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Difference between revisions of "Absolute value"

From Encyclopedia of Mathematics
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''modulus, of a real number $a$''
 
''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 $+\sqrt{x^2+y^2}$.
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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 $|z| = +\sqrt{x^2+y^2}$.
  
 
=== Properties===
 
=== Properties===
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=== Generalization===
 
=== 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]].
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A generalization of the concept of the absolute value to the case of general fields exists, cf. [[Norm on a field]] and [[Valuation]].

Latest revision as of 16:51, 30 November 2014


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 $|z| = +\sqrt{x^2+y^2}$.

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 general fields exists, cf. Norm on a field and Valuation.

How to Cite This Entry:
Absolute value. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolute_value&oldid=28836