arithmetic, and the part below the line (b) the denominator. The numerator and denominator are called the terms of the fraction. The line between them indicates division just as does the a sign. Thus is the same as a ÷ b. The denominator shows the number of parts into which the unit is divided, and the numerator shows the number of parts taken. An entire quantity or integer is one which has no fractional part, such as a + b, or a + 3xy, etc. Every integer may be considered as a fraction, the denominator of which is unity; thus are integers, and are identical with integral a+b a + 3xy and 1 1 expressions a + b and a + 3xy. We have already seen what a fractional quantity is. A mixed quantity is one which has both entire and fractional It is very important to understand the following general principles governing fractions : (1.) If the numerator of a fraction be multiplied or the denominator of a fraction divided by any quantity, the fraction is multiplied by that quantity. ac Τ a If we multiply the numerator of the fraction by c, we have b In the first fraction we have the unit divided into parts and a parts taken. In the second case we have ac parts taken. Hence since c times as many parts were taken in the second case, ac = a b b × c, i. e., multiplying the numerator of a fraction by a quantity multiplies the fraction by that quantity. Again letbe any fraction. Let the denominator b be divided b b by c. Then in the first case we have a parts of a unit divided into b parts, while in the second case we have a parts of a unit divided into parts. Consequently in the second case the number of parts will be only times as many as before, and each part will, therefore, be c times as great, or, as each part is c times as great, the whole number of parts is c times as great as before, 1 с Hence dividing the denominator of a fraction by a quantity is equivalent to multiplying the numerator by that quantity. In a similar way we may prove that : (2). If the numerator of a fraction be divided, or the denominator multiplied by any quantity, the fraction is divided by that quantity. This is also evident as the result of such an operation would, obviously, have the opposite effect to that of the principle previously demonstrated in this article. (3). If the numerator and denominator of a fraction are both multiplied, or both divided, by the same quantity, the value of the fraction is not altered. For by (1) and (2) multiplying the numerator multiplies the fraction while multiplying the denominator divides it, and dividing the numerator divides the fraction while dividing the denominator multiplies it. Hence, in the first place we multiply and divide and in the second case divide and multiply the fraction by the same quantity, which obviously does not alter its value. REDUCTION OF FRACTIONS TO LOWEST 47. A fraction is in its lowest terms when its numerator and denominator are prime to each other. It is convenient to classify into two cases. Case I. When the numerator and denominator can be readily factored by inspection. Since dividing both numerator and denominator by the same quantity (which is equivalent to cancelling equal factors in each of them) does not change the value of the fraction, we may proceed by the following rule: Resolve both numerator and denominator into their prime factors, and cancel all of these factors which are common to both. Let us take a few examples: 10a2b7c Reduce to its lowest terms: 15abc If all factors of a term are removed by cancellation, unity, 1, which is a factor of all algebraic expressions, remains in place of the cancelled term. If this occurs in the denominator, we have a case of exact division, and, as the denominator becomes 1, we have an entire quantity or integer as a result. Case II. When the numerator and denominator cannot be readily factored by inspection. Since the highest common factor of two quantities is the product of their common prime factors, we have the following rule: Divide both terms by the highest common factor of those terms. 2x 4x8x + 3 By Case III of H. C. F., the H. C. F. of both terms is - 3. Hence dividing both numerator and denominator by 2x - 3 If both terms of a fraction are multiplied quantity the value of the fraction is not altered. 6x 5 by the same Thus we may multiply both terms of a fraction by 1 without changing its This may also be understood by remembering that, as a fraction represents the quotient obtained by dividing its numerator by its denominator, if the signs of both terms are the same the fraction is positive, but if they are unlike in sign the fraction is negative. = x (for two negative signs result in a + sign when multiplied.) If either the numerator or the denominator is a polynomial and its sign is to be changed, the sign of each of its terms must be As the multiplication of quantities with like signs gives +, and the multiplication of quantities with unlike signs gives —, we If the terms of a fraction are composed of factors, changing the signs of any even number of factors does not change the value of the fraction. But changing the signs of an odd number of factors changes the sign of the fraction. The fraction a b (cd) (ef) may, therefore, be written in any of the following forms: (d—c) (f—e)' (d—c) (e-ƒ)' (d—c) (ƒ — e) REDUCTION OF FRACTIONS TO ENTIRE OR MIXED QUANTITIES. rule: Since a fraction expresses division, we have the following Divide the numerator by the denominator. 6.x2 + 4x 3 Reduce to a mixed quantity. 2x Dividing each term of the numerator by the denominator we |