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: 51b

The point $~~1-1~i~~$ is not in the Mandelbrot set.

Work Step by Step

$z = 1-1~i$ We can perform the calculation $z^2+z$: $z^2+z = (1-1~i)^2+(1-1~i)$ $z^2+z = (0-2~i)+(1-1~i)$ $z^2+z = 1-3~i$ We can perform the calculation $(z^2+z)^2+z$: $z^2+z = (1-3~i)^2+(1-1~i)$ $z^2+z = (-8-6~i)+(1-1~i)$ $z^2+z = -7-7~i$ The absolute value is $\sqrt{(-7)^2+(-7)^2} = \sqrt{98}$ which is greater than 2. Therefore, the point $~~1-1~i~~$ is not in the Mandelbrot set.

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