20 1. The Complex Numbers
r
θ
r e

Figure 1.4.1. The Polar Form r
eiθ
of a Complex Number.
Products, Powers, and Roots. Polar form is particularly useful in dealing with
products of complex numbers, since the product has such a simple expression if the
numbers are given in polar form. If z1 = r1 eiθ1 and z2 = r2 eiθ2 , then the law of
exponents implies that
(1.4.1) z1z2 = r1r2
ei(θ1+θ2)
.
Thus, z1z2 is the complex number whose norm (distance from the origin) |z1z2| is
the product of |z1| and |z2| and whose argument is the sum of the arguments of z1
and z2.
Similarly, the quotient of z1 and z2 is given by
(1.4.2) z1/z2 = (r1/r2)
ei(θ1−θ2)
.
Example 1.4.3. Find z1z2 and z1/z2 if z1 = 2 eπi/3 and z2 = 3 e2πi/3.
Solution: By (1.4.1) and (1.4.2), we have
z1z2 = 6
eπi
= −6 and z1/z2 = (2/3)
e−πi/3
= 1/3 i

3/3.
It is evident from (1.4.1) that the nth power of a complex number z = r
eiθ
is
(1.4.3)
zn
=
rn einθ
.
From this we conclude that if z = r
eiθ,
and we choose w =
r1/n eiθ/n,
then
wn
= z. Thus, w is an nth root of z. It is not the only one, however. Since z can
also be written as z =
ei(θ+2πk)
for any integer k, each of the numbers
wk =
r1/n ei(θ/n+2πk/n),
where k is an integer, is also an nth root of z. Of course, these numbers are not all
different. Those whose arguments differ by an integral multiple of are the same.
The numbers w0,w1,w2, · · · , wn−1 are all distinct, but every other wk is equal to
one of these. This proves the following theorem.
Theorem 1.4.4. If z = r
eiθ
is a non-zero complex number, then z has exactly n
nth roots. They are the numbers
r1/n ei(θ/n+2πk/n)
for k = 0, 1, 2, · · · , n 1.
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