Editing Absolute value (section)

===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.