Editing Absolute value (section)

==Definition and properties==

===Real numbers===
For any {{nowrap|[[real number]] <math>x</math>,}} the '''absolute value''' or '''modulus''' {{nowrap|of <math>x</math>}} is denoted {{nowrap|by <math>|x|</math>}}, with a [[vertical bar]] on each side of the quantity, and is defined as<ref>Mendelson, [https://books.google.com/books?id=A8hAm38zsCMC&pg=PA2 p.&nbsp;2].</ref>
<math display=block>|x| =
   \begin{cases}
     x, & \text{if }  x \geq 0 \\
     -x, & \text{if } x < 0.
   \end{cases}
 </math>

The absolute value {{nowrap|of <math>x</math>}} is thus always either a [[positive number]] or [[0|zero]], but never [[negative number|negative]]. When <math>x</math> itself is negative {{nowrap|(<math>x < 0</math>),}} then its absolute value is necessarily positive {{nowrap|(<math>|x|=-x>0</math>).}}

From an [[analytic geometry]] point of view, the absolute value of a real number is that number's [[distance]] from zero along the [[real number line]], and more generally the absolute value of the difference of two real numbers (their [[absolute difference]]) is the distance between them.<ref>{{cite book|title=Precalculus: A Functional Approach to Graphing and Problem Solving|first=Karl|last=Smith|publisher=Jones & Bartlett Publishers|year=2013|isbn=978-0-7637-5177-7|page=8|url=https://books.google.com/books?id=ZUJbVQN37bIC&pg=PA8}}</ref> The notion of an abstract [[distance function]] in mathematics can be seen to be a generalisation of the absolute value of the difference (see [[#Distance|"Distance"]] below).

Since the [[radical symbol|square root symbol]] represents the unique ''positive'' [[square root]], when applied to a positive number, it follows that
<math display=block qid=Q120645811>|x| = \sqrt{x^2}.</math>
This is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers.<ref>{{Cite book| author=Stewart, James B. | title=Calculus: concepts and contexts | year=2001 | publisher=Brooks/Cole | location=Australia  | isbn=0-534-37718-1 | page=A5}}</ref>

The absolute value has the following four fundamental properties (<math display="inline">a</math>, <math display="inline">b</math> are real numbers), that are used for generalization of this notion to other domains:

{| style="margin-left:1.6em"
|-
| style="width: 250px" |<math qid=Q120645720>|a| \ge 0 </math>
| Non-negativity
|-
|<math>|a| = 0 \iff a = 0 </math>
|Positive-definiteness
|-
|<math>|ab| = \left|a\right| \left|b\right|</math>
|[[Multiplicativeness|Multiplicativity]]
|-
|<math qid=Q120645947>|a+b| \le |a| + |b|  </math>
| [[Subadditivity]], specifically the [[triangle inequality]]
|}

Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition.  To see that subadditivity holds, first note that <math>|a+b|=s(a+b)</math> {{nowrap|where <math>s=\pm 1</math>,}} with its sign chosen to make the result positive. Now, since <math>-1 \cdot x \le |x|</math> {{nowrap|and <math>+1 \cdot x \le |x|</math>,}} it follows that, whichever of <math>\pm1</math> is the value {{nowrap|of <math>s</math>,}} one has <math>s \cdot x\leq |x|</math> for all {{nowrap|real <math>x</math>.}} Consequently, <math>|a+b|=s \cdot (a+b) = s \cdot a + s \cdot b \leq |a| + |b|</math>, as desired.

Some additional useful properties are given below.  These are either immediate consequences of the definition or implied by the four fundamental properties above.

{| style="margin-left:1.6em"
|-
| style="width:250px" |<math>\bigl| \left|a\right| \bigr| = |a|</math> 
|[[Idempotence]] (the absolute value of the absolute value is the absolute value)
|-
| style="width:250px" |<math>\left|-a\right| = |a|</math>
|[[even function|Evenness]] ([[reflection symmetry]] of the graph)
|-
|<math>|a - b| = 0 \iff a = b </math>
|[[Identity of indiscernibles]] (equivalent to positive-definiteness)
|-
|<math>|a - b|  \le |a - c| + |c - b|  </math>
|[[Triangle inequality#Example norms|Triangle inequality]] (equivalent to subadditivity)
|-
|<math>\left|\frac{a}{b}\right| = \frac{|a|}{|b|}\ </math> (if <math>b \ne 0</math>)
|Preservation of division (equivalent to multiplicativity)
|-
|<math>|a-b| \geq \bigl| \left|a\right| - \left|b\right| \bigr| </math>
|[[Reverse triangle inequality]] (equivalent to subadditivity)
|}

Two other useful properties concerning inequalities are:
{| style="margin-left:1.6em"
|-
|<math>|a| \le b \iff -b \le a \le b </math>
|-
|<math>|a| \ge b \iff a \le -b\ </math> or <math>a \ge b </math>
|}

These relations may be used to solve inequalities involving absolute values. For example:

{| style="margin-left:1.6em"
|-
|<math>|x-3| \le 9 </math>
|<math>\iff -9 \le x-3 \le 9 </math>
|-
|
|<math>\iff -6 \le x \le 12 </math>
|}

The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standard [[Metric (mathematics)|metric]] on the real numbers.

===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.