530 ALGEBRA [FRACTIONS. 5 times the same unit is dividet into 7 equal parts, and one of them taken. In any fraction the upper number, or the dividend, is called the numerator, and the lower number or divisor is called the denominator. Thus, in the fraction -, a is the numerator, and b the denominator. If the numerator be less than the denominator, such a fraction is called a proper fraction ; but if the numerator be either equal to or greater than the denominator, it is called an improper fraction ; and if a quantity be made up of an integer and a fraction, it is called a mixed quantity. m a . . a , a+x ihus, - is a proper fraction : - and - are both im- a-f x a a proper fractions ; and b 4- - is a mixed quantity. The reciprocal of a fraction is another fraction, having its numerator and denominator respectively equal to the denominator and numerator of the former. Thus, is the reciprocal of the fraction r* a 6 26. The following proposition is the foundation of the operations relating to fractions. If the numerator and denominator of a fraction be cither both multiplied or both divided by the same quan tity, the value of the resulting fraction is the same as before. To demonstrate this proposition we shall throw the definition of a fraction into a categorical form. We shall accordingly define the fraction r- as such a magnitude, that when it is multiplied by Z>, the product is a. Then since i.e. (Art. 9, Law 3), But a . r x b = a. b a 71 X T~ X ^ 72.Q&
a - x nb = n o i /T e -7 xnb = na (Dcf.) id) na nb From this proposition, it is obvious that a fraction may be very differently expressed without changing its value, and that any integer may be reduced to the form of a fraction, by placing the product arising from its multipli cation by any assumed quantity as the numerator, and the assumed quantity as the denominator of the fraction. It also appears that a fraction very complex in its form may often be reduced to another of the same value, but more simple, by finding a quantity which will divide both the numerator and denominator, without leaving a remainder. Such a common divisor, or common measure, may be either simple or compound ; if it be simple, it is readily found by inspection, but if it be compound, it may be found as in the following problem. 27. PKOB. I. To find the greatest common Measure of two Quantities. Rule 1. Range the quantities according to the power of some one of the letters, as in division, leaving out the simple divisors of each quantity. 2. Divide that quantity which is of most dimensions by the other one, and if there be a remainder, divide it by its greatest simple divisor; and then divide the last compound divisor by the resulting quantity, and if any thing yet remain, divide it also by its greatest simple divisor, and the last compoxind divisor by the resulting quantity. Proceed in this way till nothing remain, and the last divisor shall be the common measure re quired. Note. It will sometimes be necessary to multiply the dividends by simple quantities in order to make the divisions succeed. The demonstration of this proposition depends on the AXIOM, that whatever divides a number divides any mul tiple of the number ; and whatever divides two numbers divides their sum or difference. It was given by Euclid in Prop. 2, Book vii., very much as follows : Let a, b be the quantities, the smaller of which is b. Let a be divided by b, with a remainder e, b by c, with a remainder d, c by d, with no remainder, d is the greatest common measure of a and b. We have apb = c, b qc = d, c = rd. Now, (1.) d is a common measure of a and b divides c .: qc .-. qc + d .: b .: pb . . pb + c .: a- divides a and b. (2.) It is the greatest common divisor. For if not, let e be the greatest; then, since e divides a and b, it divides a and pb, . . apb . . c . . qc . . b qc . . d; i.e., e is less than d, and not greater. Cor. Every other divisor of a and b divides their greatest common measure. Observe that no fraction is in a form to be interpreted until it is reduced to its lowest terms. Ex. 1. Required the greatest common measure of the quantities a~x x 3 and a? 2a*x + ax 2 . The simple di visor x being taken out of the former of these quantities, and a out of the latter, they are reduced to a 2 a 2 and a 2 lax + x" 1 ; and as the quantity a rises to the same dimensions in both, we may take either of them as the first ,divisor : let us take that which consists of fewest terms, and the operation will stand thus : f or d .e. d - lax + 2x 2 remainder, which, divided by - 2x, is a - x)o? x~(a + x - ax Hence it appears that a x is the greatest common measure required. Ex. 2. Required the greatest common measure of 8a 2 ^ 2 - IQab 3 + 26 4 , and 9a 4 6 - 9a 3 ^ 2 + 3a 2 6 3 - Sal*. It is evident, from inspection, that b is a simple divisor of both quantities ; it will therefore be a factor of the common measure required. Let the simple divisors be now left out of both quantities, and they are reduced to 4-a 2 5ab + b 2 , and 3a 3 3a"b + ab 2 b 3 but as the second of these is to be divided by the first, it must be multiplied by 4 to make the division succeed, and the operation will stand thus : -)l2a 3 - I2a- 4a 2 - This remainder is to be divided by b, and the new divi dend multiplied by 3, to make the division again succeed, and the work will stand thus : 3a 2 + ab - 2 )1 2a 2 - 1 5al +
12a 2 + 4a6-
