In mathematics, the absolute value or modulus of a real number x, written as |x| or , is the non-negative value of x when the sign is dropped. That is, |x| = x for a positive x, |x| = −x for a negative x (in which case −x is positive), and |0| = 0.
Properties[change | change source]
Real numbers[change | change source]
The absolute value of x is always either positive or zero, but never negative. From an analytic geometry point of view, the absolute value of a real number is that number's distance from zero along the real number line. The absolute value of the difference of two real numbers is the distance between them.
The square-root notation without sign represents the positive square root. So, it follows that
which is sometimes used as a definition of absolute value.
The absolute value has the following four main properties:
Other important properties of the absolute value include:
(6) Idempotence (the absolute value of the absolute value is the absolute value) (7) Symmetry (8) Identity of indiscernibles (equivalent to positive-definiteness) (9) Triangle inequality (equivalent to subadditivity) (10) Preservation of division (equivalent to multiplicativeness) (11) (equivalent to subadditivity)
Two other useful properties related to inequalities are:
These relations may be used to solve inequalities involving absolute values. For example:
Complex numbers[change | change source]
Related pages[change | change source]
References[change | change source]
- "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-28.
- "Absolute Value". www.mathsisfun.com. Retrieved 2020-08-28.
- Weisstein, Eric W. "Absolute Value". mathworld.wolfram.com. Retrieved 2020-08-28.
- Mendelson, p. 2.
- Stewart, James B. (2001). Calculus: concepts and contexts. Australia: Brooks/Cole. ISBN 0-534-37718-1. Cite has empty unknown parameter:
|coauthors=(help), p. A5