Editing Complex number (section)

===Complex conjugate, absolute value, argument and division===

[[File:Complex conjugate picture.svg|right|thumb|upright=0.8|Geometric representation of {{mvar|z}} and its conjugate {{mvar|{{overline|z}}}} in the complex plane.]]
The ''[[complex conjugate]]'' of the complex number {{math|1=''z'' = ''x'' + ''yi''}} is defined as
<math>\overline z = x-yi.</math><ref>{{harvnb|Apostol|1981|pp=15–16}}</ref> It is also denoted by some authors by <math>z^*</math>. Geometrically, {{mvar|{{overline|z}}}} is the [[reflection symmetry|"reflection"]] of {{mvar|z}} about the real axis. Conjugating twice gives the original complex number: <math>\overline{\overline{z}}=z.</math> A complex number is real if and only if it equals its own conjugate. The [[unary operation]] of taking the complex conjugate of a complex number cannot be expressed by applying only the basic operations of addition, subtraction, multiplication and division. 

[[File:Complex number illustration modarg.svg|right|thumb|Argument {{mvar|φ}} and modulus {{mvar|r}} locate a point in the complex plane.]]
For any complex number {{math|1=''z'' = ''x'' + ''yi''}} , the product 
:<math>z \cdot \overline z = (x+iy)(x-iy) = x^2 + y^2</math>
is a ''non-negative real'' number. This allows to define the ''[[absolute value]]'' (or ''modulus'' or ''magnitude'') of ''z'' to be the square root{{sfn|Apostol|1981|p=18}}
<math display="block">|z|=\sqrt{x^2+y^2}.</math>
By [[Pythagoras' theorem]], <math>|z|</math> is the distance from the origin to the point representing the complex number ''z'' in the complex plane. In particular, the [[unit circle|circle of radius one]] around the origin consists precisely of the numbers ''z'' such that <math>|z| = 1 </math>. If <math>  z = x = x + 0i  </math> is a real number, then <math>  |z|= |x| </math>: its absolute value as a complex number and as a real number are equal. 

Using the conjugate, the [[multiplicative inverse|reciprocal]] of a nonzero complex number <math>z = x + yi</math> can be computed to be 

<math display=block>
\frac{1}{z}
= \frac{\bar{z}}{z\bar{z}}
= \frac{\bar{z}}{|z|^2}
= \frac{x - yi}{x^2 + y^2}
= \frac{x}{x^2 + y^2} - \frac{y}{x^2 + y^2}i.</math>
More generally, the division of an arbitrary complex number <math>w = u + vi</math> by a non-zero complex number <math>z = x + yi</math> equals
<math display=block>
\frac{w}{z}
= \frac{w\bar{z}}{|z|^2}
= \frac{(u + vi)(x - iy)}{x^2 + y^2}
= \frac{ux + vy}{x^2 + y^2} + \frac{vx - uy}{x^2 + y^2}i.
</math>
This process is sometimes called "[[rationalisation (mathematics)|rationalization]]" of the denominator (although the denominator in the final expression may be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.<ref>{{cite book |title=Numerical Linear Algebra with Applications: Using MATLAB and Octave |author1=William Ford |edition=reprinted |publisher=Academic Press |year=2014 |isbn=978-0-12-394784-0 |page=570 |url=https://books.google.com/books?id=OODs2mkOOqAC}} [https://books.google.com/books?id=OODs2mkOOqAC&pg=PA570 Extract of page 570]</ref><ref>{{cite book |title=Precalculus with Calculus Previews: Expanded Volume |author1=Dennis Zill |author2=Jacqueline Dewar |edition=revised |publisher=Jones & Bartlett Learning |year=2011 |isbn=978-0-7637-6631-3 |page=37 |url=https://books.google.com/books?id=TLgjLBeY55YC}} [https://books.google.com/books?id=TLgjLBeY55YC&pg=PA37 Extract of page 37]</ref>

The ''[[argument (complex analysis)|argument]]'' of {{mvar|z}} (sometimes called the "phase" {{mvar|φ}})<ref name=":2" /> is the angle of the [[radius]] {{mvar|Oz}} with the positive real axis, and is written as {{math|arg ''z''}}, expressed in [[radian]]s in this article. The angle is defined only up to adding integer multiples of <math>  2\pi </math>, since a rotation by <math>2\pi</math> (or 360°) around the origin leaves all points in the complex plane unchanged. One possible choice to uniquely specify the argument is to require it to be within the interval <math>  (-\pi,\pi]  </math>, which is referred to as the [[principal value]].<ref>Other authors, including {{harvnb|Ebbinghaus|Hermes|Hirzebruch|Koecher|Mainzer|Neukirch|Prestel|Remmert|1991|loc=§6.1}}, chose the argument to be in the interval <math>[0, 2\pi)</math>.</ref>
The argument can be computed from the rectangular form {{mvar|x + yi}} by means of the [[arctan]] (inverse tangent) function.<ref>{{cite book
|title=Complex Variables: Theory And Applications
|edition=2nd
|chapter=Chapter 1
|first1=H.S.
|last1=Kasana
|publisher=PHI Learning Pvt. Ltd
|year=2005
|isbn=978-81-203-2641-5
|page=14
|chapter-url=https://books.google.com/books?id=rFhiJqkrALIC&pg=PA14}}</ref>