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 2̇.

A pure multiple circulate is that in which several figures are continually repeated; as 2̇3̇2̇3̇, &c. marked 2̇3̇; and 5̇2̇4̇5̇2̇4̇, &c. marked 5̇2̇4̇.

A mixed single circulate is that which consists of a terminate part, and a single repeating figure; as 4̇2̇2̇, &c. or 4̇2̇. And

A mixed multiple circulate is that which contains a terminate part with several repeating figures; as 4̇5̇5̇2̇4̇.

That part of the circulate which repeats is called the repetend; and the whole repetend, supposed 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 2̇ is = 2̇9̇; and 2̇3̇ is = 2̇9̇9̇; and 5̇2̇4̇ is = 5̇9̇9̇9̇.

Dr Wallis, it appears, was the first who distinctly considered or treated of infinite circulating decimals, as he himself informs us in his Treatise of Infinites. Since his time many other authors have treated on this part of arithmetic; the principal of these, however, to whom the art is mostly indebted, are Messrs Brown, Cunn,