Introduction

The Complex Visualiser is for you to explore complex numbers in a geometrical setting. You can draw points (that is, complex numbers) on an Argand diagram and see how those points transform under various operations. Various options allow you to see how the original points and calculated points are related.

Usage

The basic usage of this program is to draw points on the complex plane (or Argand diagram).

Points can be labelled and can have guides shown. The style of the labels and guides can be either Cartesian or Polar. If Cartesian then the label is of the form $x+\mathrm{i}y$ and the guides show how it relates to $x$ and $y$ on the axes. If Polar then the label is of the form $(r,\theta )$ and the guides show how $r$ is the length and $\theta$ is the angle. Lastly, when drawing points you can choose to show only the current point or a trace of all points.

You can choose to apply an operation to your points. Some of these operations require one or more extra points, which this program calls control points. If needed, these are displayed by default as slightly larger points. If more than one is needed, they are ordered by brightness. These can be moved around by dragging them. In addition, where deemed useful then certain diagrams are shown to illustrate the operation.

The operations are as follows. In the explanations, $z$ represents the point you make by clicking, $w$ represents the result of the operation, and the $z_{\mathrm{i}}$ are the control points.

• None: No operation is applied, only the points you draw are shown.

• Add: Each point is added to the control point, So $w=z+z_1$. The diagram shows the addition parallelogram.

• Subtraction: The control point is subtracted from each point. So $w=z-z_1$. The diagram shows the subtraction parallelogram.

• Multiplication: Each point is multiplied by the control point. So $w=z\times z_1$. The diagram shows the two similar triangles: $(0,1,z)$ and $(0,z_1,w)$.

• Division: Each point is divided by the control point. So $w=z/z_1$. The diagram shows the two similar triangles: $(0,1,w)$ and $(0,z_1,z)$.

• Power: Each point is raised to the power of the control point. So $w=z^{z_1}$. Note that for most values of $z_1$ this is multi-valued. Only the first value is shown.

• Multi-Power: Each point is raised to the power of the control point. So $w=z^{z_1}$. Note that for most values of $z_1$ this is multi-valued. Five values are shown, with the principal value in the middle.

• Conjugation: Each point is conjugated, so $w=\overline{z}$.

• Exponentiation: The control point is raise to the power of each point. So $w={z_1}^z$. Note that for most values of $z_1$ this is multi-valued. Only the first value is shown.

• Multi-Exponentiation: The control point is raise to the power of each point. So $w={z_1}^z$. Note that for most values of $z_1$ this is multi-valued. Five values are shown, with the principal value in the middle.

• Roots: The roots of each point are calculated. You can choose which roots to calculate, in the range 2 to 12. All of the roots are shown, and the diagram shows the resulting regular polygon.

• Möbius: A Möbius transformation is applied to each point. This requires four control points. The transformation is $w=\frac{z_{1}z+z_2}{z_{3}z+z_4}$. By choosing the control points carefully, lots of different transformations can be viewed using this.