>> As we've seen, we can represent complex numbers in terms of
either a rectangular manifestation with a
real plus an imaginary part or z equals a plus jb.
We can also represent it in polar form,
where we have an angle Theta and orientation from the horizontal axis and
the magnitude of z being the distance from the origin out to the point.
I've redrawn here the equations
that relate the polar and rectangular coordinates.
Changing from rectangular polar,
we have the magnitude of z is equal to that from
Pythagorean's theorem and the angle is that.
Going from polar coordinates,
where the coordinate values are magnitude of z and Theta,
back to the rectangular manifestation where the coordinates are a and b.
So we go back to polar,
we have a is equal to magnitude z
cosine Theta and b is equal to the magnitude of z sine Theta.
Now, let's write the complex number z in real form,
with a real part plus an imaginary part,
but do it in terms of the polar coordinates.
In other words, z then is equal to the real part a.
Let's write it like this. a plus jb.
Which is equal to, a is equal to the magnitude of z times the cosine of
Theta plus j times the magnitude of z times the sine of Theta.
See what I've done, I've written it in
rectangular form with a real part plus j times the imaginary part.
But I've written the quantities in terms of
the polar values of z are maxi and Theta.
Now let's factor out the magnitude of z,
so this then is equal to,
give myself more room,
the magnitude of z times the cosine
of Theta plus j sine of Theta.
Now we look at that in brackets,
that cosine of Theta plus j sine Theta and we recognize that from
Euler's formula as being e to the j Theta.
Thus, another way of writing z this complex number,
what we are going to refer to as its Complex Exponential form,
which uses the polar coordinate values of magnitude z and Theta.
Thus Z is equal to magnitude of z e to the j Theta.
Why get so excited about this?
In this complex exponential form we can
use all of the properties of exponents.
In other words, say we had let's just do it over here.
Say we had one number or one expression six e to
the 3x and we want to multiply it by two e to the 1x.
Well, we know that in doing this type of thing you can be a multiply
the coefficients six times two is 12.
Because they are raised to the same base we got e to the 3x and e to the x.
When you multiply e to the 3x times e to the x,
you simply add the exponent e to the 3x plus x is 4x.
Similarly, if we wanted to divide six e to the 3x by two e to the x,
you simply divide the coefficients,
six divided by two is three and you subtract the exponents 3x minus x is 2x,
and we would have three e to the two x.
So what we're saying then is that we can represent
complex numbers in either rectangular form a plus jb
or polar form z maxi and Theta but that
polar form is equivalent to or can be
represented in this complex exponential form,
where we have the magnitude of z times e to the j angle of z.
We're going to see that this form right here,
this polar form makes
multiplying and dividing complex numbers very, very easy.
That's where we're headed now.