This material appears in section 6.5. Also, see Section 2.4 of the text for an introduction to Complex numbers.
is written in rectangular form
where
and
are real numbers and
is the imaginary unit.
The "real part" of
is the
, and the "imaginary part" is the
.
Using the rule
we can extend all the usual rules of arithmetic to complex numbers.
.
The equation is true because we are just multiplying by
, but this trick will make the denominator real:
where
is the magnitude, which is a positive real number or 0, and
is the angle.
Notice the similarity to wiritng a vector as a magniutde times a unit vector written in trig form
The idea here is the same and the computations are the same as well.
The magnitude of a complex number
(also called the norm or absolute value of
is
To compute the direction (also called the argument of
) we do the same calculation as we did in finding the direction of a vector:
to rectangular form.
You could stop here or figure out the trig functions:
then the magnitude is
and the angle
is described above.
to trig form
The real part is positive so the angle is
so
to trig form
so
applying trig identities this is equal to
So there's an easy rule for multiplying complex numbers in trig form: multiply the magnitudes and add the angles.
Conversely, to divide you should divide the magnitudes and subtract the angles:
Exponentiation is also easy, just think about what would happen if you did the multiplication rule multiple times:
Date: 2008/12/10 15:17:50