our economy.

00:08:44.550 --> 00:08:47.000
Now, agriculture is a much, much smaller

00:08:47.000 --> 00:08:49.230
part of our economy.

00:08:49.230 --> 00:08:51.000
This is just an interesting way

00:08:51.000 --> 00:08:54.000
to think about, well, okay, this is all of the capital

00:08:54.000 --> 00:08:57.000
and this both public and private capital included here.

00:08:57.000 --> 00:08:58.470
Public capital would be something

00:08:58.470 --> 00:09:01.380
that is owned by the public, by the government

00:09:01.380 --> 00:09:04.000
while private capital would be something

00:09:04.000 --> 00:09:06.660
that is owned by private individuals

00:09:06.660 --> 00:09:08.590
or corporations or whatever else

00:09:08.590 --> 00:09:11.000
but you see on average, you have had kind of a

00:09:11.000 --> 00:09:14.780
a fairly constant give or take a little bit

00:09:14.780 --> 00:09:17.530
that the total value of the capital

00:09:17.530 --> 00:09:20.000
has been around four to five times

00:09:20.000 --> 00:09:23.580
the value of the total productive output,

00:09:23.580 --> 00:09:25.310
four to five times the value

00:09:25.310 --> 00:09:29.500
of the total productive output of the country.


19 Function truth

WEBVTT
Kind: captions
Language: en

00:00:00.703 --> 00:00:03.041
- Let the function f be defined by

00:00:03.041 --> 00:00:06.302
f of x is equal to ax to the third plus b,

00:00:06.302 --> 00:00:08.773
where a and b are constants.

00:00:08.773 --> 00:00:11.638
If f of zero is equal to two,

00:00:11.638 --> 00:00:13.372
and f of two is equal to three,

00:00:13.372 --> 00:00:14.704
which of the following must be true?

00:00:14.704 --> 00:00:16.248
And it's a is greater than zero,

00:00:16.248 --> 00:00:17.310
b is greater than zero,

00:00:17.310 --> 00:00:19.913
a is less than b, so let's
look at each of these

00:00:19.913 --> 00:00:21.442
constraints right over here.

00:00:21.442 --> 00:00:26.442
So, we have f of x is equal
to ax to the third plus b.

00:00:27.369 --> 00:00:30.447
And they say f of zero, so
what is f of zero going to be?

00:00:30.447 --> 00:00:32.506
F of zero, well this
whole term is gonna be

00:00:32.506 --> 00:00:35.709
a times zero to the third, so
that's just going to be zero.

00:00:35.709 --> 00:00:38.312
So, f of zero's going to be equal to b,

00:00:38.312 --> 00:00:40.577
which they tell us is equal to two,

00:00:40.577 --> 00:00:43.645
so we know that b is equal to two.

00:00:43.645 --> 00:00:44.571
So, now we know that

00:00:44.571 --> 00:00:49.571
f of x is equal to ax
to the third plus two.

00:00:49.971 --> 00:00:52.701
Then they tell us f of two is three.

00:00:52.701 --> 00:00:53.776
Well, what is f of two?

00:00:53.776 --> 00:00:55.912
And I'll just switch colors
somewhat arbitrarily.

00:00:55.912 --> 00:00:58.181
F of two, now, is going to be

00:00:58.181 --> 00:01:02.833
a times two to the third power plus two.

00:01:02.833 --> 00:01:05.084
Well, this is going to be equal to

00:01:05.084 --> 00:01:09.033
eight a plus two, and they tell us,

00:01:09.033 --> 00:01:11.694
they tell us that that is equal to three.

00:01:11.694 --> 00:01:13.778
That is equal to three.

00:01:13.778 --> 00:01:17.776
So, we have eight a plus
two is equal to three.

00:01:17.776 --> 00:01:19.505
Now we can solve for a, actually,

00:01:19.505 --> 00:01:21.443
so let's subtract two from both sides

00:01:21.443 --> 00:01:24.714
and we get eight a is equal to,

00:01:24.714 --> 00:01:26.370
eight a is equal to one,

00:01:26.370 --> 00:01:28.039
divide both sides by eight,

00:01:28.039 --> 00:01:30.173
a is equal to one-eighth.

00:01:30.173 --> 00:01:31.767
So, which of these are true?

00:01:31.767 --> 00:01:34.846
A is greater than zero, that's true.

00:01:34.846 --> 00:01:39.366
B is greater than zero, that's true.

00:01:39.366 --> 00:01:41.969
A is less than b,
one-eighth is less than two,

00:01:41.969 --> 00:01:45.747
that's true as well,
so all three are true.


5 Equations that match constraints

WEBVTT
Kind: captions
Language: en

00:00:00.897 --> 00:00:04.849
- The number B is five less than twice

00:00:04.849 --> 00:00:06.089
the number C.

00:00:06.089 --> 00:00:06.962
Let me write that down.

00:00:06.962 --> 00:00:11.229
So B is equal to five less than

00:00:11.229 --> 00:00:12.590
twice the number C.

00:00:12.590 --> 00:00:16.267
So that's two times C minus five.

00:00:16.267 --> 00:00:19.209
It's five less than
two times the number C.

00:00:20.534 --> 00:00:24.374
The product of B and C is 45.

00:00:24.374 --> 00:00:28.657
So I could write that as
B times C is equal to 45.

00:00:29.595 --> 00:00:31.607
Which of the following pairs of equations

00:00:31.607 --> 00:00:35.889
could be used to find
the values of B and C?

00:00:36.253 --> 00:00:37.111
So let's see.

00:00:37.523 --> 00:00:40.468
So this one we have 45 is equal to BC,

00:00:40.468 --> 00:00:42.338
45 is equal to BC,

00:00:42.338 --> 00:00:43.678
so these are contenders.

00:00:43.678 --> 00:00:46.930
This one has 45C is equal to B.

00:00:46.930 --> 00:00:48.513
Well that's like, this
is a different equation.

00:00:48.513 --> 00:00:50.688
If you divided both sides by C,

00:00:50.688 --> 00:00:52.537
you would have 45 is equal to B over C,

00:00:52.537 --> 00:00:54.584
so we can knock this one out.

00:00:54.813 --> 00:00:57.515
This one has 45C is equal to B.

00:00:57.515 --> 00:00:59.021
Well once again, 45C equals B,

00:00:59.021 --> 00:01:00.849
we can knock that one out.

00:01:00.849 --> 00:01:02.774
This one has 45 is equal to BC.

00:01:02.920 --> 00:01:05.598
So now we can just focus on
the first of these equations.

00:01:06.049 --> 00:01:07.290
So let's see.

00:01:07.290 --> 00:01:09.331
B is equal to two minus 5C.

00:01:09.331 --> 00:01:11.663
No, it should be 2C minus five.

00:01:11.952 --> 00:01:13.471
So we can knock that one out.

00:01:13.678 --> 00:01:16.214
B is equal to 2C minus five.

00:01:16.260 --> 00:01:17.930
Well that's exactly what I wrote here.

00:01:18.037 --> 00:01:20.029
BC is equal to 45.

00:01:20.029 --> 00:01:21.385
That's exactly what I wrote here.

00:01:21.531 --> 00:01:23.952
Wrote over here, so this
is going to be our choice.

00:01:23.952 --> 00:01:26.527
So we can verify that this
last one isn't going to work.

00:01:26.817 --> 00:01:29.210
B is equal to two times C plus five.

00:01:29.458 --> 00:01:30.636
Well, if you distribute the two,

00:01:30.636 --> 00:01:33.217
that's like saying that B is equal to 2C

00:01:33.217 --> 00:01:35.613
plus 10, which is clearly different

00:01:35.613 --> 00:01:37.890
than this first equation
that I wrote up here.

00:01:37.890 --> 00:01:40.119
So we can knock that out as well.