Absolute value

In mathematics, the absolute value of a real number formula_1, written as or formula_2, is the value of formula_1 when the sign is dropped (or formula_1 without its sign). That is, formula_5 for a positive formula_1, formula_7 for a negative formula_1, and formula_9 
For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3 (formula_10). The absolute value of a real number may be thought of as its distance from zero. It can be defined as follows:
Because a distance is always positive, the absolute value of a number is always positive.
Similarly, the absolute value of a complex number may be thought of as its distance from the origin.
It is defined by the equation 
Properties of the absolute value.
Real numbers.
For any real number formula_13, the absolute value of formula_13 is denoted by (a vertical bar on each side of the quantity), and is defined as
The absolute value of formula_13 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.
In calculus, the absolute value function is differentiable except at 0. It is continuous everywhere.
In linear algebra, the norm of a vector is defined similarly as the distance from the tip of the vector to the origin. This is similar to the way the absolute value of a complex number is defined.
The square root notation without sign represents the "positive" square root. So, it follows that:
which is another definition of absolute value.
The absolute value has the following four main properties:
Other important properties of the absolute value include:
Two other useful properties related to inequalities are:
These relations may be used to solve inequalities involving absolute values. For example:
formula_20
Complex numbers.
For a complex number formula_21, where formula_22 is the real part of formula_23 and formula_24 is the imaginary part of formula_23,