complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them. The standard Euclidean distance between two points and in Euclidean -space is defined as: This can be seen as a generalisation, since for and 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. The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a distance function as follows: A real valued function on a set is called a metric (or a distance function) on , if it satisfies the following four axioms: {| |- |style="width:250px" | |Non-negativity |- | |Identity of indiscernibles |- | |Symmetry |- | |Triangle inequality |} Generalizations Ordered rings The definition of absolute value given for real numbers above can be extended to any ordered ring. That is, if is an element of an ordered ring R, then the absolute value of , denoted by , is defined to be: where is the additive inverse of , 0 is the additive identity, and < and ≥ have the usual meaning with respect to the ordering in the ring. Fields The four fundamental properties of the absolute value for real numbers can be used to generalise the notion of absolute value to an arbitrary field, as follows. A real-valued function on a field is called an absolute value (also a modulus, magnitude, value, or valuation) if it satisfies the following four axioms: {| cellpadding=10 |- | |Non-negativity |- | |Positive-definiteness |- | |Multiplicativity |- | |Subadditivity or the triangle inequality |} Where 0 denotes the additive identity of . It follows from positive-definiteness and multiplicativity that , where 1 denotes the multiplicative identity of . The real and complex absolute values defined above are examples of absolute values for an arbitrary field. If is an absolute value on , then the function on , defined by , is a metric