1.4. Polar Form for Complex Numbers 21
Figure 1.4.2. The Cube Roots of Unity.
Note that the n numbers
that appear in this result are evenly
spaced around the unit circle, with each successive pair separated by an angle of
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
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
= −1/2 + (
= −1/2 − (
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
3/2 + i/2),
3/2 + i/2), and
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
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.