absolute value

Absolute value is a term used in mathematics to indicate the distance of a point or number from the origin (zero point) of a number line or coordinate system. This can apply to Scalar or vector quantities. The symbol for absolute value is a pair of vertical lines, one on either side of the quantity whose absolute value is to be determined.

Suppose x is a real number. Then the absolute value of x is defined as follows:

For x = 0 or x > 0, | x | = x
For x < 0, | x | = - x

Alternatively, the absolute value of a real number x is equal to the positive square root of x ² :

| x | = ( x ² ) ^1/2

Let a + jb be a complex number, where a and b are real numbers and j is the positive square root of -1. (The symbol j is standard in engineering practice; mathematicians symbolize the positive square root of -1 as i .) Then the absolute value of a + jb , also called the modulus, is defined as follows:

| a + jb | = ( a ² + b ² ) ^1/2

The absolute value of a vector in n dimensions is defined in terms of the coordinates of its termination point, in Cartesian n -space, assuming its origin coincides with the point where all coordinate values are zero. Suppose v is a vector in n dimensions, represented by the following coordinates:

v = ( v ₁ , v ₂ , v ₃ , ..., v _n )

Then the absolute value of v is given by:

| v | = ( v ₁ ² + v ₂ ² + v ₃ ² + ... + v _n ² ) ^1/2

Also see Mathematical Symbols .