Editing Absolute value (section)

===Complex numbers===
{{Anchor|complex modulus}}[[Image:Complex conjugate picture.svg|right|thumb|The absolute value of a {{nowrap|[[complex number]] <math>z</math>}} is the {{nowrap|distance <math>r</math>}} {{nowrap|of <math>z</math>}} from the origin. It is also seen in the picture that <math>z</math> and its {{nowrap|[[complex conjugate]] <math>\bar z</math>}} have the same absolute value.]]

Since the [[complex number]]s are not [[Totally ordered set|ordered]], the definition given at the top for the real absolute value cannot be directly applied to complex numbers. However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the [[complex plane]] from the [[origin (mathematics)|origin]].  This can be computed using the [[Pythagorean theorem]]: for any complex number
<math display=block>z = x + iy,</math>
where <math>x</math> and <math>y</math> are real numbers, the '''absolute value''' or '''modulus''' {{nowrap|of <math>z</math>}} is {{nowrap|denoted <math>|z|</math>}} and is defined by<ref>{{cite book|author=González, Mario O.|title=Classical Complex Analysis|publisher=CRC Press|year=1992|isbn=9780824784157|page=19|url=https://books.google.com/books?id=ncxL7EFr7GsC&pg=PA19}}</ref>
<math display=block>|z| = \sqrt{\operatorname{Re}(z)^2 + \operatorname{Im}(z)^2}=\sqrt{x^2 + y^2},</math>
the [[Pythagorean addition]] of <math>x</math> and <math>y</math>, where <math>\operatorname{Re}(z)=x</math> and <math>\operatorname{Im}(z)=y</math> denote the real and imaginary parts {{nowrap|of <math>z</math>,}} respectively.  When the {{nowrap|imaginary part <math>y</math>}} is zero, this coincides with the definition of the absolute value of the {{nowrap|real number <math>x</math>.}}

When a complex number <math>z</math> is expressed in its [[Complex number#Polar form|polar form]] {{nowrap|as <math>z = r e^{i \theta},</math>}} its absolute value {{nowrap|is <math>|z| = r.</math>}}

Since the product of any complex number <math>z</math> and its {{nowrap|[[complex conjugate]] <math>\bar z = x - iy</math>,}} with the same absolute value, is always the non-negative real number {{nowrap|<math>\left(x^2 + y^2\right)</math>,}} the absolute value of a complex number <math>z</math> is the square root {{nowrap|of  <math>z \cdot \overline{z},</math>}} which is therefore called the [[absolute square]] or ''squared modulus'' {{nowrap|of <math>z</math>:}}
<math display=block>|z| = \sqrt{z \cdot \overline{z}}.</math>
This generalizes the alternative definition for reals: {{nowrap|<math display="inline">|x| = \sqrt{x\cdot x}</math>.}}

The complex absolute value shares the four fundamental properties given above for the real absolute value. The identity <math>|z|^2 = |z^2|</math> is a special case of multiplicativity that is often useful by itself.