naturally take their place on the right of units, then hundredths on the right of tenths, and so on; but, that the figures expressing decimal parts may not be confounded with those expressing whole units, a commat is placed on the right of units. To express, for instance, 34 units and 27 hundredths, we write 34,27. If there be no units, their place is supplied by a cipher, and the same is done for all the decimal parts, which may be wanting between those enunciated in the given number.

Thus

19 hundredths are written 0,19,

304 thousandths

3 thousandths

304

0,304,

0,003.

85. If the expressions for the above decimal fractions be compared with the following,, TO, drawn from the general manner of representing a fraction, it will be seen, that to represent in an entire form a decimal fraction, written as a vulgar fraction, the numerator of the fraction must be taken as it is, and placed after the comma in such a manner, that it may have as many figures as there are ciphers after the unit in the denominator.

Reciprocally, to reduce a decimal fraction, given in the form of a whole number, to that of a vulgar fraction, the figures that it contains, must receive, for a denominator, an unit followed by as many ciphers, as there are figures after the comma.

Thus the fractions, 0,56, 0,036, are changed into and

86. An expression, in figures, of numbers containing decimal parts, is read by enunciating, first, the figures placed on the left of the point, then those on the right, adding to the last figure of the latter the denomination of the parts, which it represents.

The number 26,736 is read 26 and 736 thousandths; the number 0,0673 is read 673 ten thousandths,

and 0,0000673 is read 673 ten millionths.

+ In English books on mathematics, and in those that have been written in the United States, decimals are usually denoted by a point, thus 0.19; but the comma is on the whole in the most general use; it is accordingly adopted in this and the subsequent treatises to be published at Cambridge.

87. As decimal figures take their value entirely from their position relative to the comma, it is of no consequence whether we write or omit any number of ciphers on their right. For instance, 0,5 is the same as 0,50; and 0,784 is the same as 0,78400; for, in the first instance, the number, which expresses the decimal fraction, becomes by the addition of a 0 ten times greater, but the parts become hundredths, and consequently on this account are ten times less than before; in the second instance, the number, which expresses the fraction, becomes a hundred times greater than before, but the parts become hundred thousandths, and, consequently, are a hundred times smaller than before. This transformation, then, becomes the same as that which takes place with respect to a vulgar fraction, when each of its terms is multiplied by the same number; and if the ciphers be suppressed, it is the same as dividing them by the same number.

88. The addition of decimal fractions and numbers accompanying them, needs no other rule than that given for the whole numbers, since the decimal parts are made up one from the other, ascending from right to left, in the same manner as whole units. For instance, let there be the numbers 0,56, 0,003, 0,958; disposing them as follows,

we find, by the rule of article 12, that their sum is 1,521. 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.

Examples for practice.

Add 4,003, 54,9, 3,21, 6,7203.

Ans. 68,8333.

Add 409,903, 107,7842, 6,1043, 10,2074.

Ans. 534,0889.

Ans. 1243,575.

Add 427, 603,04, 210,15, 3,364,,021.

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 0,250S

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

Again, let 7,564 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 is put directly under those of the given numbers.

1

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 are 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. placed 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 $4137 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.