Fractals: A Very Short Introduction by Kenneth Falconer

Author:Kenneth Falconer [Falconer, Kenneth]
Language: eng
Format: epub
ISBN: 9780199675982
Publisher: Oxford University Press
Published: 2013-12-15T06:00:00+00:00

25. (a) Addition of two complex numbers shifts one parallel to the other, (b) squaring a complex number squares the magnitude and doubles the angle

Pythagoras’ Theorem provides a straightforward way to find the magnitude of complex numbers. To get from the origin to the complex number 2 + i (the point with coordinates (2, 1)) we step a distance 2 to the right and then step a distance 1 up. But the line joining the origin to 2 + i is the hypotenuse of the right-angled triangle with these two steps as the other sides. Pythagoras’ Theorem tells us that the square of the length of this hypotenuse is 22 + 12, so that the magnitude of 2 + i is . In just the same way, the magnitude of a general complex number z = x + yi is .

We saw above that (2 + i)2 = 3 + 4i. The magnitude of 3 + 4i is which is the square of , the magnitude of 2 + i. We can also calculate the angles of these complex numbers: the angle of 3 + 4i is 53.130° which is exactly double 26.565°, the angle of 2 + i, see Figure 25(b). Squaring 2 + i gives a complex number of double the angle and the square of the magnitude. This is no coincidence, but is an instance of a remarkable property of complex numbers:

Squaring a complex number squares the magnitude and doubles the angle

Put another way, for every complex number z, the magnitude of z2 is the square of that of z, and the angle of z2 is double that of z. Verifying this property is a little involved, and more details are given in the Appendix.

Throughout this chapter we will be interested in a combination of these two operations, squaring and addition. We will be particularly concerned with the function

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