Roots of Complex Numbers

This applet investigates nth roots of complex numbers and their realtionship when drawn on a Argand Diagram.

Author and programmer: Ron Barrow

GCE Further Mathematics - Complex numbers - OCR, Edexcel, AQA, MEI FP1

   
           

Instructions below    See also:   Complex numbers 1   Complex transformations

[Applet - roots of complex numbers]

How to Use this Applet

On this Argand diagram, the complex number z is represented by a green dot, which can be dragged around the plane with your mouse to change its value. This number z is written in 3 different forms at the top left of the screen.
Also shown are the nth roots of that number, where n is a positive integer between 2 and 30. You can change n by clicking "Root+" or "Root-" at the bottom. These roots are shown as pink dots. The blue dot is the "first" root, calculated by putting r = 0 in the equation for Arg(z).

The other roots are calculated by putting
0 < r < n - 1 in the expression for the argument.
How to calculate the modulus and argument of each root is demonstrated at the bottom. As you investigate you should see a simple pattern emerge.
Here are a question you might want to think about:- Why do the nth roots of a number always add up to zero?
Play around! You should quickly develop a feel for the behaviour of complex roots.