6.2
Definition (Absolute value.)
In exercise
4.23A we showed that (for any field
![$F$](img406.gif)
in which
![$-1$](img33.gif)
is not a
square), if
![$z=a+bi=(a,b)\in\mbox{{\bf C}}_F$](img1494.gif)
, then
If we are working in
![$\mbox{{\bf C}}$](img1493.gif)
, then
![$a^2+b^2\in[0,\infty)$](img1496.gif)
and hence
![$zz^*$](img1497.gif)
has a unique
square root in
![$[0,\infty)$](img1431.gif)
, which we denote by
![$\vert z\vert$](img1498.gif)
and call the
absolute
value of
![$z$](img289.gif)
.
We note that
Also note that for
![$z\in\mbox{{\bf R}}$](img1501.gif)
, this definition agrees with our old definition of
absolute value in
![$\mbox{{\bf R}}$](img205.gif)
.
6.5
Exercise.
Prove parts b), c), d), e), f), g), h) and i) of Theorem
6.4.
A