Complex Numbers

There is a tool for inserting a complex number in the graphics view and you can also insert a number using the input bar, i is used to denote i.

If you have a complex-valued function of a complex number, you can not draw the graph of that function since this would require four dimensions. When describing complex mappings, you can instead map a set of the complex plane onto another set. In the applet above the red circle is mapped, and in the applet below the red line.

Use Locus to make complex mappings

In order to demonstrate complex mappings in GeoGebra do this;

• Make a circle and put a point Z on the circle.
• Enter another point U=1/Z, the calculation of the new point is made as if the points were complex numbers.
• Use the tool Locus, click on U and then on Z.

You can not calculate U=Z2 in this way, if you square a point the result will be the square of the distance of the point to the origin.

Note that under the mapping \(f(z)= \dfrac{1}{z}\) circles/lines are mapped onto circles/lines; this is a special case of a Möbius transformation.

Möbius transformations

A Möbius transformation is a so called conformal map, a map that preserves angles. If the domain is a rectangle, the range will be an image in which all angles are 90°, it will not however (in the general case) be a rectangle. The applet above shows Möbius transformations of three different polygons, you can change the polygons by dragging the points. Circles/lines map onto circles/lines. Half a plane is mapped onto a circle disc.

Complex roots

The equation $$x^3=1 \Leftrightarrow x^3-1=0$$ is easy to solve if you are only interested in real roots. If you, however, want to find all roots, it is a tad more complicated. In GeoGebra 4 there is a command called ComplexRoot[] that gives you all roots to a complex polynomial equation. Try to enter the code:

in the input bar.