The absolute value or modulus | x | of a real number x, is the non-negative value of x with disregard to the sign. Basically, | x | = x for a positive value of x, | x | = - x , for a negative value of x, where x is positive and | 0 | = 0. Example can be cited, the absolute value of 4 is 4 and absolute value of – 4 is also 4. The absolute value of a number is the distance between that number and zero. The generalization of absolute value of real numbers takes place in a wide variety of mathematical settings. For example, absolute value can be defined for the complex numbers, ordered rings, quaternions, vector spaces and fields. Absolute value is closely related with the notion of magnitude, norm and distance in different mathematical and physical contexts.

Any real number x, the absolute value or modulus of x can be denoted by | x | , represented using a vertical bar on either side of the quantity, and defined as

| x | = x, if x ≥ 0

| x | = - x if x < 0

The absolute value of x is always either positive or zero, but it is never negative, when x itself is negative ( x < 0 ), then the absolute value is in general positive ( | x | = - x > 0). The analytical geometric point of view, the absolute value of a real number is the distance between that number and zero along the line of real numbers, and more generalized purview of an absolute value of the difference between two real numbers is the actual distance between them. The notion of abstract distant function in mathematics is seen as a generalized view of the absolute value of the difference. The square root symbol represents an unique positive square root, when it is applied to a positive number, it is followed as | x | =√x² , which is equivalent to the above definition and can be used as an alternative definition of the absolute value of a real number. The absolute value or modulus has four fundamental properties ( a , b considered to be real numbers ), which are used to provide a generalized purview of this notion to other domains

 | a | ≥ 0 , non- negativity

| a | = 0 , positive definiteness

| ab | = | a | | b | , Multiplicavility

| a + b | ≤ | a | + | b | , Subadditivity, specifically the triangle inequality.

Non-negativity , positive definiteness and multiplicativity are apparent from the definition. Checking whether subadditivity holds, it is required to analyze one of the two alternatives of taking s, the value as either – 1 or  + 1, which guarantees that s. ( a + b ) = | a + b | ≥ 0. Now, because -1.x ≤ | x | and + 1 . x ≤ | x |, it follows that, whichever is the value of s, one has s .x ≤ | x | for all real x, subsequently | a + b | = s. ( a + b ) = s. a + s. b ≤ | a | + | b | as required. To generalize the argument to complex numbers, it is required to see the proof of triangular inequality for complex numbers.

Modulo, often it is abbreviated as “mod” is a mathematical operation. This is like a division problem, except the answer is the remainder of an integer division operation, instead of a decimal result. Further illustration provided a decimal division, modulus is not looking for the result of the division. Modulus can be defined as a function in mathematics, that is capable of changing negative numbers to a positive number. Examples like | -3 | = 3 , | - 5 | = 5, however there is no change in positive numbers to negative numbers.

The real value of absolute value functions is continuous everywhere. This is differentiable everywhere except for when x = 0. This is monotonically decreasing at the interval and monotonically increasing on the interval, since a real number and the opposite number also has the same absolute value, this is an even function and thus not invertible. The real absolute value or modulus function is a piecewise linear, convex function.

The definition of absolute value given in case of real numbers can be extended to any ordered ring. If a is an element in an ordered ring R, then the absolute value of a, is denoted using | a | can be defined as

| a | = a, if a ≥ 0

| a | = - a, if a < 0,

Where, - a is an additive inverse of a, 0 is the additive identity element and < and ≥ have the same meaning with reference to the ordering in the ring.

The four fundamental properties of the absolute value in case of real numbers are used in generalization, where the notion of absolute value to an arbitrary field, is as follows. Real valued function v on a field F is termed as an absolute value, also modulus, magnitude, value or valuation. The absolute values for real numbers is used with a minimal modification to generalize the notion for an arbitrary vector space. Real value function on a vector space V, over a field F, which is reperesented as || . || , is called absolute value, but more usually a norm.

Compositional algebra A, has an involution  x  tends to x* called as the conjugation. The product of A is an element and the conjugate x* is written as N(X) = xx* and called the norm of x. The real number R, complex numbers C and quaternions H, are all composition algebras with norms given by definite quadratic forms. The absolute value used in division algebras is given by the square root of the composition algebra form.