Powers of a complex number

Multiplying a complex number

with itself repeatedly, one can generate the powers of

$a,a^2,a^3,a^4,a^5,\ldots$

As every multiplication rotates the next power by the same angle and contracts it by the same factor, the points corresponding to these powers lie on a spiral: a logarithmic spiral.

In the following applet one can modify the number

. The applet will draw a number of powers in order. The number

of calculated powers can be controlled using the green slider.

Please enable Java for an interactive construction (with Cinderella).

Download file:

Potenz1.cdy

Complex conjugation $\hookleftarrow$ Contents $\hookrightarrow$ Complex transformations