To illustrate, begin with and substitute . The equation then becomes , which simplifies to .
Exploration.
(c) 2006 John H. Mathews, Russell W. Howell
![[Graphics:Images/ComplexNumberOrigin_gr_13.gif]](../Images/ComplexNumberOrigin_gr_13.gif){kind=small}
and
substitute ![[Graphics:Images/ComplexNumberOrigin_gr_14.gif]](../Images/ComplexNumberOrigin_gr_14.gif){kind=small}
. The
equation then becomes ![[Graphics:Images/ComplexNumberOrigin_gr_15.gif]](../Images/ComplexNumberOrigin_gr_15.gif){kind=small}
,
which simplifies to ![[Graphics:Images/ComplexNumberOrigin_gr_16.gif]](../Images/ComplexNumberOrigin_gr_16.gif){kind=small}
. and
substitute . The
equation then becomes ,
which simplifies to .![[Graphics:../Images/ComplexNumberOrigin_gr_17.gif]](../Images/ComplexNumberOrigin_gr_17.gif)
![[Graphics:../Images/ComplexNumberOrigin_gr_18.gif]](../Images/ComplexNumberOrigin_gr_18.gif)