20 1. The Complex Numbers

r

θ

r e

iθ

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 2π 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.