# 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 $+\sqrt{x^2+y^2}$. | + | 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 | + | 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=28824