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.