Absolute value facts for kids
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.
Similarly, the absolute value (or modulus) of a complex number may be thought of as its distance from the origin. It is defined by the equation
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:
() Non-negativity () Positive-definiteness () Multiplicativeness () Subadditivity
Other important properties of the absolute value include:
() Idempotence (the absolute value of the absolute value is the absolute value) () Symmetry () Identity of indiscernibles (equivalent to positive-definiteness) () Triangle inequality (equivalent to subadditivity) () Preservation of division (equivalent to multiplicativeness) () (equivalent to subadditivity)
Two other useful properties related to inequalities are:
These relations may be used to solve inequalities involving absolute values. For example:
Absolute value Facts for Kids. Kiddle Encyclopedia.