If, in the number 134,28, the point be placed between the 3 and 4, we shall have 13,428; the hundreds will become tens, the tens units, the units tenths, the tenths hundredths, and the hundredths thousandths; every part of the number having thus become ten times smaller, the result is the same as if a tenth part of it had been taken, or as if it had been divided by ten. 91. From what has been said, it will be easy to perceive the advantage, which decimal fractions have over vulgar fractions; all the multiplications and divisions, which are performed by the denominator of the latter, are performed with respect to the former, by the addition or suppression of a number of ciphers, or by simply changing the place of the comma. By adapting these modifications to the theory of vulgar fractions, we thence immediately deduce that of decimals, and the manner of performing the multiplication and division of them; but we can also arrive at this theory directly by the following considerations.

Let us first suppose only the multiplicand to have decimal figures. If the comma be taken away, it will become ten, a hundred, a thousand, &c. times greater, according to the number of decimal figures; and in this case the product given by multiplication will be a like number of times greater than the one required; the latter will then be obtained by dividing the former by ten, a hundred, a thousand, &c. which may be done by separating on the right(90) as many decimal figures, as there are in the multiplicand.

If, for instance, 34,137 were to be multiplied by 9, we must first find the product of S4137 by 9, which will be 307233; and, since taking away the comma renders the multiplicand a thousand times greater, we must divide this product by a thousand, or separate by a comma its three last figures on the right; we shall thus have 307,233.

In general, to multiply, by a whole number, a number accompanied by decimals, the comma must be taken away from the multiplicand, and as many figures separated for decimals, on the right of the product, as are contained in the multiplicand.

92. When the multiplier contains decimal figures, by suppressing the comma, it is made ten, a hundred, a thousand, &c. times greater according to the number of decimal figures. If used in this state, it will evidently give a product, ten, a hundred, a thousand, &c. times greater than that which is required, and consequently the true product will be obtained by dividing by one of these numbers, that is, by separating, on the right of it, as many decimal figures as there are in the multiplier, or by removing the comma a like number of places towards the left (90), in case it previously existed in the product on account of decimals in the multiplicand. For instance, let 172,84 be multiplied by 36,003; taking away the comma in the multiplier only, we shall have, according to the preceding article, the product 6222758,52; but, the multiplier being rendered a thousand times too great, we must divide this product by a thousand, or remove the comma three places towards the left, and the required product will then be 6222,75852, in which there must necessarily be as many decimal figures as there are in both multiplicand and multiplier.

In general, to multiply one by the other, two numbers accompa nied by decimals, the comma must be taken away from both, and as many figures separated for decimals, on the right o the product, as there are in both the factors.

In some cases it is necessary to put one or more ciphers on the left of the product, to give the number of decimal figures required by the above rule. If, for example, 0,624 be multiplied by 0,003; in forming at first the product of 624 by 3, we shall have the number 1872, containing but 4 figures, and as 6 figures must be separated for decimals, it cannot be done except by placing on the left three ciphers, one of which must occupy the place of units, which will make 0,001872.

93. It is evident (36), that the quotient of two numbers does not depend on the absolute magnitude of their units, provided that this be the same in each; if then, it be required to divide 451,49 by 13, we should observe that the former amounts to 45149 hundredths, and the latter to 1300 hundredths, and that these last numbers ought to give the same quotient, as if they expressed units. We shall thus be led to suppress the point in the first number, and to put two ciphers at the end of the second, and then we shall only have to divide 45149 by 1800, the quotient of which division will be 34 949

1300

Hence we conclude, that, to divide, by a whole number, a number accompanied by decimal figures, the comma in the dividend must be taken away, and as many ciphers placed at the end of the divisor, as the dividend contains decimal figures, and no alteration in the quotient will be necessary.

94. When both dividend and divisor are accompanied by decimal figures, we must, before taking away the comma, reduce them to decimals of the same order, by placing at the end of that number, which has the fewest decimal figures, as many ciphers 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 315452 must be divided by 23400; the quotient will be 1311238.

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 neces

sity 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, thousandth, &c. so that it may contain the divisor; as may been in the following example;

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*.

* The problem above performed with respect to decimals, is only

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 ;

Again, let the fraction be 737; the numerator must be converted into thousandths before the division can begin.

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 for the quotient required.