We denote by the set of all complex numbers whose absolute value is equal to 1.

For example, and , but .

The image of in the Cauchy-Argand plane is the unit circle: if , with real , then if, and only if .

We leave the proof to the reader; use Corollary 1.18.

whence .

Note that this argument is defined up to an additional .

In Fig 5, a value of the argument of the complex number corresponding to a point is diplayed in green.

We can now generalize this to any non zero complex number. First note that for any , we have , as the following holds:

- Take
. The absolute value of
is given by
. Then we have:

It follows that . - Now take
, then
and the following hold:

It follows that .

The following proposition is easy to understand. It explains the specific role of the coordinate axes when representing complex numbers in the Cauchy-Argand plane.

- (i)
- .
- (ii)
- is pure imaginary .

- (i)
- .
- (ii)
- Please do it yourself.

Noah Dana-Picard 2007-12-24