A decimal representationof a real numberis called a repeating decimal (or recurring decimal) if at some point it becomesperiodic: there is some finite sequence of digits that is repeated indefinitely. For example, the decimal representation of (spoken as "0.3 repeating", or "0.3 recurring") becomes periodic just after the decimal point, repeating the single-digit sequence "3" infinitely. A somewhat more complicated example is where the decimal representation becomes periodic at the second digit after the decimal point, repeating the sequence of digits "144" infinitely.

A real number has an ultimately periodic decimal representation if and only if it is a rational number. Rational numbers are numbers that can be expressed in the form a/b where a and b are integers and b is non-zero. This form is known as avulgar fraction. On the one hand, the decimal representation of a rational number is ultimately periodic because it can be determined by along division process, which must ultimately become periodic as there are only finitely many different remainders and so eventually it will find a remainder that has occurred before. On the other hand, each repeating decimal number satisfies alinear equation with integral coefficients, and its unique solution is a rational number. To illustrate the latter point, the number above satisfies the equation whose solution is