number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or
. In particular, two -adic numbers are considered to be close when their difference is
of : the higher the power, the closer they are. This property enables -adic numbers to encode
earlier work can be interpreted as implicitly using -adic numbers.
The -adic numbers were motivated primarily by an attempt to bring the ideas and techniques of
methods into number theory. Their influence now extends far beyond this. For example, the field of
of the rational numbers. The field is also given a
on the rational numbers. This metric space is
to a point in . This is what allows the development of calculus on , and it is the interaction of this analytic and
structure that gives the -adic number systems their power and utility.
The in "-adic" is a
and may be replaced with a prime (yielding, for instance, "the 2-adic numbers") or another
representing a prime number. The "adic" of "-adic" comes from the ending found in words such as
.
of the numerator by the denominator, which is itself based on the following theorem: If
The decimal expansion is obtained by repeatedly applying this result to the remainder
.
The -''adic expansion'' of a rational number is defined similarly, but with a different division step. More precisely, given a fixed
is positive. The integer
(the absolute value is small when the valuation is large). The division step consists of writing
: