Terms of a fraction may be multiplied or divided by the same quantity.

CHAPTER II.

FRACTIONS AND PROPORTIONS.

SECTION I.

Reduction of Fractions.

54. When a quotient is expressed by placing the dividend over the divisor with a line between them, it is called a fraction; its dividend is called the numerator of the fraction, and its divisor the denominator of the fraction; and the numerator and denominator of a fraction are called the terms of the fraction.

When a quotient is expressed by the sign (:) it is called a ratio; its dividend is called the antecedent of the ratio, and its divisor the consequent of the ratio; and the antecedent and consequent of a ratio are called the terms of the ratio.

55. Theorem. The value of a fraction, or of a ratio, is not changed by multiplying or dividing both its terms by the same quantity.

Proof. For dividing both these terms by a quantity is the same as striking out a factor common to the two terms of a quotient, which, as is evident from art. 35, does not affect the value of the quotient. Also multiplying both terms by a quantity is only the reverse of the preceding process, and cannot therefore change the value of the fraction or ratio.

56. The terms of a fraction can often be simplified

Greatest Common Divisor.

by dividing them by a common factor or divisor. But when they have no common divisor, the fraction is said to be in its lowest terms.

A fraction is, consequently, reduced to its lowest terms, by dividing its terms by their greatest common factor or divisor.

57. Problem. To find the greatest common divisor of several monomials.

Solution. It is equal to the product of the greatest common divisor of the coefficients, by those different literal factors which are common to all the monomials, each literal factor being raised to the lowest power which it has in either of the monomials.

58. EXAMPLES.

1. Find the greatest common divisor of 75 a3 b8 c d11 x9 and 50 a3 c2 d11 x5.

Ans. 25 a3 c dll x5.

59. Lemma. The greatest common divisor of two quantities is the same with the greatest common divisor of the least of them, and of their remainder after division.

Demonstration. Let the greatest of the two quantities be A, and the least B; let the entire part of their quotient after division be Q, and the remainder R; and let the greatest

Greatest Common Divisor.

common divisor of A and B be D, and that of B and R be E. We are to prove that

D= E.

Now since R is the remainder of the division of A by B, we have

R=A-B.Q;

and, consequently, D, which is a divisor of A and B, must divide R; that is, D is a common divisor of B and R, and cannot therefore be greater than their greatest common divisor E.

Again, we have

A=R+B.Q,

and, consequently, E, which is a divisor of B and R, must divide A; that is, E is a common divisor of A and B, and cannot therefore be greater than their greatest common divisor D.

D and E, then, are two quantities such that neither is greater than the other; and must therefore be equal.

60. Problem. To find the greatest common divisor of any two quantities.

Solution. Divide the greater quantity by the less, and the remainder, which is less than either of the given quantities, is, by the preceding article, divisible by the greatest common divisor.

In the same way, from this remainder and the divisor a still smaller remainder can be found, which is divisible by the greatest common divisor; and, by continuing this process with each remainder and its corresponding divisor, quantities smaller and smaller are found, which are all divisible by the greatest common divisor, until at length the common divisor itself must be attained.

Greatest Common Divisor.

The greatest common divisor, when obtained, is at once recognised from the fact, that the preceding divisor is exactly divisible by it without any remainder.

The quantity thus obtained, must be the greatest common divisor required; for, from the preceding article, the greatest common divisor of each remainder and its divisor is the same with that of the divisor and its dividend, that is, of the preceding remainder and its divisor; hence, it is the same with that of any divisor and its dividend, or with that of the given quantities.

61. Corollary. When the remainders decrease to unity, the given quantities have no common divisor, and are said to be incommensurable or prime to each other.

62. EXAMPLES.

1. Find the greatest common divisor of 1825 and 1995.

Greatest Common Divisor.

This process may be written more neatly and concisely as follows.

2. Find the greatest common divisor of 13212 and 1851.

Ans. 3.

3. Find the greatest common divisor of 1221 and 333. Ans. 111.

63. The above rule requires some modification in its application to polynomials.

Thus it frequently happens in the successive divisions, that the term of the dividend, from which the term of the quotient is to be obtained, is not divisible by the corresponding term of the divisor. This, sometimes, arises from a monomial factor of the divisor which is prime to the dividend, and which may be suppressed.

For, since the greatest common divisor of two quantities is only the product of their common factors, it is not affected by any factor of the one quantity which is prime to the other.

Hence any monomial factor of either dividend ́or its divisor is to be suppressed which is prime to the other of these two quantities, and when there is such a factor it is readily obtained by inspection.