A fractal is a geometric figure that consists of a pattern that is repeated infinitely on a smaller and smaller scale. The most famous fractal is the Mandelbrot Set, named after the Polish-born mathematician Benoit Mandelbrot (1924-2010). To draw the Mandelbrot Set, consider the sequence of numbers below.
c, c2 + c, (c2 + c)2 + c, [(c2 + c)2 + c]2 + c, . . .
The behavior of this sequence depends on the value of the complex number c. If the sequence is bounded (the absolute value of each number in the sequence,
| a + bi | = (a2 + b2
is less than some fixed number N), then the complex number c is in the Mandelbrot Set, and if the sequence is unbounded (the absolute value of the terms of the sequence become infinitely large), then the complex number c is not in the Mandelbrot Set. Determine whether the complex number c is in the Mandelbrot Set.
(a) c = i
(b) c = 1 + i
(c) c = ˆ’2
The figure below shows a graph of the Mandelbrot Set, where the horizontal and vertical axes represent the real and imaginary parts of c, respectively.

Transcribed Image Text:

Imaginary axis Real axis -2