that measures them both, which quantity is called the Greatest Common Measure of the numerator and denominator. [102. A fraction which has either its numerator or denominator a simple algebraical quantity is easily reduced to lowest terms; for the greatest common divisor is at once found by in3a2bc spection. Thus to reduce to its lowest terms, we see 5a3b2 d that ab is the greatest common measure of the numerator and denominator, therefore the fraction required is 30 5abd But in the case of fractions having for both numerator and denominator a compound algebraical quantity the following rule is needed.] 103. The Greatest Common Measure of two compound algebraical quantities is found by arranging them according to the powers of some letter, and then dividing the greater by the less, and the preceding divisor always by the last remainder, till the remainder is nothing; the last divisor is the greatest common measure required. Let a and b be the two quantities, and let b be contained p times in a, with a remainder c; again, let c be contained q times in b, with a remainder d, and so on, till nothing remains; let d be the last divisor, and it will be the greatest common measure of a and b. b) a (p c) b (9 gc d) c (r rd 0 104. The truth of this rule depends upon these two principles; 1st. If one quantity measure another, it will also measure any multiple of that quantity. Let a measure y by the units in n, then it will measure cy by the units in nc. d. If a quantity measure two others, it will measure their sum or difference. Let a be contained m times in ~, and n times in y; then max and nay; therefore y = ma±na* (mn) a; that is, a is contained mn times in xy, or it measures ay by the units in m ± n. * [This is merely a short method of writing a+y=ma+na, and x-y=ma-na]. 105. Now it appears from the operation (Art. 103.) that a - pbc, and b-qcd; every quantity therefore, which measures a and b, measures pb, and a pb, or c; hence also it measures qc, and b - qc, or d; that is, every common measure of a and b measures d. It appears also from the division that a = pb + c, b=qc+d, c=rd; therefore d measures c, and qc, and qc+ d or b; hence it measures pb, and pb+c, or a. Every common measure then of a and b measures d, and d measures a and b; therefore d is their greatest common measure*. [Ex. 1. To find the Greatest Common Measure of a2+2a+1 a + 1 is therefore the Greatest Common Measure of the two quantities; and if they be respectively divided by it, the fraction is reduced to a + 1 and is in its lowest terms.] Ex. 2. To find the Greatest Common Measure of a1- x1 and a3- a2x-ax2+x3; and to reduce its lowest terms. * [This conclusion is more obvious when stated thus:-every common divisor of a and b is a divisor of d, but no quantity can be a divisor of d which is greater than d, therefore every common divisor of a and b is not greater than d; and since d is one of them, therefore d is the greatest.] leaving out 2a, the next divisor is a2x2. Therefore aa is the G.c.M. required; and the fraction and is in its lowest terms. is reduced to a2 + x2 a -x The quantity 2a, found as a factor in every term of one of the divisors, 2a2x2-2a4, but not in every term of the dividend, a3- a2x-ax2+ x3, must be left out; otherwise the quotient will be fractional, which is contrary to the supposition made in the proof of the rule: and by omitting this part, 2x2, no common measure of the divisor and dividend is left out, because, by the supposition, no part of 22 is found in all the terms of the dividend. (See Note 4. Appendix I.) 106. To find the Greatest Common Measure of three quantities, a, b, c, take d the Greatest Common Measure of a and b; and the Greatest Common Measure of and c is the Greatest Common Measure required. Because every common measure of a, b, and c, measures d and c; and every measure of d and c measures a, b and c (Art. 105); therefore, the Greatest Common Measure of d and c must be the Greatest Common Measure of a, b and c. 107. In the same manner the Greatest Common Measure of four or more quantities may be found. The Greatest Common Measure of four quantities, a, b, c, d, may also be found by taking a the Greatest Common Measure of a and b, and y the Greatest Common Measure of c and d; then the Greatest Common Measure of x and y will be the common measure required. [That the continued division indicated in Art. 103 will come to an end is proved as follows:-] 108. If one quantity be divided by another, and the preceding divisor by the remainder, the remainder will at length be less than any quantity that can be assigned. = For a pb+c; and b, and consequently pb, is greater than c; therefore pb+c, or a, is greater than 2c, and is a 2 greater than c; therefore from a, a quantity greater than its half has been taken; in the same manner, when c is the dividend, more than its half is taken away, and so on; but if from any quantity there be taken more than its half, and from the remainder more than its half, and so on, there will, at length, remain a quantity less than any that can be asigned (Euc. x. I.) [109. In practice the Greatest Common Measure of two or more algebraical quantities is frequently found by a more expeditious method than the preceding, as follows. Taking Ex. 2. Art. 105. and a3-a2 x − a x2+x3. Ex. Required the G. c. M. of a1- Also a3-ax-a x2 + x3 = a2 (a − x) − x2 (α − x) = (a2 — x2) (a− x); therefore a2x2 is a common factor or divisor of the proposed quantities; and since the other factors a2+2 and a have no common measure greater than 1, therefore a22 is the Greatest Common Measure required. 110. In the same manner fractions are usually reduced to their lowest terms without the application of the Rule for finding the G. C. M. of the numerator and denominator. |