The absolute value of a number is its magnitude or distance from the origin. Absolute value is defined in real numbers and in complex numbers.
Definition 

Absolute value of a real number
The absolute value of a real number n, , is defined as 
Definition 

Absolute value of a complex number
The absolute value of a complex number is defined as 
The definition may also be extended to vectors, where absolute value is often times called "magnitude".
Definition 

Magnitude of a vector
The magnitude of a vector is defined as Notice that magnitude is sometimes denoted with a double stroke absolute value

The absolute value signs are also used in set theory to indicate a cardinality, or set size. The operation measures the magnitude of a set, or in other words, the operation counts the number of elements a set contains.
Definition 

Cardinality of a set
The cardinality of a set containing seven elements is defined as 