What is the recurring number when two-thirds is expressed as a decimal?
Repeating decimal multiple of a cyclic number (that is, a multiple having the same number of digits) is a rotation. The reason for the cyclic behavior is apparent from an arithmetic exercise of long division of : the sequential remainders are the cyclic sequence {1, 3, 2, 6, 4, 5}. See also the article 142857 for more properties of this cyclic number. A fraction which is cyclic thus has a recurring decimal of even length that divides into two sequences in 9's complement form. For example 1/7 starts '142' and is followed by '857' while 6/7 (by rotation) starts '857' followed by
Decimal prolongated into an infinite expansion by adding infinitely many 0. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainder are less than the divisor, there are only a finite number of possible remainders, and after some place, the same sequence of digits must be repeated indefinitely in the quotient. That is, one has a "repeating decimal". For example, Conversely, every eventually repeating sequence of digit is the infinite decimal expansion of a rational number. This is a consequence of the fact that the recurring part of a decimal representation is,