Complex plane

This applet illustrates the complex plane (sometimes called the Argand plane in older books), which can be used to display complex numbers, in the same way the real line is used to display real numbers.

The red dot on the plane represents a complex number z. The number z is usually expressed in Cartesian form x+yi, or in Polar form r exp(i theta). On the bottom of the applet you can see both forms of expression.

There are several ways to move z around. One is simply to click on the grid. Another is to manually change the x and y co-ordinates (or the r and theta co-ordinates). Note that you have to press "Enter" in order to process any change in these co-ordinates. You do not need to enter the "pi" when altering the theta co-ordinate. The "<" and ">" buttons decrement and increment these four co-ordinates by a fixed amount.

Finally, you can move z around by pressing one of the buttons on the top of the applet. For instance, the 2z button replaces z by 2z; i.e. it doubles the complex number. The 3+2i button sends you to the complex number 3+2i no matter what z is. And so forth. Note that the 1/z button is disabled when z is zero.

To test your knowledge of complex arithmetic, place the dot at a random location, select a random button, and see if you can predict where the dot is going to go.

Next applet: Elementary complex maps

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