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Now we're going to take a few minutes and talk about complex numbers.

So this starts by discussing imaginary numbers.

Now, imaginary numbers came about from the solutions of certain simple algebraic

equations. Now, let's take a look first at this

really trivial Algebraic equation, x squared minus 4 equals 0.

And ask the question, what value is of x solve that equation?

Well I can take and put the 4 on the other side.

So I have x squared equals 4. So I'm looking for a number that if I

raise it to the second power, I get plus 4.

Well, the two numbers that solve that are plus 2 and minus 2.

If I square negative 2, I get plus 4. So these are the solutions to that simple

algebraic equation. Now, what if I just innocently changed

the sign here from minus 4 to plus 4? So now I need to find the solutions of

this new equation. So I'll rewrite this as x squared equals

minus 4 or minus 4 times plus 4. And I have to find I have to take the

square root of this. So the square root of this product is the

product of this square root. So the solutions are x equals the square

root of 4 times the square root of negative 1.

Well, I already know that the square root of 4 is plus and minus 2.

But the square root of negative 1, I have, I can't think of any real number,

there is no real number, that if I square it I'll get a negative 1.

So I'm just going to have to carry this around explicitly written as the square

root of negative 1. Now, instead of writing that all the

time, we just introduce this symbol j. Now electrical engineers use j for the

square root of negative 1. Mathematicians and physicists typically

use i for the square root of negative 1. But electrical engineers need i to

represent current. And so we use j to keep from getting

confused. So x, then the solution of x squared plus

4. The solutions are x equals plus or minus

2j. So we had to invent j, the unit imaginary

number to solve certain types of algebraic equations.

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Now, j has a very interesting cyclic property, that's something reminiscent of

the cyclic property of the derivatives of sine and cosine.

Now, j to the first power is j. j squared, that is negative 1.

j cubed is j times negative 1 or negative j.

j to the fourth is 1, because -1, j to the fourth is like j squared times j

squared, so negative 1 times negative 1 is plus 1.

And then j to the fifth, is j times this 1.

So that's j. So if I start at j then 1, 2, 3, 4

multiplied, multiplying by j four more times gets me back to j.

So it has this sort of cyclic property. Now complex numbers are composed of real

numbers and imaginary numbers. And a complex number has a real part, x

and an imaginary part, y. And if I have some complex number z and I

can indicate, take the real part of z by just writing this, and that's x, and the

imaginary part of z is y. So those are complex numbers.

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Now let's take a minute, and look at a little bit of complex number arithmetic.

This really is a very simple thing. You just have to keep the real parts and

the imaginary parts separated, so if you look at addition, if I have a plus jb

plus c plus jd, I just add the real parts and add the imaginary parts and keep them

separate. So there's the, there's how you add two

complex numbers. When I multiply them, I just have to go

through and do all the multiplication. So this is two factors here, essentially.

So I have ac and then I have jb and a jd, so this minus bd, so the real parts of

this. And then I have the cross terms, j times

bc and j times ad. So there's the how you multiply it.

Two complex numbers. Now another thing that we're going to

need to use is the so called complex conjugate.

So if I have some complex number a plus j times b, picking the complex conjugate is

just changing the sign, the sign, sign, in front of j, everywhere you see it.

So it's as simple as going in and replacing j with minus j everywhere.

That's the complex conjugate. So we if we have a number z, a complex

number a plus jb, we represent the complex conjugate by a little star,

that's a minus jb. Now, that has the interesting and useful

property that, if I multiply a complex number by its complex conjugate, I get a

squared plus b squared. Now that, is denoted by z with these

vertical bars squared. This is the modulus squared of that

complex number. And you'll see in a second, the nice

geometric interpretation of that when we start plotting complex numbers.

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So let's start graphing complex numbers and looking at some of the properties

that emerge from looking at them that way.

Now let's start with a complex number z. Call it a plus jb, and then let's plot

it. the real part on the horizontal axis and

the imaginary part on the vertical axis. And so, each complex number is a point

somewhere in this so called complex plain.

And so, a plus jb is over a units. That's the real part, and then it's up b

units along the imaginary axis, so here's the point z.

Now, we can represent that the coordinates of this point, in a couple of

different ways. I can use the Cartesian coordinates, a

and b, to tell me how to get to this point.

But I can also use polar coordinates, and so I can indicate the distance from the

origin and the angle that that line is pointing.

And so is kind of your, your your range and your bearing.

So it tells you what the angle and how far to go.

And it will get you to this same point. Now what I want to do is take and look at

the relationship between these two types of coordinates to indicate the same

complex number. Now the, we're going to find that the,

first of all, the r here is the so called modulus of the complex number.

So I have this complex number a + jb, at some distance r away from the origin.

And that distance r is denoted as the modulus.

Now, let's take first a look at this. If I, now, I now have r and phi and I

have a and b to represent the same location in the complex plane, and just

looking at this graph here and doing a little trigonometry, you see at this sine

of this angles is the opposite over the hypotenuse.

That's b over r. So b is r sine phi.

And looking at the other side here, the cosine of phi is the adjacent over the

hypotenuse. That's a over r, so I can write a equals

r cosine phi. Now, I can also, using the Pythagorean

theorem notice that r squared, this is a right triangle here, has base a, and a

long side b. The hypotenuse of this right triangle r

is related to these other two sides by r squared is a squared plus b squared, or r

is square root of a squared plus b squared.

And finally, I can take the ratio of these two sine, I'll take b over a, so

that's r sine phi over r cosine phi, and the r is cancel, and so by taking this

ratio I get tangent of phi. Tangent is sine over cosine.

It's b over a. Or, I can write that as phi is the angle

whose tangent is b over a, or the arc tangent of b over a.

So, now I have these two transformation equations that show me how to go between

the polar coordinates and the rectangular Cartesian coordinates for a complex

number. So if I tell you r and phi, you can

calculate what a and b are. Or conversely if, if you're told a and b,

you can calculate r and phi. So this is just a convenient way

sometimes it's more convenient to represent complex numbers in polar form,

sometimes in Cartesian form. And this is how you transform between the

two.