Again, let there be the numbers 19,35, 0.3, 48,5, and 110,02, which contain also whole units; they will be disposed thus;

In general, the addition of decimal numbers is performed like that of whole numbers, care being taken to place the comma in the sum, directly under the commas in the numbers to be added.

89. The rules prescribed for the subtraction of whole numbers apply also, as will be seen, to decimals. For instance, let. 0,3697 be taken from 0,62; it must first be observed, that the second number, which contains only hundredths, while the other contains ten-thousandths, can be converted into ten-thousandths by placing two ciphers on its right (87), which changes it into 0,6200.

The operation will then be arranged thus;

0,6200

0,3697

Difference 0,2503

and, according to the rule of article 17, the difference will be 0,2503.

Again, let 7,364 be taken from 9,1457; the operation being disposed thus ;

the above difference is found. It would have been just as well if no cipher had been placed at the end of the number to be subtracted, provided its different figures had been placed under the corresponding orders of units or parts, in the upper line.

In general, the subtraction of decimal numbers is performed like that of whole numbers, provided that the number of decimal figures, in the two given numbers, be made alike, by writing on the right of that which has the least, as many ciphers as are necessary; and that the comma in the difference be put directly under those of the give

The methods of proving addition and subtraction of decimals are the same as those for the addition and subtraction of whole numbers.

90. As the comma separates the collections of entire units. from the decimal parts, by altering its place, we necessarily change the value of the whole. By moving it towards the right, figures, which were contained in the fractional part, are made to pass into that of whole numbers, and consequently the value of the given number is increased. On the contrary, by moving the comma towards the left, figures which were contained in the part of whole numbers, are made to pass into that of fractions, and consequently the value of the given number is diminished.

The first change makes the given number, ten, a hundred, a thousand, &c. times greater than before, according as the comma is removed one, two, three, &c. places towards the right, because for each place that the comma is thus removed, all the figures advance with respect to this comma one place towards the left, and consequently assume a value ten times greater than they had before.

If, for example, in the number 134,28, the point be placed

between the 2 and the 8, we shall have 1342,8, the hundreds will have become thousands, the tens hundreds, the units tens, the tenths units, and the hundredths tenths. Every part of the number having thus become ten times greater, the result is the same as if it had been multiplied by ten.

The second change makes the given number ten, a hundred, a thousand, &c. times smaller than it was before, according as the comma is removed one, two, three, &c. places towards the left; because for each place that the comma is thus removed, all the figures recede, with respect to this comma, one place further to the right, and consequently have a value ten times less than they had before.

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 34137 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 172,34 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 accompanied by decimals, the comma must be taken away from both, and as

many figures separated for decimals, on the right of 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 re. quired 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 1300, the quotient of which division will be 34

949

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