DECIMAL ARITHMETIC

DECIMAL ARITHMETIC, the art of computing by decimal fractions. See ARITHMETIC.

Circulating Decimals, called also recurring or repeating decimals, are those in which a figure or several figures are continually repeated. They are distinguished into single and multiple, and these again into pure and mixed.

A pure single circulate is that in which one figure only is repeated; as .222, &c. and is marked thus $\backslash overline\{2\}$.

A pure multiple circulate is that in which several figures are continually repeated; as .232323, &c. marked $\backslash overline\{23\}$; and .524524, &c. marked $\backslash overline\{524\}$.

A mixed single circulate is that which consists of a terminate part and a single repeating figure; as 4.222, &c. or $4\backslash overline\{2\}$. And

A mixed multiple circulate is that which contains a terminate part with several repeating figures; as $45\backslash overline\{524\}$.

That part of the circulate which repeats is called the repetend; and the whole repetend, supposed to be infinitely continued, is equal to a vulgar fraction whose numerator is the repeating number or figures, and its denominator the same number of nines; so $\backslash overline\{2\}$ is $\backslash frac\{2\}\{9\}$; and $\backslash overline\{23\}$ is $\backslash frac\{23\}\{99\}$; and $\backslash overline\{524\}$ is $\backslash frac\{524\}\{999\}$.

Dr Wallis of Oxford was the first who distinctly considered or treated of infinite circulating decimals, as he himself informs us in his Treatise of Infinites.