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EXAMPLE 19.15
√
We will find the fourth roots of 1 + i. One argument of 1 + i is π/4, and |1 + i|= 2, so
1 + i = 2 e in which k can be any integer. The fourth roots are the four numbers
1/2 i(π/4+2kπ)
1/8 πi/16
1/8 i(π/4+4π)/4
1/8 i(π/4+6π)/4
1/8 i(π/4+2π)
2 e , 2 e , 2 e , and 2 e .
Other choices for k simply reproduce these numbers. The fourth roots of 1+i also can be written
as
π π

2 1/8 cos + i sin ,
16 16

9π 9π
2 1/8 cos + i sin ,
16 16

17π 17π
2 1/8 cos + i sin ,
16 16
and

25π 25π
2 1/8 cos + i sin .
16 16
EXAMPLE 19.16
The nth roots of 1 are called the nth roots of unity. They appear in many contexts: for example,
in the development of the fast Fourier transform. Since 1 has a magnitude of 1 and an argument
is 0, the nth roots of unity are the n numbers
e 2πki/n for k = 0,1,··· ,n − 1.
For example, the fifth roots of unity are
1, e 2πi/5 , e 4πi/5 , e 6πi/5 , and e 8πi/5 .
These are the numbers
2π 2π 4π 4π

1, cos + i sin , cos + i sin ,
5 5 5 5

6π 6π 8π 8π
cos + i sin , and cos + i sin .
5 5 5 5
If plotted as points in the plane, the nth roots of unity form vertices of a regular polygon with
vertices on the unit circle and having one vertex at (1,0). Figure 19.7 shows the fifth roots of
unity displayed in this way.

y

x

FIGURE 19.7 Fifth roots of unity.

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October 15, 2010 18:5 THM/NEIL Page-691 27410_19_ch19_p667-694