Page 795 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition
P. 795
generated by rotating this root through increments of radians . From Figure 10.3.6, we see that the fourth
roots of 1 are
Figure 10.3.6
The fourth roots of 1.
Complex Exponents
We conclude this section with some comments on notation.
In more detailed studies of complex numbers, complex exponents are defined, and it is shown that
where e is an irrational real number given approximately by (11)
this result is given in Exercise 18.) …. (For readers who have studied calculus, a proof of
It follows from 11 that the polar form
can be written more briefly as
(12)
EXAMPLE 5 Expressing a Complex Number in Form 12
In Example 1 it was shown that
From 12 this can also be written as
It can be proved that complex exponents follow the same laws as real exponents, so if
are nonzero complex numbers, then
