REPEATING AND CIRCULATING

DECIMALS.

NUMERATION.

A Decimal is said to repeat or circulate when it may be carried to any extent by the repetition of the same figure or figures as, 0.333, &c. is called a repeating decimal; 0.372372, &c. is called a circulating decimal; and they are generally expressed by placing a point over the repeating figure of a repetend, and the same over the first and last repeating figures of a circulate. Thus, we may express

0.333, &c. by 0.3, and 0.372372 by 0.372.

A pure repetend or circulate is a number which contains only the repeating figure or figures, as the above.

The mixed repetend or circulate is a number in which other figures, either integral or decimal, précede the repeating figure or figures;

as, 3.703

30.46372.

Similar repetends or circulates are those which commence at the same place, either of integers or decimals: as,

4.23 and 0.35, or 5.164 and 4.272.

Conterminous circulates are those which contain the same number of figures; as,

2.826 and 4.064, or 0.378 and 0.4265.

Therefore, similar and conterminous circulates are those which commence at the same place, and contain the same number of figures; as,

3.176 and 4.862, or 0.36342 and 0.01527.

All repeating and circulating decimals are produced, either from fractions having as many nines in the denominator as there are places in the repetend or circulate, or from the equivalent lowest terms of such fractions.

REDUCTION.

To reduce repeating and circulating decimals to fractions.

CASE 1.

When the repetend or circulate is pure.

Rule. Express it as the numerator of a fraction, and place as many nines for the denominator as the repetend or circulate contains figures.

Observe. When the highest place of figures in the repetend or circulate does not commence in the place of 10 ths, so many ciphers must be annexed to the numerator when higher, or to the denominator when lower, as that place is distant from the place of 10 ths.

When the repetend or circulate is mixed.

Rule. Express the fractional value of the figures preceding the repetend or circulate, and to the numerator express the fractional value of the repetend or circulate, and reduce the whole to a simple fraction.

N. B. The numerator of the result may be formed by subtracting the finite figures from the whole of the given figures, and the denominator will contain as many nines as the repetend or circulate contains figures, placing ciphers as before to either the numerator or denominator when the repetend or circulate does not commence in the place of 10 ths.

3. 0.123, 0.74,

2. 5.1, 3.12,

0.0037, and 52.3809.

4. 4.216, 4.216, 53.627, and 53.627.

N. B. These Exercises are to be proved by reducing the fractions to decimals.

To make dissimilar repetends similar and conterminous.

Rule. Make the first repeating or circulating figure in each commence at the lowest place in which the given repetends or circulates commence; then, with circulates, find the least common multiple of the number of figures, and continue each circulate to that number of places.

When finite numbers are, for the purposes of Addition or Subtraction, required to be expressed as similar and conterminous circulates, they must be continued with ciphers, with the first of which the common circulate must commence, unless the finite parts of the other number extend to a lower place of decimals.

MULTIPLICATION.

Rule. Reduce the two terms to their equivalent fractions, find their product, and reduce it to a decimal.

N. B. When one term is a finite number and the other is a repeating decimal, the latter may be made the multiplier, and parts may be taken for the repetend.

To multiply 23.15 by 0.018.

23.15 = *313° = 457o, and 0.018 == rio

999

2570 X 710 = $1400.420966

EXERCISES.

Ex. 15. Multiply 23.25 by 4.2 and by 0.5.

16.

17.

18.

6.345 by 21 and by 4.04.

4.26 by 4.16 and by 3.02.
7.1 by 2.02 and by 5.36.

DIVISION.

Rule. Reduce the two terms to their equivalent fractions, find their quotient, and reduce it to a decimal.

N. B. If the divisor is finite, extend the dividend as far as may be required, and use the divisor as in common Division.

If the divisor is a repetend, multiply both terms by 9 and proceed as above.

If the divisor and dividend contain the same number of places of circulates, the division is performed by the division of the numerators of their equivalent fractions; but if the number of the places is unequal, we may extend each to their common multiple, and proceed as before.