# Absolute value

From Encyclopedia of Mathematics

*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=35189