A decimal expansion is natural number followed by an infinite sequence of digits (or rather, a sequence whose values are constrained to the natural numbers less than 10) that is used in representing a non-negative real number. By extension, all real numbers can then be represented by signing decimal expansions with or , although positive numbers are generally left unsigned.
A decimal expansion is denoted by:
And represents the real number that is the limit to the following:
Due to the infinite nature of decimal expansions, expressions of such are in all practicality limited to those decimal expansions that terminate (see below), or at the very least have a discernible pattern.
A decimal expansion is said to terminate if there is a positive integer such that for all . In this case, the decimal representation can be represented finitely as .
A decimal expansion is eventually repeating if there are positive integers and such that for all . In this case, the following representations will suffice:
A case for using decimal expansions can be made by observing the following two facts:
- Any decimal expansion represents a unique real number (that is, the limit always exists).
- Any nonnegative real number can be represented by some decimal expansion (and thus, any real number can be represented by some signed decimal expansion).