REPETEND

, in Arithmetic, denotes that part of an infinite decimal fraction, which is continually repeated ad infinitum. Thus in the numbers 2.13 13 13 &c. the figures 13 are the Repetend, and marked thus 1^.3^..

These Repetends chiefly arise in the reduction of vulgar fractions to decimals. Thus, 1/3 = 0.333 &c = 0.3^.; and 1/6 = 01666 &c = 0.16^.; and 1/7 = 0.142857 142857 &c = 0.1^.42857^.. Where it is to be observed, that a point is set over the figure of a single Repetend, and a point over the first and last figure when there are several that repeat.

Repetends are either single or compound.

A Single Repetend is that in which only one figure repeats; as 0.3^., or 0.6^., &c.

A Compound Repetend, is that in which two or more figures are repeated; as .1^.3^., or .2^.15^., or .1^.42857^., &c.

Similar Repetends are such as begin at the same place, and consist of the same number of figures: as .3^. and .6^., or 1.3^.41^. and 2.1^.56^..

Dissimilar Repetends begin at different places, and consist of an unequal number of figures.

To find the finite Value of any Repetend, or to reduce it to a Vulgar Fraction. Take the given repeating figure or figures for the numerator; and for the denominator, take as many 9's as there are recurring figures or places in the given Repetend. .

Hence it follows, that every such infinite Repetend has a certain determinate and finite value, or can be expressed by a terminate vulgar fraction. And consequently, that an infinite decimal which does not repeat or circulate, cannot be completely expressed by a finite vulgar fraction.

It may farther be observed, that if the numerator of a vulgar fraction be 1, and the denominator any prime number, except 2 and 5, the decimal which shall be equal to that vulgar fraction, will always be a Repetend, beginning at the first place of decimals; and this Repetend must necessarily be a submultiple, or an aliquot part of a number expressed by as many 9's as the Repetend has figures; that is, if the Repetend have six figures, it will be a submultiple of 999999; if four figures, a submultiple of 9999 &c. From whence it follows, that if any prime number be called p, the series 9999 &c, produced as far as is necessary, will always be divisible by p, and the quotient will be the Repetend of the decimal fraction = 1/p.

RESIDUAL Figure, in Geometry, the figure remaining after subtracting a less from a greater.

Residual Root, is a root composed of two parts or members, only connected together with the sign — or minus. Thus, a — b, or 5 — 3, is a residual root; and is so called, because its true value is no more than the residue, or difference between the parts a and b, or 5 and 3, which in this case is 2.

RESIDUUM of a Charge, in Electricity, first discovered by Mr. Gralath, in Germany, in 1746, is that part of the charge that lay on the uncoated part of a Leyden phial, which does not part with all its electricity at once; so that it is afterwards gradually diffused to the coating.