We know that one full revolution of the circle occurs

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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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