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as will make it terminate at the same place of decimals as the other, because then the suppression of the comma renders both the same number of times greater.

For instance, let 315,432 be divided by 23,4, this last must be changed into 23,400, and then 315432 must be divided by 23400; the quotient will be 13.

Thus, to divide, one by the other, two numbers accompanied by decimal figures, the number of decimal figures in the divisor and dividend must be made equal, by annexing to the one that has the least, as many ciphers as are necessary; the point must then be suppressed in each, and the quotient will require no alteration.

95. As we have recourse to decimals only to avoid the necessity of employing vulgar fractions, it is natural to make use of decimals for approximating quotients that cannot be obtained exactly, which is done by converting the remainder into tenths, hundredths, thousandths, &c. so that it may contain the divisor; as may be seen in the following example;

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When we come to the remainder 949, we annex a cipher in order to multiply it by ten, or to convert it into tenths; thus forming a new partial dividend, which contains 9490 tenths and gives for a quotient 7 tenths, which we put on the right of the units, after a comma. There still remains 390 tenths, which we reduce to hundredths by the addition of another cipher, and form a second dividend, which contains 3900 hundredths, and

gives a quotient, 3 hundredths, which we place after the tenths. Here the operation terminates, and we have for the exact result 34,73 hundredths. If a third remainder had been left, we might have continued the operation, by converting this remainder into thousandths, and so on, in the same manner, until we came to an exact quotient, or to a remainder composed of parts so small, that we might have considered them of no importance.

It is evident, that we must always put a comma, as in the above example, after the whole units in the quotient, to distinguish them from the decimal figures, the number of which must be equal to that of the ciphers successively written after the remainders.*

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96. The numerator of a fraction, being converted into decimal parts, can be divided by the denominator as in the preceding examples, and by this means the fraction will be converted into decimals. Let the fraction, for example, be, the operation is performed thus;

*The problem above performed with respect to decimals, is only a particular case of the following more general one; To find the value of the quotient of a division, in fractions of a given denomination; to do this we convert the dividend into a fraction of the same denomination by multiplying it by the given denominator. Thus, in order to find in fifteenths the value of the quotient of 7 by 3, we should multiply 7 by 15, and divide the product, 105, by 3, which gives thirty-five fifteenths, or 35, for the quotient required.

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Again, let the fraction be; the numerator must be converted into thousandths before the division can begin.

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* It may also be proposed to convert a given fraction into a fraction of another denomination, but smaller than the first, for instance, into seventeenths, which will be done by multiplying 3 by 17 and dividing the product by 4. In this manuer we find seventeenths, orand of a seventeenth; but of 1 is equivalent to The result then, 1, is equal to 3, wanting 3

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This operation and that of the preceding note depend on the same principle, as the corresponding operation for decimal fractions.

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97. However far we may continue the second division, exhibited above, we shall never obtain an exact quotient, because the fraction cannot, like, be exactly expressed by decimals.

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The difference in the two cases arises from this, that the denominator of a fraction, which does not divide its numerator, cannot give an exact quotient, except it will divide one of the numbers 10, 100, 1000, &c. by which its numerator is successively multiplied; because it is a principle, which will be found demonstrated in Algebra, that no number will divide a product except its factors will divide those of the product; now the numbers 10, 100, 1000, &c. being all formed from 10, the factors of which are 2 and 5, they cannot be divided except by numbers formed from these same factors; 8 is among these, being the product of 2 by 2 by 2.

Fractions, the value of which cannot be exactly found by decimals, present in their approximate expression, when it has been carried sufficiently far, a character which serves to denote them; this is the periodical return of the same figures.

If we convert the fraction into decimals, we shall find it 0,324324.... and the figures 5, 2, 4, will always return in the same order, without the operation ever coming to an end.

Indeed, as there can be no remainder in these successive divisions, except one of the series of whole numbers, 1, 2, 3, &c. up to the divisor, it necessarily happens, that, when the number of divisions exceeds that of this series, we must fall again upon some one of the preceding remainders, and consequently the partial dividends will return in the same order. In the above example three divisions are sufficient to cause the return of the same figures; but six are necessary for the fraction, because in this case we find, for remainders, the six numbers which are below 7, and the result is 0,1428571... The fraction leads only to 0,3333.

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98. The fractions, which have for a denominator any number of 9s, have no significant figure in their periods except 1;

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and so with the others, because each partial division of the numbers 10, 100, 1000, &c. always leaves unity for the remainder.

Availing ourselves of this remark, we pass easily from a periodical decimal, to the vulgar fraction from which it is derived. We see, for example, that 0,33333 ..... amounts to the same as 0,11111 ..... multiplied by 3, and as this last decimal is the development of, or reduced to a decimal, we conclude, that the former is the development of multiplied by 3, or, or lastly,

When the period of the fraction under consideration consists of two figures, we compare it with the development of, and with that of, when the period contains three figures, and

so on.

If we had, for example, 0,324324, it is plain that this fraction may be formed by multiplying 0,001001 ..... by 324; if we multiply then, of which 0,001001..... is the development, by 324, we obtain 33, and dividing each term of this result by 27, we come back again to the fraction.

In general, the vulgar fraction, from which a decimal fraction arises, is formed by writing, as a denominator, under the number, which expresses one period, as many 9s, as there are figures in the period.

If the period of the fraction does not commence with the first decimal figure, we can for a moment change the place of the point, and put it immediately before the first figure of the period and beginning with this figure, find the value of the fraction, as if those figures on the left were units; nothing then will be necessary except to divide the result by 10, 100, 1000, &c. according to the number of places the point was moved towards the right.

For instance, the fraction 0,324,141...., is first to be written 32,4141....; the part 0,4141 .... being equivalent to, we shall have 32, which is to be divided by 100, because the point was moved two places towards the left; it will consequently become and, or by reducing the two parts to the same denominator, and adding them, , a fraction which will reproduce the given expression.

32

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