# 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 .

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