Classify whether the following numbers are in the Mandelbrot Set. I : $- 1$ II : $- i$ III : $3 i$
A)Only numbers I and III are in the Mandelbrot Set.
B)Only number II is in the Mandelbrot Set.
C)Only numbers II and III are in the Mandelbrot Set.
D)Only numbers I and II are in the Mandelbrot Set.
E)None are in the Mandelbrot Set.

Question

Quizplus · Accepted Answer

The Mandelbrot set is defined for complex numbers using the iteration $z_{n+1} = z_n^2 + c$, starting with $z_0 = 0$, where $c$ is the complex number in question. A number $c$ is in the Mandelbrot set if the sequence does not tend to infinity. For $-1$, the sequence oscillates between 0 and -1, so it's in the set. For $-i$, the sequence also remains bounded, so it's in the set. However, for $3i$, the sequence rapidly grows in magnitude and tends to infinity, so it's not in the set.

Classify Whether the Following Numbers Are in the Mandelbrot Set −1- 1−1

Classify Whether the Following Numbers Are in the Mandelbrot Set $- 1$