In arithmetic, repeating decimal is a way of representing a rational number. Thus, a decimal representation of a number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely. For example, the decimal representation of 1/3 = 0.3333333… or 0.3 (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 3227/555 = 5.8144144144…, where the decimal representation becomes periodic at the second digit after the decimal point, repeating the sequence of digits "144" indefinitely.
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 a common fraction. On the one hand, the decimal representation of a rational number is ultimately periodic, as explained below. On the other hand every real number which has an eventually periodic decimal expansion is a rational number. In other words the numbers with eventually repeating decimal expansions are exactly the rational numbers (i.e.: those that can be expressed as ratios).
A decimal representation written with a repeating final 0 is said to terminate before these zeros. Instead of "1.585000…" one simply writes "1.585". The decimal is also called a terminating decimal. Terminating decimals represent rational numbers of the form k/(2n5m). For example, 1.585 = 317/200 = 317/(2352). A terminating decimal can be written as a decimal fraction: 317/200 = 1585/1000. However, a terminating decimal also has a second representation as a repeating decimal, obtained by decreasing the final (nonzero) digit by one and appending an infinitely repeating sequence of nines, a phenomenon students typically find puzzling (see List of common misconceptions#Mathematics). 1 = 0.999999… and 1.585 = 1.584999999… are two examples of this. This type of repeating decimal can be obtained by long division if one uses a modified form of the usual division algorithm.
A decimal that is neither terminating nor repeating represents an irrational number (which cannot be expressed as a fraction of two integers), such as the square root of 2 or the number π. Conversely, an irrational number always has a non-repeating decimal representation.
Read more about Repeating Decimal: Fractions With Prime Denominators, Reciprocals of Composite Integers Coprime To 10, Reciprocals of Integers Not Co-prime To 10, Converting Repeating Decimals To Fractions, Repeating Decimals As An Infinite Series, Multiplication and Cyclic Permutation, Other Properties of Repetend Lengths
Famous quotes containing the words repeating and/or decimal:
“The echo is, to some extent, an original sound, and therein is the magic and charm of it. It is not merely a repetition of what was worth repeating in the bell, but partly the voice of the wood; the same trivial words and notes sung by a wood-nymph.”
—Henry David Thoreau (1817–1862)
“It makes little sense to spend a month teaching decimal fractions to fourth-grade pupils when they can be taught in a week, and better understood and retained, by sixth-grade students. Child-centeredness does not mean lack of rigor or standards; it does mean finding the best match between curricula and children’s developing interests and abilities.”
—David Elkind (20th century)