## Quick Reference

Any real number *a* between 0 and 1 has a decimal representation, written. *d*_{1}*d*_{2}*d*_{3}…, where each *d*_{i} is one of the digits 0, 1, 2,…, 9; this means that

*a*=*d*_{1}×10^{−1}+*d*_{2}×10^{−2}+*d*_{3}×10^{−3}+⋯.

This notation can be extended to enable any positive real number to be written as

*c*_{n}*c*_{n−1}…*c*_{1}*c*_{0}. *d*_{1}*d*_{2}*d*_{3}…

using, for the integer part, the normal representation *c*_{n}*c*_{n−1}…*c*_{1}*c*_{0} to base 10 (see base). If, from some stage on, the representation consists of the repetition of a string of one or more digits, it is called a recurring or repeating decimal. For example, the recurring decimal .12748748748…can be written .̇12748̇, where the dots above indicate the beginning and end of the repeating string. The repeating string may consist of just one digit, and then, for example, .16666…is written .16̇. If the repeating string consists of a single zero, this is generally omitted and the representation may be called a terminating decimal.

The decimal representation of any real number is unique except that, if a number can be expressed as a terminating decimal, it can also be expressed as a decimal with a recurring 9. Thus .25 and .249̇ are representations of the same number. The numbers that can be expressed as recurring (including terminating) decimals are precisely the rational numbers.

*Subjects:*
Mathematics.

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