In algebra, an absolute value (also called a valuation, magnitude, or norm,[1][a] is a function which measures the "size" of elements in a field or integral domain.
More precisely, if D is an integral domain, then an absolute value is any mapping |x| from D to the real numbers R satisfying: It follows from these axioms that |1| = 1 and |−1| = 1.
Ostrowski's theorem states that the nontrivial places of the rational numbers Q are the ordinary absolute value and the p-adic absolute value for each prime p.[3] For a given prime p, any rational number q can be written as pn(a/b), where a and b are integers not divisible by p and n is an integer.
Given an integral domain D with an absolute value, we can define the Cauchy sequences of elements of D with respect to the absolute value by requiring that for every ε > 0 there is a positive integer N such that for all integers m, n > N one has |xm − xn| < ε. Cauchy sequences form a ring under pointwise addition and multiplication.
The domain D is embedded in this quotient ring, called the completion of D with respect to the absolute value |x|.
Another theorem of Alexander Ostrowski has it that any field complete with respect to an Archimedean absolute value is isomorphic to either the real or the complex numbers, and the valuation is equivalent to the usual one.