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The absolute value (or modulus or magnitude) of a complex number z = r eiφ is defined as |z| = r. Algebraically, if
z = a + bi, then
Absolute Value Properties
For all complex numbers z and w the following can be checked:
|if and only if|
By defining the distance function d(z, w) = |z − w| we turn the set of complex numbers into a metric space and we can therefore talk about limits and continuity.
The complex conjugate of the complex number z = a + bi is defined to be . As seen in the figure, is the "reflection" of z about the real axis.The following can be checked:
|if and only if z is real|
if z is non-zero.