36 75. It is important to observe, that any division, whether it can be performed in whole numbers or not, may be indicated by a fractional expression; &, for instance, expresses evidently the quotient of 36 by 3, as well as 12, for being contained three times in unity, 36 will be contained 3 times in 36 units, as the quotient of S6 by s must be. 76. It may seem preposterous to treat of the multiplication and division of fractions before having said any thing of the manner of adding and subtracting them; but this order has been followed, because multiplication and division follow as the immediate consequences of the remark given in the table of article 55, but addition and subtraction require some previous preparation. It is, besides, by no means surprising, that it should be more easy to multiply and divide fractions, than to add and subtract them, since they are derived from division, which is so nearly related to multiplication. There will be many opportunities, in what follows, of becoming convinced of this truth; that operations to be performed on quantities are so much the more easy, as they approach nearer to the origin of these quantities. We will now proceed to the addition and subtraction of fractions. 77. When the fractions on which these operations are to be performed have the same denominator, as they contain none but parts of the same denomination, and consequently of the same magnitude or value, they can be added or subtracted in the same manner as whole numbers, care being taken to mark, in the result, the denomination of the parts, of which it is composed. It is indeed very plain, that and make, as 2 quantities and 3 quantities of the same kind make 5 of that kind, whatever it may be. Also, the difference between and is, as the difference between 3 quantities and 8 quantities, of the same kind, is 5 of that kind, whatever it may be. Hence it must be concluded, that, to add or subtract fractions, having the same denominator, the sum or difference of their numerators must be taken, and the common denominator written under the result. 78. When the given fractions have different denominators, it is impossible to add together, or subtract, one from the other, the parts of which they are composed, because these parts are of different magnitudes; but to obviate this difficulty, the fractions are made to undergo a change, which brings them to parts of the same magnitude, by giving them a common denominator. 10 For instance, let the fractions be and ; if each term of the first be multiplied by 5, the denominator of the second, the first will be changed into 19; and if each term of the second be multiplied by 3, the denominator of the first, the second will be changed into ; thus two new expressions will be formed, having the same value as the given fractions (56). This operation, necessary for comparing the respective magnitudes of two fractions, consists simply in finding, to express them, parts of an unit sufficiently small to be contained exactly in each of those which form the given fractions. It is plain, in the above example, that the fifteenth part of an unit will exactly measure and of this unit, because contains five 15ths, and contains three 15th. The process, applied to the fractions and, will admit of being applied to any others. In general, to reduce any two fractions to the same denominator, the two terms of each of them must be multiplied by the denominator of the other. 79. Any number of fractions are reduced to a common denominator, by multiplying the two terms of each by the product of the denominators of all the others; for it is plain that the new denominators are all the same, since each one is the product of all the original denominators, and that the new fractions have the same value as the former ones, since nothing has been done except multiplying each term of these by the same number (56). Examples. Reduce and to a common denominator. Ans. 37, 38. 70 30 70' Reduce and to a common denominator. Ans. §6 10 Reduce, 2, and to a common denominator. Ans. 20, 45, 48 Reduce 3 4 ,,, and to a common denominator. 1890 Ans. 630 1800 1750 3150 3150 3130' 3130 The preceding rule conducts us, in all cases, to the proposed end; but when the denominators of the fractions in question are not prime to each other, there is a common denominator more simple than that which is thus obtained, and which may be shown to result from considerations analogous to those given in the preceding articles. If, for instance, the fractions were,, ,, as nothing more is required, for reducing them to a common denominator, than to divide unity into parts, which shall be exactly contained in those of which these fractions consist, it will be sufficient to find the smallest number, which can be exactly divided by each of their denominators, 3, 4, 6, 8; and this will be discovered by trying to divide the multiples of 3 by 4, 6, 8; which does not succeed until we come to 24, when we have only to change the given fractions into 24ths of an unit. To perform this operation we must ascertain successively how many times the denominators, 3, 4, 6, and 8, are contained in 24, and the quotients will be the numbers, by which each term of the respective fractions must be multiplied, to be reduced to the common denominator, 24. It will thus be found, that each term of must be multiplied by 8, each term of by 6, each term of by 4, and each term of by 3; the fractions will then become,,, . 21 Algebra will furnish the means of facilitating the application of this process. 80. By reducing fractions to the same denominator, they may be added and subtracted as in article 77. 81. When there are at the same time both whole numbers and fractions, the whole numbers, if they stand alone, must be converted into fractions of the same denomination as those which 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 12 and 3, and gives for the result 1. To add 32 to 53, the whole numbers must be reduced to fractions, of the same denomination as those which accompany them, which reduction gives 23 and 49; with these results the sum is found to be 550, or 84. If, lastly, were to be subtracted from Arith. 7 34, the operation would be reduced to taking from 13, and the remainder would be 42 82. The rule given, for the reduction of fractions to a common denominator supposes, that a product resulting from the successive multiplication of several numbers into each other, does not vary, in whatever order these multiplications may be performed; this truth, though almost always considered as selfevident, 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, and the product of 5 by 7 is 35, as well as the product of 1 by 35; but 3, or any other number, being only an assemblage of several units, the same property will belong to it, as to each of the units of which it consists; that is, the product of 3 by 5 and by 7, obtained in either way, being the triple of the result given by unity, when multiplied by 5 and 7, must necessarily be the same. It may be proved in the same manner, that were it required to multiply 3 by the product of 5, 7, and 9, it would consist in multiplying 3 by 5, then this product by 7, and the result by 9, and so on, whatever might be the number of factors. 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 be changed, because this product is equivalent to 5, multiplied by the product of the numbers 3 and ; 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 small |