Fundamental theorem of algebra for quadratic

WEBVTT
Kind: captions
Language: en

00:00:00.578 --> 00:00:02.586
Voiceover:Let's say that
we have the function

00:00:02.586 --> 00:00:06.333
F of X being defined by the
second degree polynomial,

00:00:06.333 --> 00:00:12.180
5 X squared plus 6 X plus 5.

00:00:12.180 --> 00:00:14.771
The fundamental theorem of algebra

00:00:14.771 --> 00:00:18.234
tells us that because this
is a second degree polynomial

00:00:18.234 --> 00:00:21.673
we are going to have exactly 2 roots.

00:00:21.673 --> 00:00:23.449
Or another way of thinking about it,

00:00:23.449 --> 00:00:25.649
there's exactly 2 values for X

00:00:25.649 --> 00:00:29.194
that will make F of X equal 0.

00:00:29.194 --> 00:00:31.038
So I encourage you to pause this video

00:00:31.038 --> 00:00:35.937
and try to figure out what
those 2 values of X are.

00:00:35.937 --> 00:00:37.921
So when I, if we're
looking for the X values,

00:00:37.921 --> 00:00:40.105
that would make this
expression equal to 0.

00:00:40.105 --> 00:00:42.745
That's essentially trying
to solve this equation.

00:00:42.745 --> 00:00:48.557
At 5 X squared plus 6
X plus 5 is equal to 0.

00:00:48.573 --> 00:00:50.998
And there's no obvious way I can think of

00:00:50.998 --> 00:00:52.829
factoring this right
over here so I'll just

00:00:52.829 --> 00:00:55.566
resort to the quadratic formula.

00:00:55.566 --> 00:00:58.766
So the quadratic formula tells us that X,

00:00:58.766 --> 00:01:00.886
where X is a solution to this equation,

00:01:00.886 --> 00:01:04.784
is going to be equal to negative B.

00:01:04.784 --> 00:01:06.558
I'll try to color code it.

00:01:06.558 --> 00:01:11.438
Negative B, this is right,
this is B right over here.

00:01:11.438 --> 00:01:16.022
Negative B, so negative 6, plus or minus,

00:01:16.022 --> 00:01:21.150
plus or minus the square
root of B squared,

00:01:21.150 --> 00:01:28.710
of B squared, minus 4 times A,

00:01:28.710 --> 00:01:37.094
times A, times C, times A times C.

00:01:37.094 --> 00:01:45.006
All of that, all of that over
all of that over 2 times A.

00:01:45.006 --> 00:01:48.614
All of that over 2 times A.

00:01:48.614 --> 00:01:50.334
So what does this simplify to?

00:01:50.334 --> 00:01:54.526
This is going to be equal
to so I'll go back to,

00:01:54.526 --> 00:01:58.326
this is going to be equal to negative 6.

00:01:58.326 --> 00:02:00.286
I'll try to keep it color coded still.

00:02:00.286 --> 00:02:05.558
Negative 6 plus or minus,
plus or minus the square root.

00:02:05.558 --> 00:02:06.854
Now, what is this going to be equal to?

00:02:06.854 --> 00:02:10.694
This is 36 minus 100.

00:02:10.694 --> 00:02:17.382
So negative 64 all of that over 2 times 5.

00:02:17.382 --> 00:02:19.710
All of that over 10.

00:02:19.710 --> 00:02:21.270
Now this is interesting.

00:02:21.270 --> 00:02:24.902
We are trying to take the square
root of a negative number.

00:02:24.902 --> 00:02:26.462
Or another way of thinking about it,

00:02:26.462 --> 00:02:31.038
B squared minus 4 A C is less than 0.

00:02:31.038 --> 00:02:35.790
B squared minus 4 A C which is often,

00:02:35.790 --> 00:02:40.542
often times called the
discriminant of this quadratic.

00:02:40.542 --> 00:02:43.598
This is less than 0.

00:02:43.598 --> 00:02:45.326
So this is less than 0.

00:02:45.326 --> 00:02:47.150
This part of the quadratic
formula we're going

00:02:47.150 --> 00:02:49.334
to try to take the square
root of a negative number.

00:02:49.334 --> 00:02:51.014
This is going to be a negative number.

00:02:51.014 --> 00:02:53.934
So we're going to result
in an imaginary number.

00:02:53.934 --> 00:02:57.462
So what that gives us
are not 2 real roots,

00:02:57.462 --> 00:03:00.510
but 2 non real complex roots.

00:03:00.510 --> 00:03:02.390
So this is going to be equal to

00:03:02.390 --> 00:03:08.334
negative 6 plus or minus 8 i,

00:03:08.334 --> 00:03:11.606
the square root of negative 64 is 8 i.

00:03:11.606 --> 00:03:15.070
If we extend the principle
square root function

00:03:15.070 --> 00:03:18.006
to imaginary numbers or complex numbers.

00:03:18.006 --> 00:03:20.191
All of that over 10.

00:03:20.191 --> 00:03:23.222
Or we could say X is equal to,

00:03:23.222 --> 00:03:26.342
let's see if we find a
greatest common divisor.

00:03:26.342 --> 00:03:28.686
If we try to reduce this,
so we're going to have

00:03:28.686 --> 00:03:30.446
let's see they're all divisible by 2

00:03:30.446 --> 00:03:33.606
so it's negative 3 over 5,
that's the same thing as

00:03:33.606 --> 00:03:41.102
negative 6 over 10, plus or
minus 4 over 5, 4 over 5 i.

00:03:41.102 --> 00:03:43.230
Or you could say that the 2 roots,

00:03:43.230 --> 00:03:45.730
which are non real complex roots,

00:03:45.730 --> 00:03:51.534
so it's X is equal to
negative 3/5 plus 4/5 i,

00:03:51.534 --> 00:03:52.606
that's one of the roots.

00:03:52.606 --> 00:03:54.198
And then the other root is X

00:03:54.198 --> 00:03:58.886
is equal to negative 3/5 minus 4/5 i.

00:03:58.886 --> 00:04:02.741
Notice, we satisfied the
fundamental theorem of algebra.

00:04:02.741 --> 00:04:05.158
We have 2 roots, they're non real,

00:04:05.158 --> 00:04:06.494
but the fundamental theorem of algebra

00:04:06.494 --> 00:04:07.830
says, "hey look we're just going to have

00:04:07.830 --> 00:04:10.297
"at least if we're Nth degree polynomial,

00:04:10.297 --> 00:04:12.298
"we're going to have n complex roots."

00:04:12.298 --> 00:04:15.098
They could be real or
they could be non real

00:04:15.098 --> 00:04:16.266
and we see that right over there.

00:04:16.266 --> 00:04:18.250
And we also see that they are conjugates.

00:04:18.250 --> 00:04:19.762
And the quadratic formula kinda

00:04:19.762 --> 00:04:21.867
sets up a situation where we are,

00:04:21.867 --> 00:04:24.778
especially if this ends
up being less than 0

00:04:24.778 --> 00:04:26.707
and so this when you take the square root

00:04:26.707 --> 00:04:28.322
you get an imaginary number

00:04:28.322 --> 00:04:30.994
you see where those conjugates appear.

00:04:30.994 --> 00:04:33.802
Now let's verify graphically
that this is indeed the case.

00:04:33.802 --> 00:04:36.490
That this does not have any real roots.

00:04:36.490 --> 00:04:39.050
So let's get a calculator out.

00:04:39.050 --> 00:04:41.506
So let me go into graph mode.

00:04:41.506 --> 00:04:43.482
So Y is equal to.

00:04:43.482 --> 00:04:45.898
Let me clear what I must have
been doing out here before.

00:04:45.898 --> 00:04:50.206
So Y1 is equal to 5 times X squared,

00:04:50.206 --> 00:04:55.690
plus 6 X plus 5 and then let me

00:04:55.690 --> 00:04:58.850
set a reasonable range here.

00:04:58.850 --> 00:05:00.322
So I don't know.

00:05:00.322 --> 00:05:02.498
Actually, I know very little
about this function right over.

00:05:02.498 --> 00:05:06.242
So I'm going to set minimum
range at a negative 10,

00:05:06.242 --> 00:05:09.202
X max I don't know positive 10.

00:05:09.202 --> 00:05:12.098
X scale is 1, let's see Y max let's see,

00:05:12.098 --> 00:05:15.994
this thing actually does
get pretty big pretty fast.

00:05:15.994 --> 00:05:20.253
So let's say Y max is and the Y scale,

00:05:20.253 --> 00:05:22.586
I don't know I'll say 100.

00:05:22.586 --> 00:05:25.738
100, the scale I'll make it 10.

00:05:25.738 --> 00:05:27.690
Each of those notches are going to be 10.

00:05:27.690 --> 00:05:30.314
Y minimum, we want to see the X axis

00:05:30.314 --> 00:05:34.004
to make sure it doesn't cross
it so let's go at negative,

00:05:34.004 --> 00:05:36.482
I don't know, negative 20.

00:05:36.482 --> 00:05:37.850
And now let's graph this.

00:05:37.850 --> 00:05:40.738
I'm hoping that I've actually
captured it and I have.

00:05:40.738 --> 00:05:42.714
So there you have it, you see that

00:05:42.714 --> 00:05:45.802
this thing does not intersect the X-axis.

00:05:45.802 --> 00:05:49.931
And in fact, we could,
we could zoom in on it.

00:05:49.931 --> 00:05:52.251
Let's see it's a little bit
hard to, it's a little actually,

00:05:52.251 --> 00:05:55.747
let me just change the range a little bit.

00:05:55.747 --> 00:05:57.803
So let me just change the range.

00:05:57.803 --> 00:06:02.115
So let's make our X min, let's make it 5,

00:06:02.115 --> 00:06:07.619
let's make our X max, whoops
not 50, let's make that 5.

00:06:07.619 --> 00:06:11.227
And let's make, negative 5 and positive 5.

00:06:11.227 --> 00:06:15.627
And let's see, let's make our Y max,

00:06:15.627 --> 00:06:20.340
let's make our Y max equal
to 20 and then our Y scale,

00:06:20.340 --> 00:06:21.901
I don't know we could make it 2.

00:06:21.901 --> 00:06:24.132
Okay I think this will
get us much closer in,

00:06:24.132 --> 00:06:28.004
there we see, it does
not intersect the X axis.

00:06:28.004 --> 00:06:30.177
This does not have any real roots

00:06:30.177 --> 23:59:59.999
but it has 2 non real complex roots.


Factoring sum of squares

WEBVTT
Kind: captions
Language: en

00:00:00.618 --> 00:00:03.114
Voiceover:Long ago in
algebra class, we learned

00:00:03.114 --> 00:00:07.482
to factor things like X
squared minus Y squared.

00:00:07.482 --> 00:00:09.526
We saw that this was a
difference of squares

00:00:09.526 --> 00:00:15.159
that you could factor this
as X plus Y, times X minus Y.

00:00:15.159 --> 00:00:16.775
This is a little bit of a refresher,

00:00:16.775 --> 00:00:18.327
you can multiply these two together

00:00:18.327 --> 00:00:20.679
to verify you get X
squared minus Y squared.

00:00:20.679 --> 00:00:22.391
In fact, let's just do that for fun.

00:00:22.391 --> 00:00:26.501
X times X is X squared, X times negative Y

00:00:26.501 --> 00:00:31.126
is negative XY, Y times X is positive XY,

00:00:31.126 --> 00:00:36.119
and then Y times negative
Y is minus Y squared.

00:00:36.119 --> 00:00:38.295
These two middle terms cancel out

00:00:38.295 --> 00:00:42.198
and your left with X
squared minus Y squared.

00:00:42.198 --> 00:00:44.437
What I want to tackle in this video

00:00:44.437 --> 00:00:47.623
is something that we didn't
know how to factor before

00:00:47.623 --> 00:00:50.969
and that's the sum of the squares.

00:00:50.969 --> 00:00:55.863
So if we were to factor X,
let me use a brighter color,

00:00:55.863 --> 00:01:00.311
if we want to try to factor
X squared plus Y squared,

00:01:00.311 --> 00:01:04.255
before we knew about imaginary numbers,

00:01:04.255 --> 00:01:06.838
complex numbers, we didn't
know how to factor this.

00:01:06.838 --> 00:01:09.894
But now that we do,
what I want to try to do

00:01:09.894 --> 00:01:11.430
and I encourage you
actually to pause the video

00:01:11.430 --> 00:01:15.384
and try to do it before me,
is to try to express this

00:01:15.384 --> 00:01:21.040
as a difference of squares,
using the imaginary unit I.

00:01:21.040 --> 00:01:22.310
So let's try to do this.

00:01:22.310 --> 00:01:24.470
So the first we want to
write as a difference

00:01:24.470 --> 00:01:27.207
of squares, so X square
we'll just keep as X squared,

00:01:27.207 --> 00:01:30.281
but now I want to write
this part, I want to write

00:01:30.281 --> 00:01:36.252
the second part, right over
here, as subtracting a squares.

00:01:36.390 --> 00:01:37.418
I want to subtract a square.

00:01:37.497 --> 00:01:41.072
So we could write this part as subtracting

00:01:41.072 --> 00:01:44.671
negative Y squared, obviously you subtract

00:01:44.671 --> 00:01:48.031
a negative that's the
same thing as adding.

00:01:48.031 --> 00:01:50.191
Now how does this help us?

00:01:50.191 --> 00:01:51.551
Well this is the same thing as subtracting

00:01:51.551 --> 00:01:55.582
negative one times Y squared.

00:01:55.582 --> 00:02:00.191
If we wanted to write this
whole thing as a square,

00:02:00.191 --> 00:02:05.311
how would we do it?

00:02:05.311 --> 00:02:07.183
Well we have Y squared and

00:02:07.183 --> 00:02:09.295
what's negative one the square of?

00:02:09.295 --> 00:02:16.095
Well we know by definition,
negative one is equal to

00:02:16.095 --> 00:02:20.463
I squared or that I squared
is equal to negative one.

00:02:20.463 --> 00:02:23.919
Let's rewrite it that way,
so this is going to be equal

00:02:23.919 --> 00:02:28.590
to X squared minus,
instead of negative one,

00:02:28.590 --> 00:02:35.888
I'll write that as I squared,
minus I squared Y squared.

00:02:35.888 --> 00:02:40.511
All I did was replace this
negative one with an I squared.

00:02:40.511 --> 00:02:42.319
And now this is interesting.

00:02:42.319 --> 00:02:43.807
I think you see where this is going,

00:02:43.807 --> 00:02:45.287
but I'll just make it very explicit.

00:02:45.287 --> 00:02:56.565
This is now X squared minus IY squared.

00:02:56.627 --> 00:03:00.431
And just like that, using
I, I've been able to write

00:03:00.508 --> 00:03:03.715
this sum of squares as
a difference of squares.

00:03:03.715 --> 00:03:06.338
And now we can factor
just the exact same way

00:03:06.338 --> 00:03:08.451
we factored this original expression.

00:03:08.451 --> 00:03:12.632
This thing is going to be equal to

00:03:12.632 --> 00:03:32.188
X plus IY times X minus IY.

00:03:32.911 --> 00:03:35.379
And we can verify that
if you multiply these two

00:03:35.379 --> 00:03:37.651
expressions together, you're going to get

00:03:37.651 --> 00:03:38.789
X squared plus Y squared.

00:03:38.789 --> 00:03:45.204
Let's do that, X times X is X squared.

00:03:45.204 --> 00:03:53.461
X times negative IY is
negative IXY, and then

00:03:53.461 --> 00:04:01.269
IY times X is positive
IXY, and then finally

00:04:01.269 --> 00:04:04.420
let me do this in a
color I'm not using yet,

00:04:04.420 --> 00:04:09.077
finally, IY times negative IY is equal to

00:04:09.077 --> 00:04:13.127
negative I squared Y squared.

00:04:13.127 --> 00:04:15.909
Now these middle two terms cancel out,

00:04:15.909 --> 00:04:18.933
I squared is negative one, and so we have

00:04:18.933 --> 00:04:22.972
X squared minus negative one Y squared,

00:04:22.972 --> 00:04:25.772
so this is subtracting a negative,

00:04:25.772 --> 00:04:26.971
the same thing as adding a positive,

00:04:26.971 --> 00:04:34.269
and so this simplifies to
X squared plus Y squared.

00:04:34.269 --> 00:04:36.635
So hopefully this now
gives you an appreciation

00:04:36.635 --> 00:04:41.692
of using the complex imaginary unit I,

00:04:41.692 --> 00:04:44.251
how you can actually
factor this expression

00:04:44.251 --> 00:04:48.251
into essentially the product
of two complex numbers.


Fundamental theorem of algebra

WEBVTT
Kind: captions
Language: en

00:00:01.580 --> 00:00:05.236
Voiceover:The fundamental
theorem of algebra.

00:00:05.285 --> 00:00:12.757
Fundamental, I'll write
it out, theorem of algebra

00:00:12.942 --> 00:00:17.284
tells us that if we have
an nth-degree polynomial,

00:00:17.406 --> 00:00:18.533
so let's write it out.

00:00:18.533 --> 00:00:22.670
So let's say I have the function p of x

00:00:22.670 --> 00:00:26.197
and it's defined by an
nth-degree polynomial.

00:00:26.197 --> 00:00:32.929
So let's say it's a x to the
n plus b x to the n minus one

00:00:32.929 --> 00:00:38.853
and you just go all the way to
some constant term at the end.

00:00:38.888 --> 00:00:40.829
So this is an nth-degree polynomial.

00:00:40.829 --> 00:00:44.618
The fundamental theorem
of algebra tells us

00:00:44.618 --> 00:00:48.174
that this nth-degree
polynomial is going to have

00:00:48.174 --> 00:00:55.146
n exactly n roots, or another
way to think about it,

00:00:55.146 --> 00:00:58.331
there are going to be
exactly n values for x,

00:00:58.331 --> 00:01:01.750
which will make this polynomial,

00:01:01.750 --> 00:01:06.810
make this expression on the
right, be equal to zero.

00:01:06.810 --> 00:01:09.800
So at first you might say,
"Okay, that makes sense."

00:01:09.800 --> 00:01:13.628
You've seen second-degree polynomials,

00:01:13.628 --> 00:01:15.980
whose graphs might look
something like this.

00:01:17.787 --> 00:01:20.606
So, that's the y axis
and that's the x axis.

00:01:20.889 --> 00:01:24.573
We know a second-degree polynomial
would define a parabola,

00:01:24.573 --> 00:01:28.512
so it might look something like
this and you could buy that.

00:01:28.512 --> 00:01:33.961
Okay, this is a second-degree,
that's second degree,

00:01:33.961 --> 00:01:36.960
and you see that this function

00:01:36.960 --> 00:01:38.634
equals zero at exactly two places.

00:01:38.634 --> 00:01:41.922
It has exactly two roots.

00:01:42.728 --> 00:01:47.658
It has two roots, so that seems consistent

00:01:47.658 --> 00:01:50.762
with the fundamental theorem of algebra,

00:01:50.762 --> 00:01:51.870
and you could also imagine

00:01:51.870 --> 00:01:54.249
a third-degree polynomial
looking like this.

00:01:55.182 --> 00:01:56.981
So that's my y axis.

00:01:56.981 --> 00:01:59.812
This is my x axis.

00:02:00.157 --> 00:02:01.642
You could imagine a
third-degree polynomial

00:02:01.642 --> 00:02:03.311
looking something like this.

00:02:03.550 --> 00:02:06.711
Bam, bam, bam, and it just keeps going.

00:02:07.960 --> 00:02:09.432
And here you see it's a
third-degree polynomial,

00:02:09.432 --> 00:02:13.790
and you'll see it has
one, two, three roots.

00:02:13.790 --> 00:02:15.613
And I could have a
fourth-degree polynomial.

00:02:15.626 --> 00:02:17.140
Maybe it looks something like this,

00:02:17.470 --> 00:02:20.380
where it goes something like this,

00:02:20.380 --> 00:02:21.673
and you say, "Okay, that makes sense."

00:02:21.673 --> 00:02:24.726
It will have one, two, three, four roots.

00:02:24.921 --> 00:02:26.928
But then you might
start to remember things

00:02:26.928 --> 00:02:29.387
that don't always behave in this way.

00:02:29.781 --> 00:02:32.788
For example, many, many, many times

00:02:32.788 --> 00:02:36.766
we've seen parabolas, we've
seen second-degree polynomials

00:02:36.766 --> 00:02:40.173
that look more like this, where they don't

00:02:40.173 --> 00:02:43.702
seem to intersect the x axis.

00:02:43.826 --> 00:02:47.263
So this seems to conflict with

00:02:47.263 --> 00:02:49.468
the fundamental theorem of algebra.

00:02:49.468 --> 00:02:50.710
The fundamental theorem of algebra says

00:02:50.710 --> 00:02:54.578
if we have a second-degree polynomial

00:02:54.578 --> 00:02:59.000
then we should have exactly two roots.

00:02:59.771 --> 00:03:01.400
Now, this is the key.

00:03:01.400 --> 00:03:03.657
The fundamental theorem of algebra,

00:03:03.657 --> 00:03:05.773
it extends our number system.

00:03:05.773 --> 00:03:08.177
We're not just talking about real roots,

00:03:08.177 --> 00:03:10.602
we're talking about complex roots,

00:03:10.602 --> 00:03:13.710
and in particular, the
fundamental theorem of algebra

00:03:13.710 --> 00:03:16.349
allows even these
coefficients to be complex.

00:03:16.787 --> 00:03:19.882
And so when we're looking
at these first examples,

00:03:19.882 --> 00:03:21.339
these were all real roots,

00:03:21.339 --> 00:03:24.797
and real numbers are a
subset of complex numbers.

00:03:24.921 --> 00:03:26.963
So here you had two real roots.

00:03:26.963 --> 00:03:28.826
Here you had three real roots.

00:03:28.826 --> 00:03:31.214
In this orange function,
you had four real roots.

00:03:31.787 --> 00:03:33.657
In this yellow function,

00:03:33.657 --> 00:03:35.531
this yellow parabola right over here,

00:03:35.531 --> 00:03:39.849
the second-degree polynomial,
we have no real roots.

00:03:39.849 --> 00:03:42.714
That's why you don't see
it intersecting the x axis,

00:03:42.714 --> 00:03:47.299
but we will have two complex roots.

00:03:47.299 --> 00:03:54.154
So this one right over here
will have two complex roots.

00:03:54.154 --> 00:03:57.404
And the complex roots, the
non-real complex roots,

00:03:57.404 --> 00:04:00.212
because really real numbers are
a subset of complex numbers,

00:04:00.212 --> 00:04:02.822
these always come in pairs,

00:04:02.822 --> 00:04:05.320
and we'll see that in future videos.

00:04:05.320 --> 00:04:09.290
So for example, if you have
a third-degree polynomial,

00:04:09.290 --> 00:04:11.157
It might look something like this.

00:04:11.992 --> 00:04:14.582
A third-degree polynomial
might look something like this,

00:04:14.582 --> 00:04:18.118
where it has one real root,

00:04:18.118 --> 00:04:20.423
but then the fundamental
theorem of algebra

00:04:20.423 --> 00:04:22.805
tell us it necessarily has two other roots

00:04:22.805 --> 00:04:27.144
because it is a third degree, so we know

00:04:27.144 --> 00:04:33.219
that the other two roots must
be non-real, complex roots.

00:04:33.219 --> 00:04:35.792
Now, could you have a
situation where you have a

00:04:35.792 --> 00:04:39.834
third-degree polynomial
with three complex roots?

00:04:40.182 --> 00:04:49.275
So, can you have three
non-real complex roots?

00:04:49.275 --> 00:04:53.121
Is this possible for a
third-degree polynomial?

00:04:53.121 --> 00:04:56.887
Well, the answer is no
because complex roots,

00:04:56.887 --> 00:04:59.161
as we'll see in the next few videos,

00:04:59.161 --> 00:05:01.440
always come in pairs.

00:05:01.440 --> 00:05:02.634
They're coming in pairs

00:05:02.634 --> 00:05:06.395
where they are conjugates of each other.

00:05:06.395 --> 00:05:13.352
So, you could have a
fourth-degree polynomial

00:05:13.352 --> 00:05:15.880
that has no real roots, for example.

00:05:15.880 --> 00:05:18.132
Something might look something like this.

00:05:18.221 --> 00:05:22.655
In this case, you would have
two pairs of complex roots

00:05:22.655 --> 00:05:26.493
or you would have four
non-real complex roots

00:05:26.515 --> 00:05:28.127
and you could group them into two pairs

00:05:28.127 --> 00:05:29.998
where in each pair you have conjugates,

00:05:29.998 --> 00:05:32.406
and we'll see that in the next video.


Powers of complex numbers

WEBVTT
Kind: captions
Language: en

00:00:00.513 --> 00:00:03.854
I have the complex number
cosine of two pi over three,

00:00:03.854 --> 00:00:07.292
or two thirds pi, plus
i sine of two thirds pi

00:00:07.292 --> 00:00:09.313
and I'm going to raise
that to the 20th power.

00:00:09.700 --> 00:00:12.388
What I want to do is first
plot this number in blue

00:00:12.388 --> 00:00:14.652
on the complex plane, and
then figure out what it is

00:00:14.652 --> 00:00:17.296
raised to the 20th power
and then try to plot that.

00:00:17.564 --> 00:00:19.249
I encourage you to
pause this video and try

00:00:19.249 --> 00:00:21.329
this out on your own
before I work through it.

00:00:21.982 --> 00:00:25.924
Let's first focus on this
blue complex number over here.

00:00:26.234 --> 00:00:27.716
It's clearly written in polar form.

00:00:27.716 --> 00:00:35.827
The angle is two thirds pi
or two pi over three radians.

00:00:36.357 --> 00:00:40.808
And it's magnitude of this
complex number is clearly one.

00:00:40.808 --> 00:00:42.559
To make that a little clearer
you could write it in the

00:00:42.559 --> 00:00:45.359
pure polar form where you
have its magnitude out front.

00:00:45.359 --> 00:00:48.200
It's cosine of two over three pi

00:00:48.420 --> 00:00:54.592
plus i sine of two over three pi.

00:00:55.302 --> 00:00:57.061
You could write it just like that.

00:00:57.293 --> 00:00:59.512
When you look at that the
angle is two over three pi.

00:00:59.512 --> 00:01:01.246
That would get us, let's see.

00:01:01.466 --> 00:01:03.403
This is zero, this is pi, we're going

00:01:03.403 --> 00:01:04.939
to go two thirds of the way to pi.

00:01:04.939 --> 00:01:07.794
Each of these is one, two, three, four,

00:01:07.794 --> 00:01:11.062
five, six, seven, eight, nine, 10, 11, 12.

00:01:11.062 --> 00:01:16.063
Each of these is pi over
12 so we're going to go,

00:01:16.371 --> 00:01:19.090
two thirds of the way would
be eight pis over twelve.

00:01:19.090 --> 00:01:22.313
One, two, three, four,
five, six, seven, eight.

00:01:23.075 --> 00:01:28.105
The way I was able to reason
through that is two thirds pi

00:01:28.105 --> 00:01:32.185
is the same thing as eight pi over twelve.

00:01:32.185 --> 00:01:33.684
Each of these segments is pi over 12

00:01:33.684 --> 00:01:35.111
so I just counted eight of them.

00:01:35.403 --> 00:01:37.764
That's that number, but now let's

00:01:37.764 --> 00:01:42.137
try to raise it to the 20th power.

00:01:44.655 --> 00:01:47.545
To do that we're going
to use Euler's formula.

00:01:48.135 --> 00:01:50.606
Euler's formula, you might remember,

00:01:50.606 --> 00:01:54.609
tells us that e to the i theta is equal to

00:01:54.609 --> 00:02:00.390
cosine of theta plus i sine of theta.

00:02:00.868 --> 00:02:03.402
You see right over here
this is already written

00:02:03.402 --> 00:02:06.087
in that form where theta is two thirds pi.

00:02:06.430 --> 00:02:08.204
We can rewrite what we have in blue

00:02:08.204 --> 00:02:16.808
here as e to the two thirds pi i.

00:02:17.499 --> 00:02:19.714
Then of course we're raising
that to the 20th power.

00:02:22.277 --> 00:02:24.448
This simplifies things
dramatically because

00:02:24.667 --> 00:02:26.948
here if I tried to
multiply this thing times-

00:02:26.948 --> 00:02:29.932
If I had 20 of these things
and I multiplied them together

00:02:29.932 --> 00:02:32.263
that would get really, really,
really hairy really fast,

00:02:32.263 --> 00:02:34.519
but here I can just use
exponent properties.

00:02:35.035 --> 00:02:37.772
This is going to be the
same thing as e to the,

00:02:37.772 --> 00:02:39.972
if I raise something to
exponent and then raise that to

00:02:39.972 --> 00:02:42.523
an exponent I can just take
the product of the exponents.

00:02:43.053 --> 00:02:50.364
This is e to the 20 times
two over three pi i,

00:02:50.644 --> 00:03:00.035
which is equal to e to
the 40 over three pi i.

00:03:00.755 --> 00:03:05.256
Now this is this number
raised to the 20th power

00:03:05.256 --> 00:03:08.528
but this is an awfully
large angle right here.

00:03:08.848 --> 00:03:11.098
If we're thinking of 40 over three pi,

00:03:11.098 --> 00:03:12.338
let's just try to digest this.

00:03:12.338 --> 00:03:17.289
40 over three pi, this
is the same thing as-

00:03:17.850 --> 00:03:21.005
Let's see, 40 divided
three is 13 and one third.

00:03:21.005 --> 00:03:26.301
This is the same thing as
13 and one third times pi.

00:03:26.613 --> 00:03:30.379
We know that going two pi radians gets you

00:03:30.379 --> 00:03:32.896
around the unit circle
once, so this is going

00:03:32.896 --> 00:03:37.050
over six times around
the unit circle to get-

00:03:37.972 --> 00:03:39.492
Or around, I should say,
not the unit circle,

00:03:39.492 --> 00:03:43.832
going six times around, going in circles

00:03:43.832 --> 00:03:46.130
in order to get to the point we want to.

00:03:46.487 --> 00:03:49.945
In order to simplify this a
little bit let me subtract

00:03:49.945 --> 00:03:54.037
the largest multiple of
two pi that I could figure,

00:03:54.037 --> 00:03:56.908
to get this in as small
of a form as possible.

00:03:57.188 --> 00:04:01.638
We know an angle, if we have
some angle it's equal to that

00:04:01.856 --> 00:04:07.497
angle plus some multiple of
two pi where k is any integer.

00:04:07.497 --> 00:04:08.810
k could also be negative, we could

00:04:08.810 --> 00:04:10.782
be subtracting a multiple of two pi.

00:04:11.107 --> 00:04:13.750
Let me subtract, let's see.
The largest multiple of

00:04:13.750 --> 00:04:18.581
two pi that I could subtract
here is going to be 12 pi.

00:04:19.014 --> 00:04:21.639
Let me subtract 12 pi from this.

00:04:22.029 --> 00:04:25.607
If I subtract 12 pi, I'll do it down here.

00:04:25.857 --> 00:04:31.250
13 and one third pi minus 12 pi.

00:04:31.250 --> 00:04:32.050
Remember, I'm just trying to subtract the

00:04:32.283 --> 00:04:34.283
largest multiple of two pi that I can.

00:04:41.785 --> 00:04:44.237
13 and one third minus
twelve is one and one third.

00:04:44.237 --> 00:04:47.223
That's going to be one and one third pi,

00:04:47.223 --> 00:04:50.192
or we could write it as four thirds pi.

00:04:50.801 --> 00:04:53.939
This complex number is going to be

00:04:54.177 --> 00:05:00.006
equivalent to e to the four thirds pi i.

00:05:00.395 --> 00:05:03.705
This makes it much simpler and
much easier for me to plot.

00:05:03.924 --> 00:05:07.096
Four thirds pi, or the same
thing as one and one third pi.

00:05:07.501 --> 00:05:10.973
This would be pi, and now
we have to just go another

00:05:10.973 --> 00:05:13.360
one third pi, and each of these are 12ths.

00:05:13.568 --> 00:05:18.252
If we go four 12ths pi. Sorry,
each of these are pi over 12,

00:05:18.519 --> 00:05:20.220
so we go four pi over 12.

00:05:20.220 --> 00:05:24.221
One, two, three, four
gets us right over there.

00:05:26.852 --> 00:05:29.007
This number raised to
the 20th power is this,

00:05:29.007 --> 00:05:30.351
which is equivalent to this,

00:05:30.351 --> 00:05:32.241
which we've plotted right over there.

00:05:33.011 --> 00:05:36.789
What if we wanted to take it
to, let's say the 21st power.

00:05:37.227 --> 00:05:40.709
Then we would increase
the angle by another

00:05:40.709 --> 00:05:43.913
two pi over three or eight pi over 12.

00:05:44.321 --> 00:05:47.365
We'd increase the angle by
one, two, three, four, five,

00:05:47.365 --> 00:05:50.741
six, seven, eight. And we
would go right over there.

00:05:51.273 --> 00:05:52.913
How does this make conceptual sense?

00:05:53.209 --> 00:05:56.729
The number to the first
power was right over here,

00:05:56.960 --> 00:05:59.117
that was our original number
is blue right over here.

00:05:59.382 --> 00:06:00.977
If you raise it to the second power then

00:06:00.977 --> 00:06:03.805
you're increasing the
angle by two thirds pi,

00:06:03.805 --> 00:06:05.242
you're increasing the angle to go there.

00:06:05.525 --> 00:06:07.150
You raise it to the
third power, you increase

00:06:07.150 --> 00:06:09.137
the angle by two thirds
pi, you go over there.

00:06:09.647 --> 00:06:11.055
Fourth power you get back here.

00:06:11.055 --> 00:06:13.585
Fifth, sixth, seventh, eighth, ninth,

00:06:13.585 --> 00:06:18.038
10th, 11th, 12th, 13th,
14th, 15th, 16th, 17,

00:06:18.038 --> 00:06:23.163
18, 19, 20th power gets
us right over there.


Dividing complex numbers in polar and exponential form

WEBVTT
Kind: captions
Language: en

00:00:00.465 --> 00:00:03.001
Voiceover:So this kind of
hairy looking expression,

00:00:03.001 --> 00:00:04.882
we're just dividing one complex number,

00:00:04.882 --> 00:00:08.005
written in blue, by
another complex number.

00:00:08.005 --> 00:00:09.327
This first complex - actually,

00:00:09.327 --> 00:00:11.020
both of them are written in polar form,

00:00:11.020 --> 00:00:12.694
and we also see them plotted over here.

00:00:12.694 --> 00:00:14.808
This first complex number, seven times,

00:00:14.808 --> 00:00:16.294
cosine of seven pi over six,

00:00:16.294 --> 00:00:19.101
plus i times sine of seven pi over six,

00:00:19.101 --> 00:00:20.503
we see that the angle,

00:00:20.503 --> 00:00:22.906
if we're thinking in polar
form is seven pi over six,

00:00:22.906 --> 00:00:25.991
so if we start from
the positive real axis,

00:00:25.991 --> 00:00:30.176
we're gonna go seven pi over six.

00:00:30.176 --> 00:00:31.958
So we're gonna go seven pi over six,

00:00:31.958 --> 00:00:34.136
all the way to that
point right over there.

00:00:34.136 --> 00:00:35.585
And then from the origin,

00:00:35.585 --> 00:00:37.339
we're going to step seven out,

00:00:37.339 --> 00:00:38.715
or seven away from the origin.

00:00:38.715 --> 00:00:43.510
So one, two, three, four, five, six, seven

00:00:43.510 --> 00:00:47.407
to get to that point right over there.

00:00:47.407 --> 00:00:49.173
And then the second number,

00:00:49.173 --> 00:00:52.736
the angle is seven pi over four.

00:00:52.736 --> 00:00:57.036
So that takes us all the way around here.

00:00:57.036 --> 00:01:00.329
It takes us all the way -
so I should do it smaller

00:01:00.329 --> 00:01:04.559
since that has a smaller distance,

00:01:04.559 --> 00:01:08.968
so let me - so let's
say we start over here,

00:01:08.968 --> 00:01:10.623
so we're going to go all the way over here

00:01:10.623 --> 00:01:12.152
to that point right over here,

00:01:12.152 --> 00:01:14.736
and it's distance from the origin is one.

00:01:14.736 --> 00:01:17.760
You can imagine that there's
a one in front of that.

00:01:17.760 --> 00:01:19.424
And we want to divide the two,

00:01:19.424 --> 00:01:20.981
and I encourage you to
pause this video and

00:01:20.981 --> 00:01:22.421
try to do this on your own,

00:01:22.421 --> 00:01:25.625
and then plot the resulting
number when you divide

00:01:25.625 --> 00:01:29.619
this blue complex number
by this green one.

00:01:29.619 --> 00:01:31.185
Well, as you might have realized,

00:01:31.185 --> 00:01:33.128
if you somehow just try
to divide it straight up,

00:01:33.128 --> 00:01:36.403
it can get quite hairy,
and the way to tackle it,

00:01:36.403 --> 00:01:38.654
is to write it in another form.

00:01:38.654 --> 00:01:40.751
And what might have jumped out at you is

00:01:40.751 --> 00:01:43.694
that exponential form would
be much, much simpler.

00:01:43.694 --> 00:01:46.319
And the way that we convert
this to exponential form

00:01:46.319 --> 00:01:50.992
is to recognize that this
business right over here

00:01:50.992 --> 00:01:53.185
is the same thing, this comes straight out

00:01:53.185 --> 00:02:01.933
of Euler's formula, this is
e, to the seven pi over six i.

00:02:01.933 --> 00:02:05.243
Seven pi over six i.

00:02:05.243 --> 00:02:07.276
That's this expression right over here.

00:02:07.276 --> 00:02:10.272
So this entire top complex
number can be rewritten

00:02:10.272 --> 00:02:17.796
as seven e, to the seven pi over six i.

00:02:17.796 --> 00:02:21.682
And this bottom complex
number can be rewritten

00:02:21.682 --> 00:02:27.244
as one times, we didn't
really have to write the one,

00:02:27.244 --> 00:02:29.043
but this bottom part
right over here is going

00:02:29.043 --> 00:02:33.326
to be the same thing as e,
to the seven pi over four i.

00:02:33.326 --> 00:02:36.960
Seven pi over four i,

00:02:36.960 --> 00:02:40.046
and this comes straight
out of Euler's formula.

00:02:40.046 --> 00:02:41.854
And when you write it in
this way, then we can just

00:02:41.854 --> 00:02:44.995
use exponent properties to simplify it.

00:02:44.995 --> 00:02:48.270
We have the same base, and
so we can just subtract

00:02:48.270 --> 00:02:51.006
this exponent from that
exponent right over there.

00:02:51.006 --> 00:02:53.101
So this is going to be equal to seven

00:02:53.101 --> 00:02:55.676
divided by one, is just seven.

00:02:55.676 --> 00:03:01.417
So it's going to be seven
to the, or seven times e.

00:03:01.417 --> 00:03:03.371
Let me do this in a brighter color.

00:03:03.371 --> 00:03:11.576
Seven times e, to the seven pi over six i,

00:03:11.576 --> 00:03:16.158
minus seven pi over four i.

00:03:16.158 --> 00:03:21.577
Minus seven pi over four ith power.

00:03:21.577 --> 00:03:23.776
So what is this going to be equal to?

00:03:23.776 --> 00:03:25.508
Well if I have seven pi
over six of something

00:03:25.508 --> 00:03:27.722
and I subtract seven pi
over four of that thing,

00:03:27.722 --> 00:03:29.585
how many do I have left over?

00:03:29.585 --> 00:03:31.807
Well, let's see, I'm essentially -

00:03:31.807 --> 00:03:34.912
let's just rewrite all of this.

00:03:34.912 --> 00:03:35.843
This is really just about

00:03:35.843 --> 00:03:37.907
subtracting fractions at this point.

00:03:37.907 --> 00:03:39.412
So if I were to rewrite this, let's see,

00:03:39.412 --> 00:03:41.874
if I were to rewrite it
with a denominator of 12,

00:03:41.874 --> 00:03:43.415
then I'll have a common denominator.

00:03:43.415 --> 00:03:45.843
It's my least common
multiple of six and four.

00:03:45.843 --> 00:03:50.748
So this one I can rewrite
as 14 pi i over 12.

00:03:50.748 --> 00:04:00.323
Or I can write it as 14 pi
i over 12, and then minus -

00:04:00.323 --> 00:04:01.654
so I multiplied the
numerator and denominator

00:04:01.654 --> 00:04:13.795
by three, minus 21 pi i over 12, as well.

00:04:13.795 --> 00:04:14.927
Right? I just multiplied numerators

00:04:14.927 --> 00:04:17.258
and denominators by three right over here.

00:04:17.258 --> 00:04:19.317
And so this is going to be equal to,

00:04:19.317 --> 00:04:22.234
I'm going to have, 12 in the denominator,

00:04:22.234 --> 00:04:25.923
14 pi i, minus 21 pi i,

00:04:25.923 --> 00:04:33.013
is going to be negative seven pi i.

00:04:33.013 --> 00:04:38.879
So, this thing is equal to seven e,

00:04:38.879 --> 00:04:46.872
to the negative seven pi i, over 12.

00:04:46.872 --> 00:04:51.486
Now let's see if we can do a
decent job of plotting this.

00:04:51.486 --> 00:04:53.051
So let's see, each of these increments

00:04:53.051 --> 00:04:55.354
that they've done, in
each of these quadrants,

00:04:55.354 --> 00:04:59.098
we have one, two, three, four, five, six.

00:04:59.098 --> 00:05:03.032
So they split this quadrant
into six equal angles,

00:05:03.032 --> 00:05:06.156
and this quadrant is pi over two,

00:05:06.156 --> 00:05:09.603
so each of these are pi over 12.

00:05:09.603 --> 00:05:13.972
Each of these little
subangles are pi over 12.

00:05:13.972 --> 00:05:17.715
So we're going to have
negative seven pi over 12.

00:05:17.715 --> 00:05:19.370
So we're going to go in
the negative direction.

00:05:19.370 --> 00:05:21.810
We're going to go
counter-clockwise, or sorry,

00:05:21.810 --> 00:05:23.025
we're going to go clockwise.

00:05:23.025 --> 00:05:24.896
So we're going to go, let me start here,

00:05:24.896 --> 00:05:32.887
so if I go one, two, three,
four, five, six, seven,

00:05:32.887 --> 00:05:41.885
eight, nine, 10, 11, 12,
gets us right over there.

00:05:41.885 --> 00:05:45.169
Did I do that right? That
doesn't - let me make sure

00:05:45.169 --> 00:05:48.247
I got that - negative seven,

00:05:48.247 --> 00:05:52.106
so each of these are pi
over - let me make sure.

00:05:52.106 --> 00:05:55.706
So this whole thing is
pi, so we have one -

00:05:55.706 --> 00:05:58.611
let me make sure, this whole thing is pi.

00:05:58.611 --> 00:06:01.257
We have one, two, three, four, five, six,

00:06:01.257 --> 00:06:03.482
seven, eight, nine, 10, 11, 12.

00:06:03.482 --> 00:06:04.858
So each of these are a twelfth of that,

00:06:04.858 --> 00:06:06.361
so it's pi over 12, that's right.

00:06:06.361 --> 00:06:08.322
Then we want to go negative seven of them.

00:06:08.322 --> 00:06:10.743
Negative one, negative
two, negative three,

00:06:10.743 --> 00:06:12.444
negative four, negative
five, negative six,

00:06:12.444 --> 00:06:14.890
negative seven, oh ya, I just kept going

00:06:14.890 --> 00:06:16.976
all the way to negative 12 the last time.

00:06:16.976 --> 00:06:22.961
So this is the angle, this
is our angle right over here.

00:06:22.961 --> 00:06:26.049
And our distance that we
go out from the origin

00:06:26.049 --> 00:06:31.100
is seven, so we go out
one, two, three, four,

00:06:31.100 --> 00:06:37.629
five, six, seven, so we
come out right over there.

00:06:37.629 --> 00:06:39.374
So this complex number divided by that

00:06:39.374 --> 00:06:42.749
complex number is equal
to this complex number,

00:06:42.749 --> 00:06:46.790
seven times e, to the
negative seven pi i over 12.

00:06:46.790 --> 00:06:48.419
And if we wanted to now write this

00:06:48.419 --> 00:06:50.531
in polar form, we of course could.

00:06:50.531 --> 00:06:51.971
We could say that this is the same thing

00:06:51.971 --> 00:06:58.757
as seven, times cosine of
negative seven pi over 12,

00:06:58.757 --> 00:07:08.263
plus i sine of negative seven pi over 12.


Rectangular to polar form of complex number

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:03.365
Let's say that I have the complex number z

00:00:03.365 --> 00:00:05.581
and in rectangular form we can write it

00:00:05.581 --> 00:00:12.348
as negative three plus two i.

00:00:12.348 --> 00:00:14.837
So first let's think about where this is

00:00:14.837 --> 00:00:16.532
on the complex plane.

00:00:16.532 --> 00:00:26.324
So, this is our imaginary axis
and that is our real axis.

00:00:28.000 --> 00:00:30.551
And let's see, the real
part is negative three,

00:00:30.551 --> 00:00:35.726
so we could go one, two, three
to the left of the origin.

00:00:35.726 --> 00:00:37.432
Let me space that out a
little bit more evenly.

00:00:37.432 --> 00:00:42.416
So one, two, three to
the left of the origin.

00:00:42.416 --> 00:00:44.283
So negative three.

00:00:44.283 --> 00:00:45.778
And then we have two i's.

00:00:45.778 --> 00:00:48.007
We're going to go up two.

00:00:48.007 --> 00:00:50.880
So up two in the imaginary direction.

00:00:50.880 --> 00:00:53.713
So one, two just like that.

00:00:53.713 --> 00:00:57.434
So z is real part negative three,

00:00:57.434 --> 00:00:59.920
imaginary part two.

00:00:59.920 --> 00:01:04.864
z would be right over
here in the complex plane.

00:01:04.864 --> 00:01:08.424
So that is z right over there.

00:01:08.424 --> 00:01:11.048
Now what I want to think
about are other ways

00:01:11.048 --> 00:01:14.777
to essentially specify the location of z.

00:01:14.777 --> 00:01:16.384
And in particular,

00:01:16.384 --> 00:01:19.112
instead of giving the
real and imaginary parts,

00:01:19.112 --> 00:01:20.481
essentially the coordinates here,

00:01:20.481 --> 00:01:22.976
let's think about giving a direction

00:01:22.976 --> 00:01:24.816
and a distance to get to z.

00:01:24.816 --> 00:01:26.825
So for example we could give

00:01:26.825 --> 00:01:29.033
the distance from the origin to z,

00:01:29.033 --> 00:01:35.537
so let's call this distance r,

00:01:35.537 --> 00:01:38.480
but that distance by itself
isn't enough to say where z is.

00:01:38.480 --> 00:01:40.539
You have to say in what
direction do you have

00:01:40.539 --> 00:01:43.209
to go a distance of r to get to z.

00:01:43.209 --> 00:01:47.643
and so to specify the direction,

00:01:47.720 --> 00:01:52.406
we will have this angle,
theta, in radians,

00:01:52.406 --> 00:01:54.422
this angle between the positive real axis

00:01:54.422 --> 00:01:57.245
and this line right over here,

00:01:57.245 --> 00:02:02.190
this line or the segment that
connects the origin and z.

00:02:02.190 --> 00:02:05.086
So if someone gave you this
angle and this distance,

00:02:05.086 --> 00:02:07.478
then you could get to z.

00:02:07.478 --> 00:02:09.605
Now what I want you to do right now

00:02:09.605 --> 00:02:13.885
is pause this video and see
if you can find a relationship

00:02:13.885 --> 00:02:17.477
between r theta and
negative three and two.

00:02:17.477 --> 00:02:20.894
In fact, given this complex
number in rectangular form,

00:02:20.894 --> 00:02:25.062
can you figure out what r and theta are?

00:02:25.062 --> 00:02:28.989
Well let's think through
this a little bit,

00:02:28.989 --> 00:02:31.974
and to help us let's remind ourselves

00:02:31.974 --> 00:02:33.981
of the unit circle
definition of trig functions

00:02:33.981 --> 00:02:35.765
because we are going to
use some trig functions

00:02:35.765 --> 00:02:38.814
to relate r, theta, and
two, and negative three.

00:02:38.814 --> 00:02:41.567
So I'm going to construct
a unit circle here.

00:02:41.567 --> 00:02:45.253
So construct a unit circle.

00:02:45.253 --> 00:02:47.061
So this right here's a unit circle,

00:02:47.061 --> 00:02:48.653
a circle of radius one.

00:02:48.653 --> 00:02:52.110
So construct a unit circle.

00:02:52.110 --> 00:02:54.222
That's a unit circle.

00:02:54.222 --> 00:02:57.860
So by definition what are the horizontal

00:02:57.860 --> 00:03:01.693
and vertical coordinates of
this point right over here

00:03:01.693 --> 00:03:05.677
where this line intersects
the unit circle?

00:03:05.677 --> 00:03:08.669
Well, this is forming an angle of theta

00:03:08.669 --> 00:03:10.262
with a positive real axis

00:03:10.262 --> 00:03:13.373
and so the horizontal coordinate over here

00:03:13.373 --> 00:03:18.013
by definition is going
to be cosine of theta.

00:03:18.013 --> 00:03:20.118
Cosine of theta.

00:03:20.118 --> 00:03:22.109
That's the unit circle
definition of cosine of theta,

00:03:22.109 --> 00:03:26.262
and the vertical coordinate
is going to be sine of theta.

00:03:26.262 --> 00:03:27.997
Sine of theta.

00:03:27.997 --> 00:03:32.853
And so what would the horizontal

00:03:32.853 --> 00:03:34.773
and vertical coordinates of this point be?

00:03:34.773 --> 00:03:36.694
We obviously know they're
negative three and two,

00:03:36.694 --> 00:03:38.070
but what would they be in terms

00:03:38.070 --> 00:03:39.189
of cosine theta and sine theta?

00:03:39.189 --> 00:03:42.444
Well look, this point right over here

00:03:42.444 --> 00:03:45.973
is a radius of one away from the origin.

00:03:45.973 --> 00:03:50.886
So this distance right over here is one,

00:03:50.886 --> 00:03:53.733
but now we are r away from the origin.

00:03:53.733 --> 00:03:55.781
We're r times as far.

00:03:55.781 --> 00:03:58.213
So if we're r times as
far in that direction,

00:03:58.213 --> 00:03:59.631
then we're going to be r times as far

00:03:59.631 --> 00:04:01.501
in the vertical direction and r times

00:04:01.501 --> 00:04:03.205
as far in the horizontal direction.

00:04:03.205 --> 00:04:06.981
We're going to scale everything by r.

00:04:06.981 --> 00:04:12.510
So the horizontal coordinate
of this point right here

00:04:12.510 --> 00:04:13.942
instead of being cosine of theta

00:04:13.942 --> 00:04:16.413
is going to be r times cosine of theta.

00:04:16.413 --> 00:04:18.254
So this point right over here,

00:04:18.254 --> 00:04:19.406
which we know is negative three,

00:04:19.406 --> 00:04:22.461
is going to be equal to r cosine of theta,

00:04:22.461 --> 00:04:24.686
and by the same logic,
this point over here,

00:04:24.686 --> 00:04:25.878
the vertical coordinate,

00:04:25.878 --> 00:04:29.894
we're going to scale up sine
theta by r, we're r times as far.

00:04:29.894 --> 00:04:31.392
So this point right over here is going

00:04:31.392 --> 00:04:33.660
to be r sine theta and we already know

00:04:33.660 --> 00:04:35.922
that that's equal to two.

00:04:35.922 --> 00:04:38.562
r sine theta.

00:04:38.562 --> 00:04:42.771
So given that, can we now
figure out what r and theta are?

00:04:42.771 --> 00:04:48.618
So let's first think about
figuring out what theta is.

00:04:48.618 --> 00:04:55.683
So to do that, let's think about
some of our trig functions.

00:04:55.683 --> 00:04:57.777
So one trig function that involves

00:04:57.777 --> 00:05:02.186
sine theta and cosine
theta is tangent theta.

00:05:02.186 --> 00:05:05.041
So for example we could say tangent theta,

00:05:05.041 --> 00:05:10.497
tangent of our angle, tangent of theta,

00:05:10.497 --> 00:05:20.238
is equal to sine of theta
over cosine of theta.

00:05:20.238 --> 00:05:23.000
We could also multiply the numerator

00:05:23.000 --> 00:05:24.192
and denominator here by r.

00:05:24.192 --> 00:05:26.401
That won't change the value.

00:05:26.401 --> 00:05:30.328
So that's the same thing as r
sine theta over r cosine theta

00:05:30.328 --> 00:05:33.432
and we know r sine theta
is going to be equal to two

00:05:33.432 --> 00:05:37.584
and we know that r cosine
theta is negative three.

00:05:37.584 --> 00:05:41.736
So this whole thing is
going to be negative 2/3.

00:05:41.736 --> 00:05:44.440
Another way of thinking
about it is the tangent

00:05:44.440 --> 00:05:48.809
of theta is going to be the same thing

00:05:48.809 --> 00:05:53.791
as the slope of this line right over here.

00:05:53.791 --> 00:05:55.696
And what is the slope of that line?

00:05:55.696 --> 00:05:57.743
Well if you start at z and
you want to go to the origin

00:05:57.743 --> 00:06:02.521
you're going to go positive
three in the x direction

00:06:02.521 --> 00:06:05.493
and then you're going to go negative two,

00:06:05.493 --> 00:06:07.942
or I should say positive three
in the horizontal direction,

00:06:07.942 --> 00:06:10.695
and then you go negative two
in the vertical direction.

00:06:10.695 --> 00:06:11.957
So the slope is your change

00:06:11.957 --> 00:06:13.766
in vertical over change in horizontal.

00:06:13.766 --> 00:06:17.766
It's negative two over three.

00:06:17.766 --> 00:06:20.282
But now we can use this
to solve for theta.

00:06:20.282 --> 00:06:21.985
To solve for theta, we just
take the inverse tangent

00:06:21.985 --> 00:06:25.514
of both sides and we get theta is equal to

00:06:25.514 --> 00:06:29.635
inverse tangent of negative 2/3,

00:06:29.635 --> 00:06:32.274
of this thing right over here.

00:06:32.274 --> 00:06:36.273
So we copy and paste that and we can get

00:06:36.273 --> 00:06:38.370
our calculator out to figure
out what this actually is.

00:06:38.370 --> 00:06:40.674
So let's turn it on.

00:06:40.674 --> 00:06:43.490
Let me make sure that I am in radian mode.

00:06:43.490 --> 00:06:44.353
I am.

00:06:44.353 --> 00:06:46.618
and so I can take the inverse tangent

00:06:46.618 --> 00:06:50.545
of negative two divided by three,

00:06:50.545 --> 00:06:56.715
which gets us negative point
five eight eight zero zero,

00:06:56.715 --> 00:06:57.689
on and on and on.

00:06:57.689 --> 00:06:59.497
So if we round to the nearest
hundredth of a radian,

00:06:59.497 --> 00:07:02.810
it would be negative point five nine.

00:07:02.810 --> 00:07:10.122
So this is approximately
negative zero point five nine.

00:07:10.122 --> 00:07:12.026
Now is this the right angle?

00:07:12.026 --> 00:07:15.849
Is this the theta that we are looking for?

00:07:15.849 --> 00:07:19.026
Well this theta, negative
zero point five nine,

00:07:19.026 --> 00:07:21.681
that's going to get us over here.

00:07:21.681 --> 00:07:23.794
That's this angle right over here.

00:07:23.794 --> 00:07:27.946
That's what the inverse tangent gave us,

00:07:27.946 --> 00:07:30.865
and it makes sense because this ray is

00:07:30.865 --> 00:07:33.017
a continuation of this
ray right over here.

00:07:33.017 --> 00:07:34.626
Together they would form a line.

00:07:34.626 --> 00:07:35.954
It has the same slope.

00:07:35.954 --> 00:07:38.001
But that's not the theta
that we are looking for.

00:07:38.001 --> 00:07:39.866
The theta that we are looking for

00:07:39.866 --> 00:07:41.706
is going in the opposite direction.

00:07:41.706 --> 00:07:44.217
It's half way around the circle.

00:07:44.217 --> 00:07:46.265
So if we think in terms of radians,

00:07:46.265 --> 00:07:51.777
it's going to be this
angle plus pi radians.

00:07:51.777 --> 00:07:56.096
So the theta that we're looking for isn't

00:07:56.096 --> 00:07:58.429
going to be what our
calculator gives us for this.

00:07:58.429 --> 00:08:01.853
It's going to be this value plus pi.

00:08:01.853 --> 00:08:03.948
So theta, the theta that we care about,

00:08:03.948 --> 00:08:10.261
is going to be negative zero
point five nine, roughly, plus pi.

00:08:10.261 --> 00:08:11.605
I'm going to say approximately just

00:08:11.605 --> 00:08:13.413
because I have rounded right over here.

00:08:13.413 --> 00:08:16.021
So let's get the calculator out.

00:08:16.021 --> 00:08:18.981
So let's take our previous response,

00:08:18.981 --> 00:08:21.389
and that's just the previous answer,

00:08:21.389 --> 00:08:26.029
plus pi because we're going to
go in the opposite direction.

00:08:26.029 --> 00:08:30.037
So that gives us roughly
two point five five radians.

00:08:30.037 --> 00:08:35.101
So this is approximately
two point five five radians

00:08:35.101 --> 00:08:38.493
is our claim for what theta is.

00:08:38.493 --> 00:08:39.724
Now does that make sense,

00:08:39.724 --> 00:08:45.709
that this angle is two
point five five radians?

00:08:45.709 --> 00:08:49.220
Well, this angle if we go straight up,

00:08:49.220 --> 00:08:51.396
that's pi over two radians,

00:08:51.396 --> 00:08:52.797
and pi is three point one four,

00:08:52.797 --> 00:08:57.525
so that's going to be
one point five seven,

00:08:57.525 --> 00:08:59.613
something like that,

00:08:59.613 --> 00:09:01.469
and this is pi radians,

00:09:01.469 --> 00:09:04.396
so that's three point
one four one five nine,

00:09:04.396 --> 00:09:05.708
keep going on and on and on.

00:09:05.708 --> 00:09:08.493
So two point five nine is
indeed in the right quadrant.

00:09:08.493 --> 00:09:09.732
So that is right.

00:09:09.732 --> 00:09:13.125
This is going to be an angle
of two point five five radians.

00:09:13.125 --> 00:09:16.565
Now we just have to figure
out the length, what r is.

00:09:16.565 --> 00:09:18.900
And that we can just use
the Pythagorean theorem for.

00:09:18.900 --> 00:09:24.121
We can set up a right triangle.

00:09:24.121 --> 00:09:29.451
We know that this distance
right over here is two

00:09:29.451 --> 00:09:32.060
because this distance
right over here is two,

00:09:32.060 --> 00:09:37.084
and we know that this distance is,

00:09:37.084 --> 00:09:38.123
the coordinate here is negative three,

00:09:38.123 --> 00:09:39.952
but the distance here is just three.

00:09:39.952 --> 00:09:42.147
So we know from the Pythagorean theorem

00:09:42.147 --> 00:09:47.621
that r squared is equal to two squared

00:09:47.621 --> 00:09:54.311
plus three squared or that r is

00:09:54.311 --> 00:09:59.255
equal to the square root of
four plus nine, which is 13.

00:09:59.255 --> 00:10:00.527
And so there we have it.

00:10:00.527 --> 00:10:02.584
We have negative three,

00:10:02.584 --> 00:10:05.369
so we know r is equal
to the square root of 13

00:10:05.376 --> 00:10:07.608
and theta is two point five five radians.

00:10:07.608 --> 00:10:09.051
So using that information,

00:10:09.051 --> 00:10:14.101
we can now rewrite z in
terms of r and and our theta.

00:10:14.175 --> 00:10:17.904
So now we can say that z is equal to,

00:10:17.919 --> 00:10:21.652
instead of negative 3, we can
write it as r cosine theta,

00:10:22.217 --> 00:10:24.122
so we can write it as the square root

00:10:24.122 --> 00:10:33.363
of thirteen times the cosine of theta,

00:10:33.363 --> 00:10:36.747
which is, we're saying theta
is roughly two point five five

00:10:36.747 --> 00:10:44.251
if we round to the nearest
hundredth, plus two,

00:10:44.251 --> 00:10:45.972
we already know is r sine of theta

00:10:45.972 --> 00:10:49.339
is the square root of thirteen,

00:10:49.339 --> 00:10:55.986
and then sine of theta,
two point five five,

00:10:55.986 --> 00:11:00.034
and then all of this is going

00:11:00.172 --> 00:11:01.800
to be multiplied times i,

00:11:01.800 --> 00:11:04.208
and if we like to
simplify it a little bit,

00:11:04.208 --> 00:11:08.239
or kind of really make
clear what the r is here,

00:11:08.239 --> 00:11:09.565
we could factor it out.

00:11:09.565 --> 00:11:14.327
So we could say that z is
equal to the square root

00:11:14.327 --> 00:11:19.678
of 13 times cosine of two point five five,

00:11:19.678 --> 00:11:22.230
and once again two point five
five is an approximation,

00:11:22.230 --> 00:11:25.566
so maybe I should say
approximately equal to,

00:11:25.566 --> 00:11:35.587
cosine of two point five five plus

00:11:35.633 --> 00:11:41.287
i times sine of two point five five.

00:11:41.287 --> 00:11:44.159
And so when I rewrote z in
this format right over here,

00:11:44.159 --> 00:11:46.840
it makes it much clearer what

00:11:46.840 --> 00:11:48.511
the direction I have to go in is,

00:11:48.511 --> 00:11:51.287
the direction is two
point five five radians

00:11:51.287 --> 00:11:53.944
counterclockwise from
the positive real axis.

00:11:53.944 --> 00:11:55.864
It also says how far I need to go,

00:11:55.864 --> 00:11:58.359
I need to go square root of 13.

00:11:58.359 --> 00:12:00.721
This essentially makes the polar,

00:12:00.721 --> 00:12:05.184
it makes it clearer how we
get there in kind of a more,

00:12:05.184 --> 00:12:06.887
I guess you could say, polar mindset,

00:12:06.887 --> 00:12:09.647
and that's why this form
of the complex number,

00:12:09.647 --> 00:12:12.071
writing it this way is
called rectangular form,

00:12:12.071 --> 00:12:15.538
while writing it this
way is called polar form.


Introduction to ratios

WEBVTT
Kind: captions
Language: en

00:00:00.567 --> 00:00:01.900
Voiceover:We've got some apples here

00:00:01.900 --> 00:00:03.433
and we've got some oranges and what

00:00:03.433 --> 00:00:06.733
I want to think about
is, what is the ratio,

00:00:07.100 --> 00:00:11.900
what is the ratio of apples ...

00:00:11.900 --> 00:00:15.000
Of apples, to oranges?

00:00:15.633 --> 00:00:17.333
To oranges.

00:00:18.633 --> 00:00:21.566
To clarify what we're even talking about,

00:00:21.566 --> 00:00:24.033
a ratio is giving us the relationship

00:00:24.033 --> 00:00:26.734
between quantities of 2 different things.

00:00:27.234 --> 00:00:28.500
So there's a couple of ways that we

00:00:28.500 --> 00:00:29.800
can specify this.

00:00:29.967 --> 00:00:32.200
We can literally count
the number of apples.

00:00:32.233 --> 00:00:34.200
1, 2, 3, 4, 5, 6.

00:00:34.566 --> 00:00:36.400
So we have 6 apples.

00:00:37.033 --> 00:00:39.033
And we can say the ratio is going to be

00:00:39.033 --> 00:00:43.134
6 to, 6 to ...

00:00:43.166 --> 00:00:44.733
And then how many oranges do we have?

00:00:44.766 --> 00:00:48.600
1, 2, 3, 4, 5, 6, 7, 8, 9.

00:00:48.900 --> 00:00:50.733
It is 6 to 9.

00:00:51.100 --> 00:00:55.266
The ratio of apples to oranges is 6 to 9.

00:00:55.792 --> 00:00:57.500
And you could use a different notation.

00:00:57.500 --> 00:00:58.933
You could also write it this way.

00:00:59.166 --> 00:01:00.966
6 to ...

00:01:00.966 --> 00:01:05.133
You would still read the
ratio as being 6 to 9.

00:01:05.766 --> 00:01:07.833
But we don't have to
just satisfy us with this

00:01:07.833 --> 00:01:09.934
because one way to think about ratios,

00:01:10.100 --> 00:01:11.766
especially if we're thinking about apples

00:01:11.766 --> 00:01:14.700
to oranges, is how many apples do we have

00:01:14.700 --> 00:01:16.533
for a certain number of oranges?

00:01:16.766 --> 00:01:18.100
When you think about it that way, we can

00:01:18.100 --> 00:01:20.566
actually reduce these
numbers, as you might

00:01:20.566 --> 00:01:22.466
have already thought about.

00:01:22.633 --> 00:01:26.600
Both 6 and 9 are divisible by 3.

00:01:26.967 --> 00:01:29.300
So just like we can reduce fractions,

00:01:29.300 --> 00:01:31.267
we can also reduce ratios.

00:01:31.433 --> 00:01:35.100
So if you divide 6 and 9 both by 3.

00:01:35.100 --> 00:01:37.001
6 divided by 3 is 2.

00:01:37.500 --> 00:01:39.800
6 divided by 3 is 2.

00:01:40.033 --> 00:01:42.733
And 9 divided by 3 is 3.

00:01:43.100 --> 00:01:44.833
So we could also say that the ratio

00:01:44.833 --> 00:01:48.000
of apples to oranges is 2 to 3.

00:01:48.300 --> 00:01:50.034
Or if we want to use this notation,

00:01:50.034 --> 00:01:54.266
2 to 3.

00:01:54.900 --> 00:01:56.134
2 to 3.

00:01:56.166 --> 00:01:57.233
Now, does that make sense?

00:01:57.233 --> 00:01:58.400
Well look.

00:01:58.967 --> 00:02:00.733
We divided each of these groups into 3.

00:02:01.100 --> 00:02:02.301
So one way to think about it ...

00:02:02.301 --> 00:02:06.400
If you divide this whole
total into 3 groups.

00:02:06.966 --> 00:02:09.866
So 1 group, 1 group.

00:02:10.433 --> 00:02:13.066
2 groups, 2 groups.

00:02:14.366 --> 00:02:16.266
And 3 equal groups.

00:02:16.900 --> 00:02:19.233
We see that in each of those groups,

00:02:19.233 --> 00:02:22.367
for every 2 apples, for every 2 apples,

00:02:22.367 --> 00:02:23.734
we have 3 oranges.

00:02:23.900 --> 00:02:26.066
For 2 apples we have 3 oranges.

00:02:26.233 --> 00:02:29.267
For 2 apples we have 3 oranges.

00:02:29.700 --> 00:02:32.300
So, once again, the ratio
of apples to oranges.

00:02:32.300 --> 00:02:35.000
For every 2 apples we have 3 oranges.

00:02:35.366 --> 00:02:36.767
But we could think about
things the other way

00:02:36.767 --> 00:02:38.000
around as well.

00:02:38.167 --> 00:02:40.667
We could also think about
what is the ratio ...

00:02:41.366 --> 00:02:44.066
We could also think about
what is the ratio ...

00:02:45.700 --> 00:02:48.266
Ratio, of oranges to apples?

00:02:48.700 --> 00:02:52.200
Oranges to apples.

00:02:52.566 --> 00:02:54.233
And here we would, essentially, switch

00:02:54.233 --> 00:02:55.266
the numbers.

00:02:55.833 --> 00:03:00.466
The ratio of oranges to apples.

00:03:00.633 --> 00:03:03.433
Notice, up here we said apples to oranges

00:03:03.433 --> 00:03:07.167
which is 6 to 9 or 2
to 3 if we reduce them.

00:03:07.167 --> 00:03:08.633
And here we're going to say the ratio of

00:03:08.633 --> 00:03:11.000
oranges to apples, so
we've swapped these 2.

00:03:11.100 --> 00:03:12.600
So we're going to swap the numbers.

00:03:13.033 --> 00:03:15.400
Here we have 9 oranges for every 6 apples.

00:03:15.633 --> 00:03:21.200
So we could say the ratio
is going to be 9 to 6.

00:03:22.033 --> 00:03:23.734
The ratio is 9 to 6.

00:03:24.033 --> 00:03:26.500
Or if we want to reduce
it, for every 3 oranges ...

00:03:26.500 --> 00:03:28.466
So we're going to divide this by 3.

00:03:28.500 --> 00:03:30.766
So for every 3 oranges we are going

00:03:30.766 --> 00:03:33.600
to have 2 apples.

00:03:34.100 --> 00:03:36.466
We are going to have 2 apples.

00:03:36.500 --> 00:03:38.233
So notice, this is just
exactly what we had

00:03:38.233 --> 00:03:41.300
up here, but when we had apples to oranges

00:03:41.300 --> 00:03:42.666
it was 6 to 9.

00:03:42.700 --> 00:03:44.734
6 apples for every 9 oranges.

00:03:44.833 --> 00:03:47.033
And now when it's
oranges to apples, we say

00:03:47.033 --> 00:03:48.466
it's 9 to 6.

00:03:48.568 --> 00:03:50.600
9 oranges for every 6 apples.

00:03:50.700 --> 00:03:53.766
Or we could say for
every 3 oranges we have

00:03:53.766 --> 00:03:56.533
exactly 2 apples.


Ratios as fractions

WEBVTT
Kind: captions
Language: en

00:00:00.100 --> 00:00:01.567
Voiceover:Let's think
about another scenario

00:00:01.567 --> 00:00:02.766
involving ratios.

00:00:02.766 --> 00:00:04.863
In this case, let's think about the ratio

00:00:05.570 --> 00:00:06.733
of the number of apples.

00:00:07.767 --> 00:00:12.333
Number of apples to ...

00:00:12.700 --> 00:00:14.033
Instead of taking the ratio of the number

00:00:14.033 --> 00:00:15.700
of apples to the number of oranges,

00:00:15.700 --> 00:00:17.567
let's take the number of apples to the

00:00:17.567 --> 00:00:21.333
number of fruit.

00:00:22.033 --> 00:00:25.000
The number of fruit
that we have over here.

00:00:25.033 --> 00:00:26.166
And I encourage you to pause the video

00:00:26.166 --> 00:00:28.067
and think about that on your own.

00:00:28.766 --> 00:00:30.633
Well, how many total apples do we have?

00:00:30.633 --> 00:00:32.934
We have 2, 4, 6, 8 apples.

00:00:33.300 --> 00:00:35.600
So we're going to have 8 apples.

00:00:35.833 --> 00:00:37.533
And then how much total fruit do we have?

00:00:37.700 --> 00:00:39.367
Well we have 8 apples and we have

00:00:39.367 --> 00:00:41.733
3, 6, 9, 12 oranges.

00:00:41.966 --> 00:00:44.333
So our total fruit is 8 plus 12.

00:00:44.366 --> 00:00:45.933
We have 20 pieces of fruit.

00:00:46.366 --> 00:00:49.933
So this ratio is going to be 8 to ...

00:00:50.633 --> 00:00:53.066
8 to 20.

00:00:53.566 --> 00:00:55.833
Or, if we want to write
this in a more reduced form,

00:00:55.833 --> 00:00:59.266
we can divide both of these by 4.

00:00:59.300 --> 00:01:01.600
4 is their greatest common divisor.

00:01:01.700 --> 00:01:04.200
And so this is the same thing as a ratio.

00:01:04.299 --> 00:01:10.666
8 divided by 4 is 2 and
20 divided by 4 is 5.

00:01:11.233 --> 00:01:13.433
So 2 to 5.

00:01:13.433 --> 00:01:14.633
Now, does this make sense?

00:01:14.633 --> 00:01:16.333
Well, if we divide ...

00:01:16.500 --> 00:01:19.333
If we divide everything into groups of 4.

00:01:20.433 --> 00:01:21.433
So ...

00:01:21.433 --> 00:01:23.133
Or if we divide into 4
groups, I should say.

00:01:23.300 --> 00:01:31.266
So 1 group, 2 groups,
3 groups, and 4 groups.

00:01:31.633 --> 00:01:33.100
That's the largest number of groups

00:01:33.100 --> 00:01:35.566
that we can divide these into

00:01:35.566 --> 00:01:37.233
so that we don't have to cut up the apples

00:01:37.233 --> 00:01:38.466
or the oranges.

00:01:38.566 --> 00:01:41.767
We see that in each
group, for every 2 apples

00:01:41.767 --> 00:01:45.534
we have 1, 2, 3, 4, 5 pieces of fruit.

00:01:45.900 --> 00:01:48.466
For every 2 apples we
have 5 pieces of fruit.

00:01:49.033 --> 00:01:50.433
This is actually a good opportunity for us

00:01:50.433 --> 00:01:53.866
to introduce another
way of representing ...

00:01:54.900 --> 00:01:56.700
Another way of representing
ratios, and that's

00:01:56.700 --> 00:01:58.467
using fraction notation.

00:01:58.833 --> 00:02:02.233
So we could also represent this ratio as

00:02:02.233 --> 00:02:04.601
2 over 5.

00:02:05.366 --> 00:02:08.200
As the fraction 2 over 5.

00:02:08.367 --> 00:02:10.166
Whenever we put it in
the fraction it's very

00:02:10.166 --> 00:02:13.267
important to recognize
what this represents.

00:02:13.433 --> 00:02:16.433
This is telling us the fraction of fruit

00:02:16.433 --> 00:02:18.133
that are apples.

00:02:18.233 --> 00:02:21.867
So we could say 2/5 of the fruit ...

00:02:22.700 --> 00:02:24.267
Of the fruit ...

00:02:25.100 --> 00:02:27.100
Of the number of fruit,
I guess I could say.

00:02:27.100 --> 00:02:28.668
Of number of fruit ...

00:02:29.500 --> 00:02:32.333
Of fruit is equal to the number of apples.

00:02:32.500 --> 00:02:34.033
Right, I'm just going to say 2/5 of fruit

00:02:34.033 --> 00:02:36.201
if we're just speaking
in more typical terms.

00:02:36.766 --> 00:02:40.467
2/5 of fruit are apples.

00:02:41.233 --> 00:02:45.333
Are, are apples.

00:02:45.900 --> 00:02:47.367
So, once again, this is introducing

00:02:47.367 --> 00:02:50.001
another way of representing ratios.

00:02:50.033 --> 00:02:52.633
We could say that the
ratio of apples to fruit,

00:02:52.633 --> 00:02:55.666
once again, it could be 2 to 5 like that.

00:02:56.100 --> 00:02:57.900
It could be 2, instead of putting

00:02:57.900 --> 00:02:59.300
this little colon there we could literally

00:02:59.300 --> 00:03:01.600
write out the word to.

00:03:01.966 --> 00:03:03.066
2 to 5.

00:03:03.500 --> 00:03:05.967
Or we could say it's
2/5, the fraction 2/5,

00:03:05.967 --> 00:03:09.400
which would sometimes be read as 2 to 5.

00:03:09.500 --> 00:03:11.501
This is also, when it's written this way,

00:03:11.501 --> 00:03:13.366
you could also read that as a ratio,

00:03:13.366 --> 00:03:14.566
depending on the context.

00:03:14.566 --> 00:03:16.166
In a sentence like this
I would read this as

00:03:16.166 --> 00:03:20.467
2/5 of the fruit are apples.