are to be added to them, or subtracted from them; and if the whole numbers are accompanied with fractions, they must be reduced to the same denominator with these fractions. It is thus, that the addition of four units and changes itself into the addition of and, and gives for the result 1a. To add 3 to 5, the whole numbers must be reduced to fractions, of the same denomination as those which accompany them, which reduction gives 23 and 9; with these results the sum is found to be 550, or 84. If, lastly, were to be subtracted from 31, the operation would be reduced to taking from 13, and the remainder would be 48. 82. The rule given for the reduction of fractions to a common denominator supposes, that a product resulting from the successive multiplications of several numbers into each other, does not vary, in whatever order these multiplications may be performed; this truth, though almost always considered as self-evident, needs to be proved. We shall begin with showing, that to multiply one number by the product of two others, is the same thing as to multiply it at first by one of them, and then to multiply that product by the other. For instance, instead of multiplying 3 by 35, the product of 7 and 5, it will be the same thing if we multiply 3 by 5, and then that product by 7. The proposition will be evident, if, instead of 3, we take an unit; for 1, multiplied by 5, gives 5, same. To represent in a shorter manner several successive multiplications, as of the numbers 3, 5, and 7, into each other, we shall write 3 by 5 by 7. This being laid down, in the product 3 by 5, the order of the factors, 3 and 5 (27), may be changed, and the same product obtained. Hence it directly follows, that 5 by 3 by 7 is the same as 3 by 5 by 7. The order of the factors 3 and 7, in the product 5 by 3 by 7, may also he changed, because this product is equivalent to 5, multiplied by the product of the numbers 3 and 7; thus we have in the expression 5 by 7 by 3, the same product as the preceding. By bringing together the three arrangements, 3 by 5 by 7 5 by 3 by 7 5 by 7 by 3, we see that the factor 3 is found successively, the first, the second, and the third, and that the same may take place with respect to either of the others. From this example, in which the particular value of each number has not been considered, it must be evident, that a product of three factors does not vary, whatever may be the order in which they are multiplied. If the question were concerning the product of four factors, such as 3 by 5 by 7 by 9, we might, according to what has been said, arrange, as we pleased, the three first or the three last, and thus make any one of the factors pass through all the places. Considering then one of the new arrangements, for instance this, 5 by 7 by 3 by 9, we might invert the order of the two last factors, which would give 5 by 7 by 9 by 3, and would put 3 in the last place. This reasoning may be extended without difficulty to any number of factors whatever. Decimal Fractions. 83. ALTHOUGH We can, by the preceding rules, apply to fractions, in all cases, the four fundamental operations of arithmetic, yet it must have been long since perceived, that, if the different subdivisions of a unit, employed for measuring quantities smaller than this unit, had been subjected to a common law of decrease, the calculus of fractions, would have been much more convenient, on account of the facility with which we might convert one into another. By making this law of decrease conform to the basis of our system of numeration, we have given to the calculus the greatest degree of simplicity, of which it is capable. We have seen in article 5, that each of the collections of units contained in a number, is composed of ten units of the preceding order, as the ten consists of simple units; but there is nothing to prevent our regarding this simple unit, as containing ten parts, of which each one shall be a tenth; the tenth as containing ten parts, of which each one shall be a hundredth of unity, the hundredth as containing ten parts, of which each one shall be a thousandth of unity, and so on. Procceding thus, we may form quantities as small as we please, by means of which it will be possible to measure any quantities, however minute. These fractions, which are called decimals, because they are composed of parts of unity, that become continually ten times smaller, as they depart further from unity, may be converted, one into the other, in the same manner as tens, hundreds, thousands, &c. are converted into units; thus, the unit being equivalent to 10 tenths, the tenth the hundredth 10 hundredths, 10 thousandths, it follows, that the tenth is equivalent to 10 times 10 thousandths, or 100 thousandths. For instance, 2 tenths, 3 hundredths, and 4 thousandths will be equivalent to 234 thousandths, as 2 hundreds, 3 tens, and 4 units make 234 units; and what is here said may be applied universally, since the subordination of the parts of unity is like that of the different orders of units. 84. According to this remark, we can, by means of figures, write decimal fractions in the same manner as whole numbers, since by the nature of our numeration, which makes the value of a figure, placed on the right of another, ten times smaller, tenths 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 0,304, 19 3 1000, 85. If the expressions for the above decimal fractions be compared with the following, 1, 34, 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. 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 published at Cambridge. Thus the fractions, 0,56, 0,036, are changed into and T888. 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 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. |