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**Unformatted text preview: **shrinking14 the distance from 0 to z by the factor |w|, followed up by a clockwise 15 rotation of β
z
radians. In the case of z and w from Example 11.7.3, we arrive at w by ﬁrst halving the distance
2π
from 0 to z , then rotating clockwise 3 radians.
Imaginary Axis Imaginary Axis 3i i 2i z = 4cis i 1
|w| 0 1 2 z = 2cis 3 π
6 0 1 2 π
6 z = 2cis 3 Real Axis −i π
6 −2i Real Axis Dividing z by |w| = 2.
Visualizing 1
|w| zw = 2cis π 2π
63 Rotating clockwise by Arg(w) =
z
for z = 4cis
w π
6 and w = 2cis 2π
3 2π
3 radians. . Our last goal of the section is to reverse DeMoivre’s Theorem to extract roots of complex numbers.
Definition 11.4. Let z and w be complex numbers. If there is a natural number n such that
wn = z , then w is an nth root of z .
Unlike Deﬁnition 5.4 in Section 5.3, we do not specify one particular prinicpal nth root, hence the
use of the indeﬁnite article ‘an’ as in ‘an nth root of z ’. Using this deﬁnition, both 4 and −4 are
14
15 Again, assuming |w| >...

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