So, so let's talk about complex numbers. 
I cannot emphasize enough how important 
complex numbers are in electrical 
engineering, It's really important, it's 
really fundamental, and we're going to 
exploit the properties of complex numbers 
in ways I think you're very surprised to 
see. 
So, as always we're going to define what 
a complex number is. We'll then learn how 
to add, subtract, multiply and divide 
them, and then we're going to go into the 
important topic of the two different ways 
of writing complex numbers. 
It's called Cartesian and Polar forms. 
Okay, so let's define a complex number, 
and this definition, I think, is a little 
different than you may be used to. 
A complex number is an ordered pair of 
numbers. So, a and b, an ordered pair. 
Now, what the ordered term means is that 
a, b does not equal b, 
a. 
The order matters. 
They're not the same thing. 
And similarly, if you have two complex 
numbers that equal each other, 
what that means is that a equals c and b 
equals d. 
So that's the, what the ordering has to 
do with things. 
And now we need to figure out how to do 
arithmetic with these numbers, 
what the rules of algebra are for, for 
them. 
And lets start with multiplying a complex 
number by a scalar. 
And so, when you do that, what you do is 
you multiply each component, 
the a and b component by the same scalar. 
Okay. 
I think that makes a lot of sense. 
If you add them, it's the same as adding 
the components of the 2 complex numbers. 
Again, keeping them separate. I think 
that follows. 
What isn't so obvious is where, where the 
multiplication and reciprocal formulas 
come from. 
Well it turns that these are absolutely 
the right algebraic rules to use when 
multiplying and dividing and we'll see 
what that is in just a second. 
But first, I want to point out something. 
This notation a comma b you should have 
seen before in analytic geometry. 
That's how you put notation you use to 
represent the location of a point in the 
plane. 
So here we have a point and this is its 
location in the plane. 
Its x component is a, its y component is 
b and this is just standard notation and 
this also helps explain why the 2 
components are kept separately in this 
geometric interpretation. 
We're going to call the x component the 
real part, we're going to call the y 
component the imaginary. 
Okay, and that's just terminology, 
but where it comes from, is the following 
interpretation of the complex number. 
We're going to write a complex number in 
the following way. 
a plus jb. 
J is an indicator of which one is the Y 
component, which one is up. 
So, this a plus jb means I can get to 
this point in the plane by going out a 
along the real part axis, the x axis, if 
you will, and then going up b units in 
the y direction, the imaginary part 
direction. 
And so, this j simply represents which 
way is the vertical, the y component 
direction. 
Okay, so this is an interesting way of 
writing a vector sum, that's all it is. 
The fun comes when you define that 
indicator j to be the square root of 
minus 1. 
That's where things get interesting, and 
this helps us figure out where those 
multiplication and division rules came 
from Before I get into that, I have to 
point out, and you should be a little 
curious, why is j used to represent the 
square root of mnus 1. 
You, you probably learned it as i was the 
square root of minus 1. 
And it's true for world other than 
electrical engineers use i for the square 
root of minus 1. 
But not electrical engineers. 
We're special, we use j and there's a 
historical reason for that. 
It turns out at the beginning of the 19th 
century, the French physicist Ampere 
wrote a fundamental paper on the 
mathematical physics of electricity and 
he referred to current as [FOREIGN], for 
expressing how much current was flowing. 
And so he used the symbol i for current. 
So, according to him, i is the same as 
current. 
Well, it was toward the end of the 19th 
century that the importance of complex 
numbers came up, and so we can't use i 
because electrical engineers throughout 
the 19th century were using i for 
current, so we use the next letter in the 
alphabet. 
So, it's only electrical engineers that 
use j for the square root of minus 1. 
So, since you're in this course, you're 
going to use j over the square root of 
minus 1. 
And all I can say is, welcome to the 
electrical engineering club. 
You are now, an electrical engineering, 
if you use j for the square root of minus 
1. 
Fun to be in this exclusive group, of 
really cool people. 
Okay, so, lets go back and, look at our, 
arithmetic rules before, in the light of 
thinking about this a plus jb, a vector 
sum. 
So, now multiplication by a scalar. 
Well, things still make sense. 
If you have a scaler multiplying a sum, 
well, the scalar multiplies each term in 
the sum. 
When you have a sum of sums, you just add 
them all up, and then we're going to 
group them by their x, and y components, 
there a real part, and they're measuring 
for our components. 
That makes perfect sense. 
Now the multiplication rule is going to 
come up and make a whole lot more sense. 
So, you have a product of 2 sums. 
Well just use your normal algebraic 
rules, that you already know. 
So, A1 times A2, That's A1A2, of course, 
and then you get A1 times jB2. Well, 
that's this, that's fine. 
So, now let's say jB1 times A2 There's 
that term. 
That's fine, and the interesting one is 
jb1 times jb2. 
Now, rearranging that multiplication a 
little bit, that's got to be j squared 
times b1, b2 of j squared is the square, 
the square root of minus 1, that's minus 
1, which means that's where this term 
comes from, so in order to use the 
regular rules of algebra, when you 
represent complex numbers in this kind of 
vector form, this has to be the 
multiplication So now where this 
reciprocal rule comes from, you're 
going to have to wait a little bit. 
I'll get to that in just a second. 
Well, now we come to one of the most 
important results mathematical results 
for this course called Euler's Formula. 
Euler was an 18th century mathematician, 
the best and most famous mathematician 
after Newton. 
And he noticed the following formula. 
So take an exponential, e to the cx, 
and write it in Taylor series. 
And what Euler wondered was, what is, 
what happens when c is the square root of 
minus 1, it is, j. 
Well, he didn't know what e to the jx 
was, but one way of looking at it is to 
use the Taylor series formula and see 
what you get. 
So here are the calculations that he 
made. So he just plugged in j for ce, 
everywhere in the formula, and just did 
the usual rules that we just talked 
about. 
So let's look at the even powered terms. 
1 is 1, okay. 
This term, that's j squared times x 
squared. 
Well j squared is minus 1, so that's 
where this term comes from. 
j to the 4, that's j squared times j 
squared, so that's plus 1. 
So that's where that term comes from. 
Okay, let's look at the odd powered 
terms. 
j x, okay, there's j x. 
j cubed is j times j squared. 
So that's j times minus 1. 
So we get a minus x cubed term here. 
So, I think you can continue this on and 
what the Taylor series says is that the 
real part, the part with the x component 
consist of all of the Even powered terms, 
with an alternating sign. 
The imaginary part, consists of the odd 
powered terms, with an alternating sign. 
Well, low and behold Euler noticed, right 
away, because he's a smart mathemetician 
That it has to be true, that e to the jx 
is cosine x plus j sine x. 
Because that is the Taylor series for 
cosine. 
And that is the Taylor series for sine. 
It has to be true of all, everything is 
going to work together and hang together, 
and be a coherent mathematical 
formulation. 
So, and say another way, he showed that 
re to the j theta, has to be r cosine 
theta times jr sine theta. 
Well that should be very strike a 
familiar note, cause this looks like 
polar form. 
Remember this is a plus jb. 
Hmm, that sure looks like a polar form, 
and sure enough, you go back to geometry, 
that's exactly what it is. 
So, here's our representation of a point 
in a plane, using the regular ordered 
pair notation, 
which you can also think about a point in 
the plane in polar form. so there is a 
distance from the origin, the point is 
some distance from the origin, and at 
some angle. 
Well we know how to convert between the 
two. 
We know that the a component, the x 
component, is r cosine theta, 
and that the vertical component, the y 
component, the imaginary component here 
is r sine theta. 
So that's where the rule I derived on the 
previous slide, is all consistent with 
this. 
All consistent with this geometric 
picture of representing something 
Cartesian coordinates or polar form. 
It turns out to hop between the two, 
Euler's formula had better be true or 
else something is very very wrong. 
So, 
you know, from your geometry that you 
find the radius, just by using the 
Pythagorean theorem, and the angle is 
just the arc tangent of the ratio of the 
y and x components. 
Okay, so, we now have 2 ways of 
representing the same complex number. 
One is Cartesian form, a plus jb. One is 
Polar form, re to the j-theta and these 
rules are how you would go from one to 
another. 
Go from Polar form back to Cartesian 
using these rules, go from Cartesian form 
to Polar form using these rules. 
And we're going to do that a lot. 
It turns out that certain calculations 
are better done in one form than another, 
and you need to be able to go back and 
forth, and these are the rules that they 
come from. 
But these aren't just arbitrary rules, 
they have to do with the geometry of the 
2 d point, Okay, 
well now this might help explain how some 
of these weird and [INAUDIBLE] 
multiplication and reciprocal rules come 
from. 
Well let's check multiplication rule when 
things are written in polar form. 
So we have two polar form complex 
numbers, and a little bit of 
rearrangement gives us this. 
And note that you're multiplying 2 
exponentials together, their exponents 
add. 
So that's where this comes from. 
Okay, well. 
This is just a special, form. 
I'm going to convert it to Cartesian form 
using our rules. 
Okay, that's where that comes from. 
And now I'm going to use the cosine of a 
sum of angles, and the sine of a sum of 
our angles, 
formulas that you'd learn in 
trigonometry. 
So, that's where this is. 
This term corresponds to the cosine of 
the sum of angles, 
and this turn, comes from the sine of of 
a sum of angles, 
and we're using the J to keep us, 
straight about which one goes where. 
And I've rearranged things a little bit 
to put the r1 with theta 1, and the r2 
with the theta 2. 
Well, identifying terms, that is just a 
1, that's just a 2, etc., and sure enough 
we come up with the same application that 
we derived for Cartesian form. 
So that's, that works fine. 
Now the reciprocal rule can be derived 
rather easily. 
So we're writing Z in, in polar form, 
that's going to be that whole reciprocal 
of an, this is 1 over r times e to the 
minus j theta, 
because, e to the x, and then take the 
reciprocal of that, why is e to the minus 
x. 
So, that, this has to be true. 
Well, now convert that to Cartesian form. 
So, e to the j theta is just cosine of 
theta, I'm sorry, cosine of minus theta 
is cosine of theta That's where that 
comes from. 
The sine of minus theta is minus the sine 
of theta, and that's where this comes 
from. 
So, that's really quite nice, and now you 
see where this comes from, because if I 
multiply top and bottom of this 
expression by r, 
we see that R cosine theta, well that's 
A, and R sine theta is E, and R squared 
is just A squared plus B squared. 
So that's where that reciprocal rule 
comes from. 
It's most easily derived in polar form, 
and now everything hangs together. 
We now understand all the algebra that 
there is. 
Now there's one special thing that comes 
up in complex numbers, which is a new 
thing, and that's the complex conjugate. 
It's a very easy operation, but it's not 
algebraic. 
So, the complex conjugate of z, which we 
denote by z star, 
what it does is it flips the sign of the 
imaginary part. 
So, z is equal to a plus jb and z 
conjugate is a minus jb. 
In Cartesian form, and in polar form, 
what that does is it changes the sign of 
the angel, flips it, and that's what the 
conjugate is. 
And the conjugate, it turns out, is 
mathematically important because of 
following reasons. You can now use, 
say that z times e conjugate is equal to 
the radius squared. 
So, that was e times e conjugate, well 
that's just squared. 
So you can derive, if you will, from R, 
just from looking at Z. 
And also, we can define what's called the 
magnitude of a complex number, is equal 
to r. 
So, another word for the radius is called 
the magnitude, 
and the magnitude squared is just r 
squared is z times. 
Z conjugate. 
It's this formula that's going to be 
very, very important for us, in talking 
about, complex numbers mathematically. 
You can also express the real part and 
imaginary parts, of a complex number 
using the com, the complex conjugate. 
So, the real part of z, which we'll 
denote by this, the x component, if you 
will, is equal to z plus z, conjugate. 
So, that's a pretty easy thing to see. 
So you have a plus jb. 
I'm going to add to that a minus jb. 
Well, when you do that, these, these 
parts cancel, 
and you're left with 2 times a, 
so you divide by 2. 
And a similar thing happens when you 
subtract. 
Now what happens, is that the real parts 
cancel, and you're left with the two 
imaginary parts adding, and you have to 
divide by 2J to get back to B. 
So this equals a and this equals b. 
So now we can derive the Cartesian 
components just by referring to the 
original complex number, by simply saying 
real part, and the imaginary part, by 
saying the imaginary part and get them by 
using this formula. 
And that's going to be very useful for us 
in doing some operations on complex 
numbers. 
So, how do you use all this, to do 
calculations? 
What I tend to do is add and subtract in 
Cartesian form, 
and I multiply and divide in either form. 
It depends on the details of what I do. 
I tend to use Polar form especially for 
division, and you'll see that in a 
second. 
And then I convert between the, between 
Polar and Cartesian form, depending on 
what's going to be needed. 
So here's a little example. 
Suppose we want to divide these two 
complex numbers. 
the easiest way to see this, I could use 
the reciprocal formal, but the easier 
thing to do, is to notice you can play a 
little, 
play a little mathematical trick. 
So a plus jb divided by c plus jd, I can 
multiply top and bottom by the conjugate 
of the denominator. 
I keep writing v, I don't know why, 
that's a d. 
Okay, I'm multiplying by 1. 
I'm writing 1 in a fancy way. Well, the 
denominator, is the magnitude squared of 
that complex number, 
and then I just have to multiply the two 
top things together, where you can see 
where it comes from. 
Because 5, is 1 squared plus 2 squared, 
and there's the, conjugate part. 
And I just multiply them together, and 
when the smoke clears, you get 3-J. 
So, that's a, easy way to do a division. 
In this case in Cartesian form. 
If it's in Polar form, it's, if these 
original numbers were in Polar form, I 
would have stayed in Polar form and just 
done it kind of like, because it's very, 
very easy. 
Now another thing that comes up is the 
magnitude of a ratio. 
And what I want to point out is that the 
magnitude of a ratio is equal to the 
ratio of the magnitudes. 
And, I think this is pretty easy to see 
if you think about this in polar form, 
you can derive this yourself. 
So the magnitude of the, numerator is 
square root of 1 squared plus 1 squared. 
So you get, 2, square root of 2. 
And as we've already seen, the square 
root of 1 squared plus 2 squared is the 
square root of 5. 
And so, 
a magnitude of this come, the number is 
the square root of 2 over 5, and if you 
take, you apply, you find the magnitude 
of this number, which I asked you to do, 
I think you will get the square root of 2 
over 5. 
And just do better. 
Alright, so complex numbers, it turns 
out, are going to simplify our 
calculations immensely. 
I cannot stress that enough. 
It is extremely important for signal and 
system theory in electrical engineering. 
Why that is is not obvious yet. 
Just stay tuned, and we'll get to it very 
very quickly.