Unformatted text preview: ep process. The multiplication of |z | by |w| can be interpreted as magnifying12 the
distance |z | from z to 0, by the factor |w|. Adding the argument of w to the argument of z can be
interpreted geometrically as a rotation of β radians counter-clockwise.13 Focusing on z and w from
13 Assuming |w| > 1.
Assuming β > 0. 11.7 Polar Form of Complex Numbers 851 Example 11.7.3, we can arrive at the product zw by plotting z , doubling its distance from 0 (since
|w| = 2), and rotating 23 radians counter-clockwise. The sequence of diagrams below attempt to
geometrically describe this process.
Imaginary Axis Imaginary Axis 6i 6i 5i π
6 zw = 8cis 4i π
6 z |w| = 8cis + 5i 2π
3 4i 3i z |w| = 8cis π
6 3i 2i 2i π
6 z = 4cis i i
01 2 3 4 5 6 7 Real Axis −7 −6 −5 −4 −3 −2 −1 Multiplying z by |w| = 2. 01 2 3 4 5 Rotating counter-clockwise by Arg(w) = Visualizing zw for z = 4cis π
3 and w = 2cis Real Axis 6 7 2π
3 radians. . z
We may also visualize division similarly. Here, the formula w = ||w|| cis(α − β ) may be interpreted...
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