8.6 Periodicity 241
so that each is a cube root of the number 1:
z
01 = 1 e
01 = 11
We note thatz
1andz
2are a complex conjugate pair, withe
4π
i 231 = 1 e
− 2π
i 23. Because
of the periodicity of the exponential, the three roots can be specified by any three
consecutive values of k; conveniently as
z
k1 = 1 e
2π
ki 23, k 1 = 1 0, ± 1
such thatz
01 = 11 ,z
± 11 = 1 e
± 2π
i 23.
In general the n nth roots of the number 1 are
6(8.44)
Thus when nis odd, the only real root is+ 1 (fork 1 = 10 ) and the rest occur as complex
conjugate pairs. When nis even, two of the roots are real,± 1 (fork 1 = 10 ,n 22 ). The n
representative points lie on the vertices of a regular n-sided polygon.
EXAMPLE 8.13The six sixth roots of 1.
The six roots are
z
k1 = 1 e
2 πki 26, k 1 = 1 0, ±1, ±2, 3
or z
01 = 1 e
01 = 11
z
31 = 1 e
πi1 = 1 cos 1 π 1 + 1 i 1 sin 1 π 1 = 1 − 1
0 Exercises 45–47
ze i i
i±±==± =−±
2232
3
2
3
1
2
3
2
πππ
cos sin
ze i i
i±±==± =±
1333
1
2
3
2
πππ
cos sin
ze k
nn
kki n==
,±,± , ,± −
,±
2012 12
01
πfor
... () if is odd
,, ± , , ± − ,
2212 ... ()nn nif is evenze i i
i2434
3
4
3
1
2
3
2
== + =−−
πππ
cos sin
ze i i
i1232
3
2
3
1
2
3
2
== + =−+
πππ
cos sin
6The nth roots of a complex number were discussed by de Moivre in a Philosophical Transactionspaper of 1739.
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........................................................................................z
0z
1z
2z
3z
- 2
z
- 1
Figure 8.9