We denote by
the set of all complex numbers whose absolute value is equal to 1.
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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.
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Note that this argument is defined up to an additional
.
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|
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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:
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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.
Noah Dana-Picard 2007-12-24