Absolute value

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.