Alright guys, we got a lot to get to in this video we are
Introducing complex numbers get pumped. All right. So this is lesson number four in my unit
I believe eight lesson sequence that changes from year to year, but we're looking at the the polar unit
You'll see why in a second why this is in the polar unit
If you want to check this stuff out in the textbook before we get started or after we finish go ahead and look at pages
157-164 and 508-511. Oh those two spots in the text with that kind of address this topic. So umm...
Just because I'm doing so much in this video
And what are we gonna get to in this video? We are going to look at a new type of number
And then we're going to look at the basic operations: addition, subtraction
multiplication and division with those new types of numbers
And then we're going to look at the two different forms the rectangular form and the polar form of a complex number
A lot of this probably doesn't make sense yet, unless you of course you've talked about this with me before
But uh, it will make sense by the end of the video so without further ado, let's go ahead and jump right in
The first thing I want to do is I want to take you back to
elementary school
Maybe middle school, it kind of depends on when you started doing algebra, but I want to answer the question
What number one squared?
Gives you the number nine and
A long long time ago, forget algebra, for a second. If you actually take yourself back to elementary school when you were first learning numbers
there was only one solution to this and
That solution was a three because when you were that young the only numbers you knew about were the natural numbers
But fast-forward take yourself tooo
Late elementary school, early middle school and actually probably saw time entry school
All right. If you were then again, ask the question what number when you multiply it by itself gives you nine
You had been introduced to the
integers
and
You may have said, okay. There are two things that one squared give you nine
Positive three and negative three, right
fast-forward
Now you're in middle school probably and you are introduced to a new type of number what number one squared gives you
Not 9 because 9 is a perfect square
What about 5, what number when squared gives you 5 all right in elementary school, you'd be like nothing
you know 1 times 1, 2 times 2 is 4, 3 times 3 is 9
There's no numbers between 2 & 3 those all I know are numbers that I can count with
So there was no solution to this. But once you got to middle school, you're like, oh there's this a new type of number
There's this number in between
2 & 3 that when you square it gives you 5 and the answer was
positive the square root of 5 or
negative, the square root of 5 all
Right. So we have natural numbers. We have integers. We have what we call the real numbers, right?
These are things that you've become comfortable with right in in elementary school
You didn't you probably were not comfortable or not even aware of these irrational numbers, right, fast forward to now. All right
What if I were to pose to you the question what
when squared gives you a
negative number
All right, we need to make sense of this because there is a solution. All right. It's not a natural number
It's not an integer. It is not a real number
Right what when squared gives you -9. Okay. So I'm gonna leave this page. We're gonna come back to this question
but I want to introduce you to something called
All right, but before I do that, let's jump to the number line
All right, let's give this thing some
Arrows because it goes on forever
Left, I'll go ahead and put 0 in the middle
All right, but we got 0, we got 1, we got 2. We got 3, I'll go up to 4 in both directions
We got -1, -2, -3, -4
And what I want to do is I want to remind you guys what exactly it means to multiply by something
So let's say I have the number one on the number line
If I were to ask you the question, what does it mean to multiply?
This number one by like what is one times I?
Don't know. Let's go with, let's go three. All right, what is one times three?
You may say that 1 times 3 just means well you have one of something, and then you take that group of things and you
Have three groups of that thing so three groups of one or do you think of it as one group of three?
So you'd go out here and say that one times three is three
All right, but the way I actually want you to think about it is imagine you have
An array
and
When you multiply that number by three what you're really doing is think back to geometry, you are dilating that thing
You're dilating the distance. It is from the origin of our number line by three
So here is some sort of distance and when you multiply by three you take that distance and you dilate it
By a scale factor of three that's what the number three does
All right, my follow-up question to that is what if you take the number 1
And you multiply it by
Negative two, what does multiplying by negative two do right. So you have the number one?
and
You can think about this in many different ways, as I said with three
But if I have the number one and I multiply it by negative two
I want to think about that in terms of a dilation as well
so if I take the number one and I dilate it by negative 2 I'm going to
Scale by 2, but think about what that negative does in terms of geometry
All right that negative
Passes you through the origin
To the other side
All right. What was that exactly that multiplied by -2 isn't simply a dilation, but it's a dilation and
a
rotation of
180 degrees, right, multiplying by three is a rotation of zero degrees and a dilation of three multiplying by -2
multiplying by negative a is a dilation of the number and then a rotation of
180 degrees
on the number line, so
It's now time to reveal
What the heck this number I does right so I'm gonna get myself a new number line
I'm gonna give myself a new origin
Myself a 1 give myself a - I'm only gonna go out to 3 this time and then
Go to negative 3. Yeah, you know what let's go over the top. Let's go Igo about 4
All right, cool so I have the number one
What does it mean to multiply by this new number "i," what this number of "i's" function is
it
Dilates by a scale factor of one so I a
Number
That dilates
By a scale factor of 1
And
Rotates, right, -1 rotated 180 degrees. i rotates 90 degrees. All right
Multiplying by i, rotates a number, dilates by one rotates a number... outside of the real number line
90 degrees, sorry
What is that gonna be, counter...
Clockwise
All right, so multiplying by i
dilates by one and rotates 90 degrees counterclockwise
Outside of the real number line. Where does that take us that takes us to a
complex number
line, and that is going to be right
there
All right, so multiplying by i
Takes a ray
of unit one
and rotates it 90 degrees counterclockwise
All right, and that is
What the number i does, so on this complex number line
What is 1 times i, well 1 times anything is that thing. So
1 times i is i
We're gonna put I right here on
This complex number line. All right with that said I'm going to quickly add the
abbreviation "Re," for real and the abbreviation "Im" for
imaginary
All right, that, that word imaginary is kind of kind of a misnomer
These numbers are very real
We just call these numbers the real numbers and these numbers the imaginary numbers
But these have a function they're not made up. They have a real application in mathematics
So we have the number one
Multiplying by I gives you I and that is 90 degrees counterclockwise from one. We are now in the
Complex plane
All right, let's look at a few more of these let me give myself the first quadrant of the complex plane
All right, so let's say I have like the number two
All right, sorry the number two all right, I'm on the real number line
um what happens when I multiply by let's go with like three I
Write 2 times 3i that's going to be
6i, write 2 times 3 is 6 and then the i just hangs out, but what does that mean? Functionally that is saying
dilate
three units
And
rotate
a 90 degrees counter clockwise
Right, so I have one, two, I'm going to have one, two, three, four, five, six
I'm going to be the way up here. You could also do the dilation down here, on two three four five six
So you're going to have some sort of...
Distance here, 2, you're going to dilate that thing by six
And
Then rotate
90 degrees counterclockwise
By CCW, all right, so the final
measurement here is going to be
This number up here, what do we call this number? Well if this is I
Down here, this is going to be 2i, this is going to be 3i, this is going to be 4i
This is going to be 5i and then this will be
6i alright so multiplying by a scalar
multiple of i does the same thing at rotates 90 degrees counterclockwise, but it also scales so three
I also a very usable number has the property that when you
Multiply by 3i, you dilate and rotate. All right. I got one more for you
What happens
When I multiply the number one or any number that is, alright, by i
Twice
All right. So algebraically this is going to be 1 times. I times I that's I squared
Right, but what does this do on the number line? So let's go ahead and look at that
Here's a number line. Okay, here is a complex plane. Alright, well, I'm here
Let me just remind you that this is not the Cartesian coordinate system. This is a, this is a plane a number plane
It's, it's similar to a number line except it has two dimensions
alright, so we have
We have the unit 1 let's go ahead put right there
and
Multiplying by i has the function of rotating 90 degrees. So when I'm whenever I multiply by i
I'm going to rotate 90 degrees
Ok, so one times i is going to be i but it's also going to be up here
Right and what happens when I multiply by i, I'm going to rotate 90 degrees again
Okay, so when i multiply by i, no matter where I am in this plane multiplying by i has the function of a 90-degree rotation
counterclockwise
I'm multiplying by i again gets me over here. And what is that on the real number line?
That's negative 1
Right, wait one second, when you multiply by i twice, when you get i squared that is
functionally equivalent to multiplying by -1 a
180-degree rotation counterclockwise
Alright, so there you have it. This is one of the most important
discoveries that we need to
Address in this video and that is i squared
Functionally equivalent to multiplying by negative one. That's the big idea. I
Squared is equal to
Negative one, right, multiplying by i rotates 90 degrees counterclockwise
Multiplying by -1 rotates a 180 degrees clockwise. So, counterclockwise (error)! So multiplying by i twice
Does that function twice and it's equivalent to negative one? All right, this is the money right here. Let's go back to the original question
All right, what when squared gives you negative nine?
Well multiplying by three
twice
Dilates nine and
multiplying by I
Twice, gives you a rotation of 180 degrees, right? So 3i
Quantity squared is going to give you negative 9. Negative 9 has the function: dilate 9, rotate 180 degrees counterclockwise
3i has the operator... the, the function dilate 3, rotate 90
And if you do that twice
You get a 180 degree rotation, counterclockwise, and a dilation of 9. So x equals 3i is
A solution to this question what when squared gives you negative 9?
What when done twice rotates 180 degrees and dilates 9 3i, all right
And as well, we could just as easily
do this with -3i because -3i is going to
dilate
3 and rotate 180 degrees and then another negative 3 is going to rotate another 180 degrees. So total of 360
That's just a full rotation. So
negative 3 and positive
3i
When squared, gives you negative 9 so we now have a new
Type of number we have the natural numbers. We have the integers. We have the real numbers a long time ago
This is all you had this is all you had
All right, but then we introduced negative numbers. All right, that was completely new to you then
We introduced real numbers the numbers in between
the integers
All right
Not to mention the rational numbers right, fraction were beasts when you were guys were doing them in elementary school. Very difficult!
Now we have a new type of number the complex
Number. All right, that's a capital C boldface
alright, this is a whole new type of number that you are going to be comfortable with and is completely
valid, in terms of mathematics
So I have prepped the following
diagram for you
So the first type of numbers and I'm going to be loose loose about the following set of statements
But the first set of number that you guys learned was the natural numbers you got
You were counting, you were learning how
10 digits on your hands, you're using those to you know, count apples, whatever. Alright, that's the first set of numbers that you learned or
The natural numbers and we use a bold and to describe this
alright numbers include 1 2
My favorite number 47, right? These are numbers that you dealt with. All right the next set of numbers that you learned. All right
Later on in your schooling you learned
The
integers you learned about negative numbers, so you have negative 1
Oh, but you still have one right you 47?
But now you have negative 47 so you introduce the negative numbers
And maybe 0 maybe zeros here, maybe zeros here. It's kind of up to you
That's a whole philosophical question that you can answer on your own time
All right, all right, but then we introduced
The rational numbers right. Worst Q I've ever drawn in my life. All right, we introduced the rational number. So what is that?
That's a some sort of combination of two integers, right 1 over 2. All right
47 over 3, right?
negative
6 over 5, right, but also one was here
All right. Zeros here right negative 1 is here, right the negative numbers are still contained in the rational numbers
Maybe you see what I'm getting at here
Right the next set of numbers that you learned were the real numbers
Right and
What fell in the real numbers? Well, you had um irrational things like PI, things that cannot be expressed
as the quotient between two integers: pi, e
Right, you have the negative of the square root of 5 notice negative numbers still are here
All right, -e is a real number, right?
But one is also still a real number and negative 1 is still a real number. All right,
now we have what we call the complex numbers and we have i.
All right, we have 3i.
Right. This has the function of rotating 90 degrees counterclockwise. This down here, has the function of rotating 90 degrees counterclockwise in the
complex plane, but also dilating 3, right, negative 2i. All right. This has the function of
rotating.
What is that? That's going to be 90 degrees clockwise.
With also a dilation of 2, right, but also one is still contained in here -1 is still contained in here.
These are complex numbers. All right 1 and -1 are complex numbers. All right, i,
3i, -2i, we call those purely imaginary complex numbers.
Umm...
But they are still complex numbers and,
for another day, maybe there's some more stuff out there, right? We are going to be focusing on complex numbers,
but maybe there are
other types of numbers. All right, and this idea of number expansion doesn't end with complex,
but that's another conversation for another day.
With that said, let's proceed, and
start exploring i a little bit more. All right, so I...
All right, i is i, all right.
There's some people out there that say i is the square root of -1. No, i is its own number?
All right, treat it like it's such. You know, I always make this joke
my first name is Eddie. All right, if I pull that eye out,
right, people don't call me
edde the square root of negative 1, right? People call me Eddie.
Okay, so we don't want to rename that letter i
something else. We just want to leave it as i, but we can go ahead and say i
squared is equal to negative 1 because they have the they've functionally the same thing.
Okay, so i squared is equivalent to negative 1. We want to treat i the way it is. We want to treat it like a
new number. We don't want to lean on the the real numbers in trying to understand this stuff. We want to accept it.
Okay, i-squared is -1. All right, i cubed, well, let's go ahead and split this up. Let's do a little bit of
expansion. I-squared, - 1, this is going to be negative. So I cubed has the same function as negative
i. Well, is that true?
Right, multiplying by I 3 times going to be...go ahead and put a point there. Multiply by i,
multiply by i a second time. So this is times. I times i again times i a third time.
You're gonna get down here. All right, that's i cubed and a negative. I well,
that's just in the opposite direction at 90 degrees clockwise. So multiplying by negative I is
Equivalent to multiplying by i 3 times, nice. So i cubed is equal to negative i, all right. What's i to the fourth?
Well, that's just well that's going to be i squared
times i squared, which
is -1 times -1,
which is functionally equivalent to multiplying by 1, so no dilation at all.
We can keep doing this, right. We can do I
to the fifth,
well,
that's just I to the fourth times i and i to the fourth was 1, so this is just i. I to the
fifth its equivalent to i. I
to the sixt,  that's going to be the same thing as I to the fourth times
i squared, which is negative one. I to the 7 is
going to be the same thing as i cubed cause it's gonna be i to the fourth, which is just one times i cubed.
So I cubed that was negative i, and then i to the eighth. We can keep doing this forever.
That's going to be two sets of i to the four,
which is just 1, so this is just one. So we can keep this pattern going. So my question to you is
using this pattern. What is I
to the 98?
Right. This is going to kind of motivate the next thing that we do. So real quick figure out what I to the 98 is
alright so I don't expect you to find out what 9 is and then 10 is and 11 then 12 and 13 and so on
and so forth all the way up to 98.
I want you to find this pattern and
if you notice we can pull out as many i to the fours as we want and it won't change our
final answer. All right, cause i to the 4 is 1 and multiplying by 1 doesn't change anything.
So how many 4 is going to 98? Well,
it's just less than 100, 25 going to 100, so 20. We got, let me go back a little bit, i to the
4, 24 times and then we have a left over i squared. So what this amounts to is just pulling out how many
quadruplets we have. All right, so we have i squared, which is negative one.
All right.
We can do this as much as we want. If you want to know, what one i to the
117 is, that's not a 17, that's a 4. All right, there we go. What is that? Well, I know that 4 goes into
116, so this is just i.
All right. What is i to the 412?
All right, I know that 4 goes into this number.
So this is just going to be 1, right. We have this little trick that we can do, but it does get better.
So here is this is called the rectangular form of a complex number, but um
this is just the arbitrary form of a complex number.
Alright, so i...the number i would be when b is 1 and a is 0. Alright.
The number 3 i would be when b is 3 and a is 0. Alright, the number 1, which is a complex number,
would be when a = 1 and b = 0, right, and an
arbitrary complex number
is when a and b are nonzero. So, let's look at some arbitrary
complex number.
Right, -3 + 5i, all right, on the complex plane.
Where is this number? Well it has
three or starting negative three real parts and
five
imaginary parts, so that number is right here. All right. This is the number negative -3 + 5i in the
complex plane.
Alright and we will soon discover what this numbers functional ability is in terms of dilations and rotations.
You can start guessing now, right,
but what I want to do before I
start looking at what this is geometrically, let's do some calculations. So,
let's take -3 + 5i and
let's look at addition. What happens if I add another a complex number to it? So 2 - 4i.
All right. What is this going to be?
So, um, the way we add complex numbers is we simply add their real and imaginary components.
So -3 + 2 is going to be negative 1 and
5 plus negative 4., sorry 5i plus -4i, that's going to be
1. So we have 1i, right.
So, if -3 + 5i + (2 - 4i), that's going to be negative 1 plus i.
So, what does this look like in the complex plane...well if I take mmm, let me get a complex plane going.
So real quick, if I take the numbers
like 1
and
I don't know 3.
Okay. What is 1 plus 3 well 1 plus 3 is you take this
segment and then you take this segment, right?
And you line them up
kind of like a head to tail and
then you end up with what's 1+3, but 1 + 3 is will you take this segment right here?
Right
And you line it up and you get four.
All right 1 + 3 = 4.
So what does it mean to take negative three plus five I and add 2 to minus 4i?
Well, where is the number negative three plus five?
I have negative three plus five is one two three five is gonna be about up here, right? This is negative three plus five. I,
okay, two plus four is two and then four is gonna be done here. So there's your two plus 4i,
sorry two minus 4i all right, and then you have these segments.
All right, my picture is not perfect.
It's a little bit badly scaled, but I'm going to take this number two plus 4i and then add it to this thing,
so I'm going to take its segment and then add it to this thing and ultimately it's going to become
negative
one plus i.
We have i there. We have negative one there. So what does it mean to add two things?
You're kind of lining them up as you do with normal numbers, alright, on the number line.
It's really important that we understand what i is and what all these complex numbers are in terms of the complex plane.
All right, let's look at some more.
So negative three
plus 5i
minus
the quantity 2 minus 4i, what's that going to be?
Well, we're gonna do the same thing here, but with subtraction. We're gonna do negative 3 minus 2,
which is negative 5 and we are going to do
5i minus negative 4i, which is
positive 9i.
All right, so you can take a second to think about what that means geometrically. I'm going to proceed. All right, how about multiplication?
What is that?
Alright, we're actually gonna take a break from looking at the meaning and
we are going to just compute because there's a better way to think about the meaning of multiplication,
but we want to get to that a little bit later.
Alright, so how do we multiply complex numbers in their rectangular form? All right
Negative 3 times 2 is negative 6. We're going to distribute. All right, 5i times negative 4i that's going to be
negative 20 i
Squared and we know that i squared is just negative 1, so while I'm at it
let me go ahead and put a positive 20 there, but we have more okay, we also have 10i and,
what is that, 12i, so we have 10i
plus 12i. We have, what is that going to be, 20, sorry not equals
plus 22i and then if we simplify this further we get
14 plus 22i.
All right, so that's that's a subtraction and multiplication and I'll let you guys real quick
Think about
how might we
divide these. I keep calling them rectangular forms because this describes kind of a coordinate
in the rectangular plane.
And let's do negative 3 plus 5i divided by
2 minus 4i, so take a second to try to figure out how do we simplify this?
How do we compute this?
Alright, so if you think back to when we wanted to simplify
Quotients that had radicals. We multiply it by something called to be conjugate here. We call it the complex conjugate.
So let's multiply the denominator and the numerator.
Thus multiplying by 1 that's not changing anything.
By the conjugate of the denominator. So what does this give us, negative 3 plus 5i two plus 4i.
Let's do the numerator. We get negative 6. We're gonna get a negative 20.
That's going to be a negative 26.
We're gonna get a 5i.
And that's 10i and we're gonna get a negative
12i, that's not gonna be what is that gonna be.
2i
All right, so that's our numerator and then our denominator is going to be 4.
Alright, 2 times 2 is 4 all right negative 4i times
positive 4. I that's gonna be a negative 16. I squared aka positive 16. You know what let me
not jump so fast, I mean negative 16 I squared I squared is just negative 1
all right, and then the rest kind of disappears because we're multiplying by a
difference of squares right for negative 4i and 2 positive 4i - that's gonna go away. All right.
So what is the final answer here? We have a negative 26. We have a 2i
in the numerator.
We have a 4 - or 4 + now 16 so 20 and we're gonna get
negative 26 - 2i
divided by 20 and that simplifies to...
What is that -13/10
minus
1/10i, all right. So keep in mind we can also have fractions when it comes to complex numbers, right? All right.
Rational numbers are contained in the complex numbers.
All right, as I said before we got a lot to get to so let's go ahead and look at solving
quadratic equations using
complex numbers.
All right, so how do we solve this. Well you can factor. That's not gonna work. You can
do a lot of different things. We are going to just straight-up use the quadratic formula. So
negative B plus or minus the square root of
B squared 49 minus 4
times 1 times
15 you can might you might be seeing the problem that's popping up here. All right all over
2 times 1. All right cool. And if you caught it before like I said
we're going to have a negative number underneath the radical, right,  49 minus 60 is a negative 11 all divided by 2
All right back in the day you'd stop and you say there's no solutions right? There were no real solutions, right?
But watch this little trick.
All right negative 11 that's equal to
Negative 1 times 11 and
a negative 1 is equal to I squared and
We could take the square root of a square, all right the square root of I squared that's just I
All right. So 7 plus um now i
times the square root of 11 all over 2 and we are now a
complex number in standard form. All right, seven halves plus or minus
root 11 i.
All right, a plus or minus bi right, so we have solved this equation.
Using the fact that we now have I squared is equivalent to negative one.
All right, and you can plug this thing back in and it will verify this.
All right, that's one of the motivations for this whole thing
We want solutions to this this equation here, right?
And if I take this number here and I simply have the rule that I squared is equal to negative 1 that solves this
This equation here, right? If I plug this in I'm gonna get something minus something plus 15 and
Everything is gonna cancel out real nicely.
I'm not gonna do that in this video, but I'll leave that for you to do.
All right, let's look at what these
complex numbers do terms in terms of geometry.
So let's say I have the complex number Z, that's often the letter that we use. Let me get that out there right now.
Z is 6 minus 2i, so what the heck does this thing do? Alright, so let's give ourselves a
complex plane.
Alright and let's take the number 1. All right. This is a really good way to figure out what this this number actually does.
So if we have the number 1
Right and we want to figure out what 6 minus 2i does, let's look at 6 minus 2y so 1 2 3 4 5
6
Negative I negative 2i
Hey, this was a six-way over here right the number
Six minus 2i is way out here in the complex plane. Alright and what the number six minus 2i
does
is
It takes the number one it dilates it out
some amount and
then rotates it
some amount.
Okay, how far did it dilate out. Well we have the Pythagorean theorem. Let's figure that thing out.
All right, if this is...if this is 2 and this
is 6
Negative 2 we could say. All right, if this is 2 and this is 6 and this magnitude is
36 plus 4 square root. It's a
The number 6 minus 2i using the Pythagorean theorem has the property that it dilates square root 40.
Right and then it rotates. Well, what the heck is this angle? Right this angle is
All right the arc
Tan or the arc sine of the arc cosine or whatever
Two of these three measurements that you want. I'm gonna do tan. So that's gonna be opposite over
adjacent.
So negative 2 over 6. Okay, negative 1/3 and the arc tan of negative 1/3 is
I've switched to the forbidden degree mode. It is
negative eighteen point four three five degrees.
All right,
so this number here Z has the property that it dilates root 40 and
Rotates negative eighteen point four three five degrees and this is me graphing it in the complex plane.
This is our
rectangular form.
With that said let's look at what I believe to be the far superior form. Let's look at the polar
form of
a complex number.
I'm gonna stop there with the complex number symbol.
And we've kind of already done it. Let's look at...let me capture this.
Alright, let's describe this thing using polar
so it has a x coordinate that is 6 and it has a
Y-coordinate that's gonna be negative 2i. All right, how can we describe this thing? Well,
there's another way we can write 6, right, we can write 6 as the square root of 40 times the cosine of
-18.435 degrees.
Let me actually write a 3.
Right, and we can write negative 2 as
all right, the square root of 40 times the sine of
-18.435 degrees. Sorry. I'm running out of room here. All right, and I can pull out
Oh, there's an eye. All right, let's not forget that I've of that that's there. All right. Check your calculator
This is negative 2 this this is 6. All right, we've constructed as such that that is the case
Right cosine of negative eighteen point four three five
That is the length here on the unit circle that we scaled 6
And then if I pull out this off 40 of the square root of 40
I'm left with cosine of negative eighteen point four three five
Degrees
plus sign or
Or better yet limb for the eye up here. I sign
Negative eighteen point four three five. I
Keep writing five what's wrong with me?
And
That is kind of what we're trying to get to
Right, what is root 48 polar form, then it's its distance from the pole
So we're gonna call that our right and then we have a here cosine
Theta that's its angle with the polar x-axis here it's negative I plus I
sine that same angle and this is what we call the polar and
With this last little bit of room
I'm gonna go ahead and write it
like this our
C is
Theta and that's kind of just shorthand our sis data that's shorthand for R
cosine theta I
sine theta
All right, and this is how we'll be writing a polar numbers as we proceed
All right, because this angle is the same this angle and this angle they are the same
So I can only need to write at once our sis data. This is the polar form
This is describing this point in the complex plane
Just like six plus negative two is all right
There are some things you need to worry about not worry necessarily, but I'm this is unique, right?
If you're using the the rectangular form of a complex number the six plus negative 2i is the only way to write
This number right here
But our
Cis theta is not the only way root 40 cis(-18.435)
Degrees is not the only way to write this point
There are multiple ways to write complex numbers in polar form, right?
You could use a positive angle that has the same, the same terminal end
If I want to graph the number 4
Cosine
5 PI over 3 and
Put parentheses there. Plus i
sine
5pi over 3
I'm aka
4 cis of 5 PI over 3
If I want to graph that all I need to do
Is let me make that more straight. There we go. That makes me happy
we need to graph 4 units away from the pole and
An angle of 5 PI over 3 away from the polar axis. So 5 PI over 3 is going to be down here
And a 4, I'm just gonna go ahead and approximate
right
Here, right. So this is the number for
Cis 5 PI over 3 another way to write this would be to say 4 cis negative PI negative 2. Im sorry...
Yeah negative PI over 3. All right, this is equal to 4 cis
Negative PI over 3. Ok, so there's multiple ways to write numbers in their polar form. So that's one
Maybe you might consider a downside. I don't necessarily consider a downside but polar does have its benefits which
let's get to
One of those benefits is
This number for cis 5pi over 3 tells you the geometric function of this number
right the, the
Cartesian or sorry the rectangular form doesn't tell you immediately the function but this says
dilate
For dilate by a scale factor of 4 and rotate
Of 5 PI over 3 radians
Right, so this this is one big advantage to the polar form is it it immediately tells you the function in the complex plane of
What this number does you take some number if you multiply it by 4 6 5 PI over 3?
It's going to dilate that number by 4 and then rotate 5 PI over 3
Let's multiply two complex numbers so
Let's say we have the number
Let's go with orange oranges on our number let's go with the number 6
cis PI over 6
times
2
cis
2 PI over 3
All right
so one way to do this would be to turn them into their Cartesian form their rectangular form and then
distribute because you're going to have two components to both and then
Condense and then maybe convert back to polar
Um, but what I want to do is I'm going to just use the laws of multiplication
So real quick, this is what it's saying is 6
cosine PI over 6
plus
I sign five or six
All right, some kind of are converting to its rectangular form here. Six times cosine PI over six is going to be the
Real part and six times (i)sine PI over six if we were to just simplify this like actually calculate it out
We'd get rectangular numbers. All right, and then we have a two
Cosine 2 PI over 3 plus
i
sine 2 PI over 3
Cool. All right
Let's go ahead and just squeeze those sixes and twos together. Let's make them a 12 and then let's distribute all this stuff out
so something beautiful is about to happen, right so
cosine PI over 6 times cosine 2 PI over 3
That's going to just be cosine PI over 6 cosine 2 PI over 3
All right, and then we're gonna have let's go ahead and get let's go ahead and get the other two real parts
So I sine PI over 6 times I sine 2 pi over 3 that's going to be I squared
all right, I'm gonna go ahead and write negative 1
and you know
I'm not gonna I'm gonna keep this at the keep is going we have i squared and then we have sine PI over 6
sine 2 PI over 3
All right, and then we have the two complex parts or sorry they're all complex sorry, we have the to fully imaginary parts
so we have plus i
Sine
PI over 6
times cosine 2 PI over 3
all right, that's just this and this being multiplied and
Then we have plus and running out of room. Oh, I
Sine 2 pi over 3. You know what? I'm gonna write this as cosine PI over 6
Sine 2 PI over 3. I
Just took this and of this and multiply them I
Cosine PI over 6 sine 2 PI over 3 and you may actually see why I did that
Cosine cosine sine sine sine sine cosine cosine sine cosine cosine sine sine sine. It's right in front of us
It's there for the pigging. All right, so
We have 12, but now we have cosine cosine sine sine sine, right? What is this sign? It's actually a minus minus 1
All right, I squares so we have cosine cosine, sine, sine, sine that's going to be cosine.
PI over 6
Sign sine sine plus
2 PI over 3
All right
we just went from a bunch of cosines and sines to 1 cosine beautiful, and over here if we factor out that i
We have a sine cosine cosine sign that's going to be sine of the sum of the angles PI over 6 plus 2 PI over
3
All right, you may now see why multiplying by a
Multiplying in polar form is amazing because what we're gonna get here is we're going to get 12
cis
PI over 6 plus 2 PI over 3. All right 2 PI over 3 plus PI over 6 that's going to be 5 PI over
6 we're done and look what we did right we started with
6 cis 2 PI over 3
Sorry PI over 6 and
We multiplied by 2 cis 2 PI over 3 and
This ended up becoming 12
Cis
5 PI over 6 or the sum of the two angles
All right, and this like makes sense geometrically think about it 6 cosine?
isin of PI over 6 that says rotate PI over 6 and dilate 6. 2 says
Dilate 2 and then the 2 PI over 3 says rotate 2 PI over 3
so what this is saying is if you multiply a number like 1
by these two numbers
You're going to rotate PI over 6 and rotate 2 PI over 3 in total rotate PI 5 PI over 6
And you're going to dial 8 6 and dilate 2 in total dilate 12
All right multiplying and this is like kind of the formula that you need to know all right multiplying two numbers R 1
Sis theta and let goes theta 1 and R 2
CIS theta 2
All right multiplying two numbers in their polar form just says hey multiply the two magnitudes and add
The two angles
Right and that's what it means to multiply in polar form
You don't have to deal with this distribution anymore
If you are in the polar form multiplication just becomes a multiplication and a little bit of addition, right?
This is one of the huge benefits to being in polar form. You can really understand what the number is doing
It's a rotation and that and a dilation
I'll leave it to you
To prove this for all numbers
For all values of r and theta for all complex numbers Z
Alright I'll leave that as an exercise
You know, I'll also leave as an exercise so we got this one right here, let me go ahead and capture it
I'm also going to leave as an exercise for you guys to prove that dividing
Two numbers, so if I have our
Sis, theta R 1 this is theta 1 and I divided by r2
It's a SATA 2. Alright that ends up being the quotient of the two numbers
times the cosine of
Theta 1 I sine theta
2 alright the the difference of the two. Alright, so these
Are two facts of the polar form of a complex number
Let's do a quick Division one like let's show how powerful this is right if I want to divide 4
cosine PI over 4 plus
I sine
PI over 4 and I want to divide by a 2
Cosine of 5 PI over 4. Plus I sine
5pi over 4
there is no need to
Multiply by any conjugates. Sorry, I couldn't multiply by
Cosine 5 PI over 4 minus I sine 5 herbivore on the top and the bottom. Alright, but there's no need to do that
I also don't need to convert to Cartesian. Sorry
Rectangular I need a stop saying Cartesian but rectangular I do not need to do that
All I need to do is take the quotient of the magnitudes, right?
this thing says to dilate by 4 if I'm dividing this thing says to
Dilate by 1/2. All right, so all in all we've dilated by 2 in total right 4 divided by 2
All right, and then this says I'm rotate PI over 4, but then rotate in the opposite direction
5 PI over 4. So rotate PI over 4 rotate in the opposite direction 5 PI over 4 that's going to be a total of
negative PI
All right, and then what does it mean to rotate?
Negative PI. Well, that's just multiplied by negative 1 that takes us way back to the beginning of this video lesson. All right
Ok, these two numbers divided by each other is negative 2
4 times what is cosine PI over 4 that's root 2 over 2. So let me actually backtrack to root 2
Plus I'm sine of PI over 4. That's also root 2. So for it that's going to be plus I
sine
Sorry, I 2 root 2
Divided by cosine 5 PI over 4 that's going to be negative root 2 over 2. So we have a negative
2 times root 2 over 2. That's a negative and we're gonna have
The same thing over here. I
Root negative root 2. All right, these two obnoxious complex numbers of the form a plus IB
When you divide them you just get negative 2. Oh you're done. All right, and being in polar form makes that really easy
With that said I'm gonna stop there this video is kind of long. Sorry
What I do want you to do on your own time. I know I asked you guys to do this later is
prove these two formulas
All right, do what I did with the multiplication
But for arbitrary values of R and theta
and that's kind of the
Extension of this lesson is that is the proofs?
All right
we looked at I
And we didn't just say hey to square root of negative 1 please please please do not just go around saying I is the square
Root of negative 1 right eye is its own number? It has a function rotate 90 degrees counterclockwise
All right. If I do that twice, I get the number negative 1 so conveniently I squared is negative 1
All right, and then now we've looked at how to add
Subtract multiply and divide
Numbers both in their rectangular and their polar form next time. We'll look at some cool
Consequences of this some cool corollaries to being able to multiply and divide in polar form
All right, but with that said I'm gonna close up here. I hope this video was engaging
Give me tips and suggestions if you want something new. I will see you guys later
