>> We're going to continue on with our
introduction to Electrical and Computer

Engineering by moving now into
the Sinusoidal Steady-state,

we're analyzing circuits
that involve resistors,

capacitors and inductors, that
are driven by sinusoidal sources.

My name's Lee Brinton, I'm

an electrical engineering instructor
in Salt Lake Community College.

We're going to start this discussion
with a review of complex numbers.

We find that electrical engineers

use complex numbers just
a little bit differently

than are typically taught in
college algebra and trig classes.

So we're going to start
out with just going back

to remembering where
complex numbers come from,

we'll review rectangular and polar form

of complex numbers, we'll introduce to you,

I think for many of you
will be the first time,

the complex exponential
form of a complex number,

and then we'll go through
some example calculations to show how

these different forms make
certain types of calculations

easier than doing those same
calculations in other forms.

First of all, where do
complex numbers come from?

But probably, the easiest or
at least an easy way to see it is to look

at the graph of the function
y equals x squared minus 1,

it's simply a parabola
dropped down one unit.

You can say, "Well, what are
the x-intercepts there?"

To find the x-intercepts,

you of course set the equation or
the function y equal to 0 and solve for x.

So we have x squared minus 1
equals 0, add one to both sides,

you get x squared equals one,

and take the square root of both sides,

and you get x equals plus or minus 1.

So sure enough, the way x-intercepts
are at plus or minus one.

Well, now let's change
this function to simply

changing this minus sign to a plus sign,

and we can then do the same calculations.

Now, obviously, there's
no x-intercepts here.

Nonetheless, we can still set y equal
to zero and try and solve for x.

We have x squared plus 1 equals 0.

Subtract one from both sides gives
us x squared equals minus one.

Now take the square root of
both sides and we go x equals,

we're looking for a number that when
multiplied by itself equals negative one,

and in the real number system we
all know there is no such number.

There's no number that
when multiplied by itself

equals negative one in
the real number system.

Well, that's not very satisfactory,

there's got to be
a solution just because we

change the sign from minus to plus,

and all of a sudden we
don't have solutions,

now there's got to be a solution.

So we define or we invented
the solution to that equation.

We called it i, so we have x then
is equal to plus or minus i,

where i then equals is defined as,

so let's make a third line there,

i is defined as the square root
of negative one.

Now, in electrical engineering,

i is typically used to represent current,

and so electrical engineers say, "Well,

let's let j equal i or j then is equal
to the square root of negative one."

Thus we have this new type of number,

it's called an imaginary number,

rather than unfortunate choice
of term from my perspective,

because it's no more imaginary
than the real numbers,

but you get my point,

they're numbers that exist just
not in the real number system.

In fact, we're going to show that
we need to introduce a new domain.

To do so, we already
introduced a new domain,

is known as the imaginary number line,

and we can plot imaginary
numbers on a number line

just like we can plot real numbers
on the real number line.

Now, it becomes a little bit more
interesting when we look at this equation.

Now what we've done is
we've shifted the axis of

symmetry of the parabola off of the y-axis

over a distance of two units to the right.

Let's take this and set that equal to zero.

We have x minus minus 2 quantity
squared plus 1 equals 0.

Let's read my one go, plus 1 equals 0.

Subtract one from both sides,

we get x minus 2 squared equals negative 1.

Take the square root of both sides,

we get x minus 2 equals plus or minus i,

x then equals two plus i,

and x equals 2 minus i.

Now we have x representing
not a pure real number,

not a pure imaginary number,

but a combination of the two

and when you've got a real number
plus an imaginary number,

we call that a complex number.

As we know, when solving equations that

involve real numbers, real coefficients,

that lead to imaginary solutions,

those solutions come as complex conjugates,

2 plus i and 2 minus i are said to be
conjugates of each other because they

differ only by the sign on
the imaginary part of the number.