In mathematics, the absolute value or modulus of a real number is the number without the sign. The absolute value of 2 is 2, the absolute value of -2 is also 2. This notation is to express a number's distance from zero on a number line. The absolute value of 10 would be 10 since the number 10 is 10 numbers away from zero, same follows with negatives.
The modulus of a complex number is given by
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:
() 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 concerning inequalities are:
These relations may be used to solve inequalities involving absolute values. For example: