Complex Numbers: Absolute Value, Conjugate, Distance Formulas and Facts

Absolute value:

The absolute value (or modulus or magnitude) of a complex number z = r e^iφ 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

Distance:

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.

Conjugate:

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.

To find out the absolute value or the square root of a complex number use our complex numbers calculator!