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.
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