A complex number is a number of the form
, where a and b are real numbers. a is called the
real part and b is called the imaginary
part; the notation is
You add, subtract, and multiply complex numbers in the obvious ways:
The conjugate of complex number is obtained by
flipping the sign of the imaginary part. The conjugate of
is denoted
or sometimes
. Thus,
You can divide by (nonzero) complex numbers by "multiplying the top and bottom by the conjugate":
With these operations, the set of complex numbers forms a field.
Example.
The norm of a complex number is
Note that
When a complex number is written in the form , it's said to be
in rectangular form. There is another form for
complex numbers that is useful: The polar form
. In this form, r and
have the same
meanings that they do in polar coordinates.
DeMoivre's formula relates the polar and rectangular forms:
This key result can be proven, for example, by expanding both sides in power series. It's really useful, as you'll see in the examples below.
Observe that
Thus, is a complex number of norm 1.
Example. Convert to polar form.
Let (or
).
Then
Example. ( A trick with
Demoivre's formula) Find .
It would be tedious to try to multiply this out. Instead, I'll try to
write the expression in terms of for a
good choice of
.
Example. ( Proving trig identities) Prove the angle addition formulas:
I have
Equating real and imaginary parts on the left and right sides, I get
Example. ( Computing
integrals) Compute .
Note that . Thus,
Copyright 2005 by Bruce Ikenaga