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

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Engineering by moving now into
the Sinusoidal Steady-state,

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we're analyzing circuits
that involve resistors,

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capacitors and inductors, that
are driven by sinusoidal sources.

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My name's Lee Brinton, I'm

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an electrical engineering instructor
in Salt Lake Community College.

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

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We find that electrical engineers

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use complex numbers just
a little bit differently

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than are typically taught in
college algebra and trig classes.

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

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to remembering where
complex numbers come from,

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we'll review rectangular and polar form

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of complex numbers, we'll introduce to you,

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

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the complex exponential
form of a complex number,

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

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these different forms make
certain types of calculations

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easier than doing those same
calculations in other forms.

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First of all, where do
complex numbers come from?

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

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

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it's simply a parabola
dropped down one unit.

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

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To find the x-intercepts,

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

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

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you get x squared equals one,

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and take the square root of both sides,

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and you get x equals plus or minus 1.

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

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Well, now let's change
this function to simply

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changing this minus sign to a plus sign,

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and we can then do the same calculations.

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

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

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We have x squared plus 1 equals 0.

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

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

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

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

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There's no number that
when multiplied by itself

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equals negative one in
the real number system.

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Well, that's not very satisfactory,

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there's got to be
a solution just because we

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change the sign from minus to plus,

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

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now there's got to be a solution.

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

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

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where i then equals is defined as,

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so let's make a third line there,

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i is defined as the square root
of negative one.

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Now, in electrical engineering,

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i is typically used to represent current,

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and so electrical engineers say, "Well,

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

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Thus we have this new type of number,

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it's called an imaginary number,

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rather than unfortunate choice
of term from my perspective,

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

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but you get my point,

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

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

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To do so, we already
introduced a new domain,

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is known as the imaginary number line,

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and we can plot imaginary
numbers on a number line

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

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

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

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symmetry of the parabola off of the y-axis

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over a distance of two units to the right.

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

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

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

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Subtract one from both sides,

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we get x minus 2 squared equals negative 1.

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Take the square root of both sides,

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we get x minus 2 equals plus or minus i,

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x then equals two plus i,

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and x equals 2 minus i.

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Now we have x representing
not a pure real number,

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not a pure imaginary number,

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but a combination of the two

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

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we call that a complex number.

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As we know, when solving equations that

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involve real numbers, real coefficients,

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that lead to imaginary solutions,

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those solutions come as complex conjugates,

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

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