Decimal representation

A decimal representation of a non-negative real number r is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:

Any real number can be approximated to any desired degree of accuracy by rational numbers with finite decimal representations.

The decimal expansion of non-negative real number x will end in zeros (or in nines) if, and only if, x is a rational number whose denominator is of the form 2n5m, where m and n are non-negative integers.

Some real numbers have decimal expansions that eventually get into loops, endlessly repeating a sequence of one or more digits:

Every time this happens the number is still a rational number (i.e. can alternatively be represented as a ratio of an integer and a positive integer). Also the converse is true: The decimal expansion of a rational number is either finite, or endlessly repeating.

Every decimal representation of a rational number can be converted to a fraction by converting it into a sum of the integer, non-repeating, and repeating parts and then converting that sum to a single fraction with a common denominator.

The exponents are the number of non-repeating digits after the decimal point (3) and the number of repeating digits (4).