# Roots of Complex Numbers

This applet investigates `n`th 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

TweetInstructions below See also: Complex numbers 1 Complex transformations

## 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 *n*th 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 *n*th roots of a number always add up to zero?

Play around! You should quickly develop a feel for the behaviour of complex roots.