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DIVIDENDS OF DIFFERENT MAGNITUDES LEAVE DIFFERENT REMAINDERS.

For every dividend is a number of 10s. If, then, it be supposed that two dividends of different magnitudes leave the same remainder, let the less be taken from the greater, and there will remain a number of 10s, which the divisor will exactly measure. But each dividend contains fewer 10s than the divisor contains units; and the same must of course be true of their difference. Therefore, according to the principle just laid down, it is absurd to suppose that the divisor will measure this difference.

Now, in reducing a Fraction whose denominator is prime to 10, if our successive remainders are different, our dividends must be so too, (since the dividends are formed by annexing cyphers to the remainders,) and these, in turn will produce different remainders again; so that we never can obtain the same remainder, nor of course the same dividend, until the number in the numerator, with which we began, recurs as a remainder. That is, the decimal will not repeat, until the remainder left, is equal to the original numerator. But every infinite decimal, obtained by division, must repeat. Therefore, we must necessarily obtain, after a time, the original numerator, as a remainder. Hence the principle,

WHEN THE DENOMINATOR OF A FRACTION IS PRIME TO 10, THE CORRESTONDING DECIMAL WILL BE INFINITE, AND WILL BEGIN TO REPEAT WHEN THE NUMERATOR RECURS AS A REMAINDER.

NOTE. In this case the repetend begins in the first place of decimals, and is, of course, a pure or simple Repetend.

If the numerator be 1, the decimal repeats when the remainder becomes 1. Of course, if we were, then, to take the original numerator, with all the cyphers which have been annexed to it since the divisor commenced, and from this to subtract the remainder 1, the denominator would exactly measure the number left. This number would be a series of 9s. For example, in reducing, the remainder 1 occurs after annexing six cyphers. Take, then, 1,000,000 and subtract this remainder, 1, from it, and there is left 999,999, which 7 will exactly measure. It will be seen, that there are just as many 9s as there were cyphers annexed, and of course, as there are decimal places in the repetend. Hence, when the denominator of a Fraction is prime to 10, to determine of how many places the corresponding pure circulate will consist,

DIVIDE A SERIES OF 98, BY THE DENOMINATOR UNTIL NO REMAINDER IS LEFT. THE NUMBER OF 98 USED WILL SHOW THE NUMBER OF PLACES IN THE CIRCULATE.

CASE III. MIXED REPETENDS, When the denominator is not itself prime to 10, but contains a factor, which is prime to 10, we shall best understand the nature of the corresponding decimal, by separating the denominator into two factors; one of which shall be the entire part that is prime to 10. The other part, of course, must be composed of 2s separately, or 5s separately, as factors; or of 2s and

5s multiplied together. Now, in the reduction, instead of dividing by the whole denominator at once, we may divide successively by these two factors; employing that which is prime to 10, last. Dividing by the factor composed of 2s, 5s, &c., will produce a finite decimal as a quotient. In dividing this quotient by the other factor, it is evident, that, (whatever figures we first obtain,) when we arrive at the end of the significant figures in the number divided, we shall be in the same situation as above, when reducing a Fraction whose denominator was simply prime to 10. Hence, the circulate will begin where the previously found finite decimal ended. Hence, also, the resulting decimal will be a mixed repetend. Hence, the general principle,

WHEN THE DENOMINATOR OF A FRACTION IS ENTIRELY COMPOSED OF FACTORS OF 10, THE CORRESPONDING DECIMAL WILL BE FINITE ; WHEN IT IS PRIME TO 10, A PURE REPETEND; WHEN IT IS NOT PRIME, BUT CONTAINS A FACTOR WHICH IS PRIME TO 10, A MIXED REPETEND.

From what has been said, the pupil will easily be able to deter mine where the repetend will begin in the latter case, and also, of how many places it will consist. For, it will begin after as many places as there are 10s, 5s, or 2s, factors in the denominator, and will extend as many places as it requires 9s for the factor prime to 10, to measure.

Hence, the general rule for determining the nature of the decimal, corresponding to any Vulgar Fraction.

I. DIVIDE THE GIVEN DENOMINATOR BY 10 AS OFTEN AS POSSIBLE, AND AFTERWARDS BY 2 OR 5 AS OFTEN AS POSSIBLE WITHOUT REMAINDER. IF THE LAST QUOTIENT BE 1, THE DECIMAL WILL BE FINITE, AND WILL CONTAIN AS MANY PLACES AS THERE WERE DIVISIONS.

II. IF THE LAST QUOTIENT BE NOT 1, THE DECIMAL WILL BE INFINITE. THEREFORE, DIVIDE A SERIES OF 98 BY THAT QUOTIENT, UNTIL NO REMAINDER IS LEFT. THE NUMBER OF 98 USED WILL SHOW THE NUMBER OF PLACES IN THE REPETEND; WHICH WILL BEGIN AFTER AS MANY PLACES AS YOU AT FIRST PERFORMED DIVISIONS BY 2,5 OR 10.

NOTE. The Fraction, it must be recollected, must be in its lowest terms.

EXAMPLES FOR PRACTICE.

17 18

13

144

1. Let the Fraction be, 128, 230, 31, 38, 1726' 1128 2. Let the Fraction be 720'43'

3

94

90' 8450 3912 1987

18 235 2345 1763

139 12598 781200

3. Let the Fraction be T25 172 83 84509876000

4. Let the Fraction be 187

543267 17943814532 389 188 8884 083 3 3 3 4 4 4 4 5

§ LXX. It is evident from what has been said, that any numbe. prime to 10, will divide exactly a series of 9s. Now, if the quotient, obtained by such a division, be multiplied into the dividing number, the same series of 9s will be produced again. This quo. tient is evidently the circulate, which would be obtained by reducing

a Fraction, whose numerator is 1, and whose denominator, the dividing number. Now, if both terms of this Fraction were to be multiplied by this quotient, or circulate, the Fraction, without being altered in value, (§ XLI.) would be changed to one whose numerator would be the circulate itself, and whose denominator, just as many nines as the circulate has places. To illustrate by an example. 999999÷7=142857, and reduced to a decimal=142857. 12 142337=143857. Hence, if the circulate .142857 had been given us, and it had been required to find its value in a vulgar Fraction, (that is, to find what vulgar Fraction would produce it,) we might have made the circulate itself the numerator of a Fraction, and just as many nines as it had places, the denominator, thus, 48837, and this, reduced to its lowest terms, would have given us the original Fraction, would give us twice as great a cir

142857

999999,

culate,, three times as great, and so on.

Hence, to change any pure circulate, or repetend, to its equivalent Vulgar Fraction,

MAKE THE GIVEN REPETEND THE NUMERATOR, AND THE DENOMINATOR AS MANY 98, AS THERE ARE PLACES IN THE REPETEND.

In mixed repetends, the principle is similar. If a repetend begin in the place of hundredths, its value will be the same as before, except that it will be ten times smaller than if it began, like a simple repetend, in the place of tenths. For when it begins in the first, or tenths' place, it is a Fraction of a unit, or whole number. But when it begins in the second, or hundredths' place, it is a Fraction of a tenth. Its value is therefore decreased ten fold. Thus .3 is 3 of a whole number; but .03 is of a tenth 3 of 3%. After finding the Vulgar Fraction, as before, then, we are obliged to divide it by 10, or, (which, (§ LIII., is the same thing,) multiply its denominator by 10; which is done by annexing a cypher. In like manner, if it had begun a place lower still, we should have been obliged to annex two cyphers, and so on; annexing always as many cyphers as there are places between the separatrix, and the first figure of the repetend, or as there are finite places in the decimal. Hence, to find the value of the circulating part of a mixed repetend,

MAKE THE REPETEND THE NUMERATOR; AND FOR THE DENOMINATOR, ANNEX AS MANY CYPHERS AS THERE ARE FINITE PLACES, TO AS MANY 98 AS THERE ARE PLACES IN THE CIRCULATE.

The mode of finding the value of the finite part has already been given. (§ LVIII.) Hence to find the whole value of a mixed repetend, in a Vulgar Fraction,

FIND THE VALUES OF THE FINITE AND CIRCULATING PARTS SEPARATELY, AND ADD THEM TOGETHER.

By the above rule, find the values of .104 .839 .61407 .93 .815 .7 .6311 .9831 .4762 .83141 .98764 .30942 .111333 7778889 .63115438 .71324285.

§ LXXI. A single repetend may evidently be regarded as a compound repetend, consisting of as many places as we choose to make it. Thus, 3 is a single repetend. Giving it two places, thus, 33, or three, thus, .333, it becomes a compound repetend. So .97 may be made .97777 or 977777, &c. We may also make a single repetend begin later, or at a lower place, reserving its higher figures as finite decimals. Thus, .6 may be made .66, or 666, &c.

We may also make the same changes upon compound repetends; making them begin later, thus, .46 changed to .464 or .46464, &c. or extending the number of places, thus, .379 changed to .37979, or to .3797979, &c.; or making both changes at once, thus .432 changed to 4324324, &c. In changing compound repetends, we must be careful to place the points so that the repeating figures, when extended beyond the last point, shall recur in the same order as before. Thus, .862 is inaccurately changed to .86286, and .813 to .81313, the latter numbers, in each case, representing different repetends from those given. By moving the points in this manner, we may render any two repetends similar. Thus .135 and .861 are dissimilar; but .1351 and .861 are similar. Having made repetends similar, they may also be rendered conterminous, by extending their fig. ures to a number of places, that is a common multiple of the numbers of places in the given repetends. For, on making the number of places in any repetend, twice, three times, or any number of times as great as before, it will still repeat in the same order as at first. The least common multiple is most convenient.

The foregoing examination of the nature of circulating decimals, will afford us some useful rules for conducting arithmetical operations upon them. Single repetends, in their true value, have been seen to be ninths. Hence, for ADDITION when there are FINITE DECI

MALS, AND Single repetENDS, or SINGLE REPETENDS ONLY,

1. MAKE THE REPETENDS CONTERMINOUS, EXTENDING THEM ONE PLACE BEYOND THE LONGEST FINITE DECIMAL; CARRY FOR 9 INSTEAD OF 10 FROM THE RIGHT HAND COLUMN; AND IN OTHER RESPECTS add as USUAL. THE RIGHT HAND FIGURE OF THE SUM WILL BE A REpetend.

In case there are compound repetends, having made them conterminous, it is evident that, if they were extended farther still, (as they might be,) there would often be something to carry from those figures so extended; and this number carried would be the same as that carried forward from the first place of the repetends. Hence, when there are FINITE DECIMALS, AND CIRCULATES, or CIRCULATES ONLY,

II. MAKE THE REPETENDS SIMILAR AND CONTERMINOUS, COMMENCING THEM ONE PLACE BELOW THE LONGEST FINITE DECIMAL; CARRY TO THE RIGHT HAND FIGURE OF THE SUM, THE SAME NUMBER THAT IS CARRIED FORWARD FROM THE FIRST PLACE OF THE REPETENDS; AND IN OTHER RESPECTS ADD AS USUAL. THE SUM OF THE CIRCULATING FIGURES, WILL BE THE CIRCULATE OF THE AMOUNT.

NOTE. If the circulate found by adding be a series of 9s, it becomes equal to the denominator of its equivalent vulgar Fraction, and of course its value is 1; which may be added to the next higher place, and the repetend neglected.

Multiplication is a repeated addition. Hence, for MULTIPLICATION, when the MULTIPLIER IS FINITE, and the MULTIPLICAND A SINGLE REPETEND,

I. MULTIPLY AS USUAL, CARRYING FOR FROM THE PRODUCT OF THE REPETEND. THE RIGHT HAND FIGURE OF EACH PARTIAL PRODUCT WILL BE A REPETEND. THE TOTAL PRODUCT MUST THEREFORE BE FOUND BY RULE I. FOR ADdition.

That is, the partial products must be made conterminous, before adding, and, in adding we must carry for 9 from the right hand column. The last figure of the total product will then be the repetend. When the MULTIPLIER IS FINITE, and the MULTIPLICAND A CIRCULATE,

II. MULTIPLY AS USUAL, CARRYING TO THE RIGHT HAND FIGURE OF EACH PARTIAL PRODUCT, THE NUMBER WHICH IS CARRIED FORWARD FROM THE FIRST PLACE OF THE CIRCULATE. EACH PARTIAL PRODUCT WILL HAVE AS MANY CIRCULATING FIGURES AS THE MULTIPLICAND. THE TOTAL PRODUCT MUST THEREFORE BE FOUND BY RULE II. FOR ADDITION.

The case in which the multiplier is a repetend, remains to be considered. Here we have no means of proceeding decimally with accuracy. Hence, when the MULTIPLIER is a repetend,

III. CHANGE THE MULTIPLIER TO ITS EQUIVALENT VULGAR FRACTION; MULTIPLY BY ITS NUMERATOR, AND DIVIDE BY ITS DENOMINATOR.

NOTE. If the multiplicand be also infinite, this rule anticipates a case of division to be mentioned. (Rule 1.) It may however be mentioned here, that, when, in ordinary cases, we should annex cyphers in dividing, we must annex the repeating figures of the number divided instead of them; and so continue to do until the decimal repeats. This will determine the number of figures in the final circulate.

In Subtraction it is evident, that, when the repetends are similar and conterminous, if that of the subtrahend be greater than that of the minuend, we shall be obliged to borrow 1 from the next higher place. And this would also be the case, if the repetends were extended below the last point. This would make the remainder 1 less. Hence, for SUBTRACTION, when EITHER OR BOTH NUMBERS ARE REPETENDS,

PREPARE THE NUMBERS AS IN ADDITION, AND SUBTRACT AS USUAL, OBSERVING TO DIMINISH THE REMAINDER BY 1, WHEN THE REPETEND OF THE SUBTRAHEND IS GREATER THAN THAT OF THE MINUEND.

In division, when the dividend is infinite, it is plain, that since the repeating figures may be extended to any distance, those figures will occupy the places of the cyphers which would otherwise be

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