1.4. Polar Form for Complex Numbers 21

1

−1/2+i 3/2

−1/2−i 3/2

Figure 1.4.2. The Cube Roots of Unity.

Note that the n numbers

ei(θ/n+2πk/n)

that appear in this result are evenly

spaced around the unit circle, with each successive pair separated by an angle of

2π/n.

The nth roots of the number 1 play a special role. They are called the nth roots

of unity. If we apply the above theorem in the special case where r = 1 and θ = 0,

it tells us that the nth roots of unity are the numbers

(1.4.4)

e2πki/n

for k = 0, 1, 2, · · · , n − 1.

Example 1.4.5. Find the cube roots of unity.

Solution: In the particular case where n = 3, (1.4.4) tells us that the roots of

unity are

e0

= 1,

e2πi/3

= −1/2 + (

√

3/2)i, and

e4πi/3

= −1/2 − (

√

3/2)i.

Example 1.4.6. Find the cube roots of 2i.

Solution: Since 2i = 2 eπi/2, Theorem 1.4.4 implies that the cube roots of 2i

are

21/3 eπi/6

=

21/3(

√

3/2 + i/2),

21/3 ei(π/6+2π/3)

=

21/3 e5πi/6

=

21/3(−

√

3/2 + i/2), and

21/3 ei(π/6+4π/3)

=

21/3 e3πi/2

=

−21/3i.

The Logarithm. If z = r eiθ, and log r is the natural logarithm of the positive

number r, then the law of exponents implies that

z =

elog r+iθ

.

Thus, it would make sense to define log z to be log r +iθ = log |z|+i arg z. There is

a problem with this, however. There are infinitely many possible choices for arg z,

and so log z is not well defined, just as arg z is not well defined.

The solution to this problem is to restrict θ = arg z to lie in a specific half-open

interval of length 2π.