## Trigonometry (11th Edition) Clone

Published by Pearson

# Chapter 8 - Complex Numbers, Polar Equations, and Parametric Equations - Section 8.4 De Moivre's Theorem: Powers and Roots of Complex Numbers - 8.4 Exercises - Page 383: 51c

#### Answer

The point $~~-0.5~i~~$ is not in the Mandelbrot set.

#### Work Step by Step

$z = -0.5~i = -\frac{i}{2}$ We can perform the calculation $z^2+z$: $z^2+z = (-\frac{i}{2})^2+(-\frac{i}{2})$ $z^2+z = -\frac{1}{4}-\frac{~i}{2}$ We can perform the calculation $(z^2+z)^2+z$: $z^2+z = (-\frac{1}{4}-\frac{~i}{2})^2-\frac{i}{2}$ $z^2+z = (-\frac{3}{16}+\frac{i}{4})-\frac{i}{2}$ $z^2+z = -\frac{3}{16}-\frac{i}{4}$ We can perform the calculation $[(z^2+z)^2+z]^2+z$: $z^2+z = (-\frac{3}{16}-\frac{i}{4})^2-\frac{i}{2}$ $z^2+z = (-\frac{7}{256}+\frac{3i}{32})-\frac{i}{2}$ $z^2+z = -\frac{7}{256}-\frac{13i}{32}$ The absolute value is less than 2, and if we continue doing this calculation, the absolute value seems to be getting smaller and smaller. Therefore, the point $~~-0.5~i~~$ is not in the Mandelbrot set.

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