The graph of the absolute value function for real numbers
The absolute value of a number may be thought of as its distance from zero.
The absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a positive x, |x| = −x for a negative x (in which case −x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.
The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.
Video: Absolute Value of Integers
Absolute value of real numbers
For any real number x, the absolute value or modulus of x is denoted by |x| (a vertical bar on each side of the quantity) and is defined as
The absolute value of x is thus always either positive or zero, but never negative, since x < 0 implies −x > 0.
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, and more generally the absolute value of the difference of two real numbers is the distance between them.
Indeed, the notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference (see "Distance" below).
Video: Definition of the absolute value of a real number
Since the square root symbol represents the unique positive square root (when applied to a positive number), it follows that
is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers.
Properties of absolute value
The absolute value has the following four fundamental properties (a, b are real numbers), that are used for generalization of this notion to other domains:
Non-negativity
Positive-definiteness
Multiplicativity
Subadditivity, specifically the triangle inequality
Video: Absolute value properties
Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition.
Some additional useful properties are given below. These are either immediate consequences of the definition or implied by the four fundamental properties above.
Idempotence (the absolute value of the absolute value is the absolute value)
Evenness (reflection symmetry of the graph)
Identity of indiscernibles (equivalent to positive-definiteness)
Triangle inequality (equivalent to subadditivity)
(if )
Preservation of division (equivalent to multiplicativity)
Reverse triangle inequality (equivalent to subadditivity)
Two other useful properties concerning inequalities are:
or
These relations may be used to solve inequalities involving absolute values. For example:
The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standard metric on the real numbers
Distance and its relation to absolute value
The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them.
Video: Determine the Distance Between Two Points
The standard Euclidean distance between two points
and
in Euclidean n-space is defined as:
This can be seen as a generalisation, since for real, i.e. in a 1-space, according to the alternative definition of the absolute value,
and for and complex numbers, i.e. in a 2-space,
The above shows that the "absolute value"-distance, for real and complex numbers, agrees with the standard Euclidean distance, which they inherit as a result of considering them as one and two-dimensional Euclidean spaces, respectively.
How to arrange absolute values from the smallest to the largest
This video shows how to arrange absolute values from the smallest to the largest.
Examples to find absolute values
This video shows examples to find absolute values.
Comparing absolute values
This video shows how to compare absolute values.
Finding absolute value of a complex number
This video shows how to find absolute value of a complex number.
Let's Review
The absolute value of a number may be thought of as its ____ from ____ .
The absolute value of the difference of two real numbers is the ____ between them.
The absolute value of a real or complex number is the distance from that number to the origin, along the ____ ____ ____ , for real numbers, or in the ____ ____ , for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the ____ between them.
Answer
The absolute value of a number may be thought of as its distance from zero.
The absolute value of the difference of two real numbers is the distance between them.
The absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them.
(if 
)
or 
real, i.e. in a 1-space, according to the alternative definition of the absolute value, 
and 
complex numbers, i.e. in a 2-space,
