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''modulus, of a real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010370/a0103701.png" />''
 
  
The non-negative number, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010370/a0103702.png" />, which is defined as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010370/a0103703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010370/a0103704.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010370/a0103705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010370/a0103706.png" />. The absolute value (modulus) of a complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010370/a0103707.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010370/a0103708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010370/a0103709.png" /> are real numbers, is the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010370/a01037010.png" />. Absolute values obey the following relations
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010370/a01037011.png" /></td> </tr></table>
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010370/a01037012.png" /></td> </tr></table>
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''modulus, of a real number $a$''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010370/a01037013.png" /></td> </tr></table>
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The non-negative number, denoted by $a$, which is defined as follows: If $a\geq0$, $|a|=a$; if $a<0$, $|a|=-a$. The absolute value (modulus) of a complex number $z=x+iy$, where $x$ and $y$ are real numbers, is the number $+\sqrt{x^2+y^2}$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010370/a01037014.png" /></td> </tr></table>
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=== Properties===
  
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Absolute values obey the following relations
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* $|a| = |-a|$,
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* $|a|-|b|\leq |a+b| \leq |a| + |b|$,
 +
* $|a|-|b|\leq |a-b| \leq |a| + |b|$,
 +
* $|a\cdot b|=|a|\cdot |b|$,
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* if $b\ne0$ then $\left|\frac{a}{b}\right| = \frac{|a|}{|b|}$,
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* $|a|^2 = |a^2| = a^2$ (only for real numbers).
 +
 +
=== Generalization===
 
A generalization of the concept of the absolute value to the case of arbitrary fields exists, cf. [[Norm on a field|Norm on a field]].
 
A generalization of the concept of the absolute value to the case of arbitrary fields exists, cf. [[Norm on a field|Norm on a field]].

Revision as of 14:25, 23 November 2012


modulus, of a real number $a$

The non-negative number, denoted by $a$, which is defined as follows: If $a\geq0$, $|a|=a$; if $a<0$, $|a|=-a$. The absolute value (modulus) of a complex number $z=x+iy$, where $x$ and $y$ are real numbers, is the number $+\sqrt{x^2+y^2}$.

Properties

Absolute values obey the following relations

  • $|a| = |-a|$,
  • $|a|-|b|\leq |a+b| \leq |a| + |b|$,
  • $|a|-|b|\leq |a-b| \leq |a| + |b|$,
  • $|a\cdot b|=|a|\cdot |b|$,
  • if $b\ne0$ then $\left|\frac{a}{b}\right| = \frac{|a|}{|b|}$,
  • $|a|^2 = |a^2| = a^2$ (only for real numbers).

Generalization

A generalization of the concept of the absolute value to the case of arbitrary fields exists, cf. Norm on a field.

How to Cite This Entry:
Absolute value. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolute_value&oldid=17245