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M and C the greatest number of times it occurs in either M or C. Therefore, M' will contain each factor of A, B, and C the greatest number of times it occurs in A or B or C; and that is what is required to be proved.

125. For an understanding of the remaining chapters in this book. complete theories of the greatest common divisor and the least common multiple are not necessary. Such discussions will be found in works on the theory of equations. However, the solution of the examples illustrating the principles discussed in the preceding sections is a valuable exercise in the fundamental operation of algebra.

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12. a (x —b) (x — c), b(c−x) (x -− a), c(a — x) (b −x).

-

13. 1+y+y, 1−y+y2, 1+ y2+y^.

14. (1-x), (1 — x)2, (1—x)3.

15. (a+c)2= b2, (a + b)2 — c2, (b+c)2 — a2.

16. 25+2x+x3+8a2+16x+8 and 25-4x+xo — 4.

17.

18.

19.

x-4x3+x2-4 and x3 +2 ̧2 −6 x — 12.

2x2+2x−1, 3 x3 — 4 x +1, and 2x3-3x+1.
2-x-2, x3-4x2+3, and a3-3x2+2.

20. x2-3x+2, x3 −6 x2 + 11x-6, and x2−5x+6.

CHAPTER X

FRACTIONS

For

126. If a and b are two integral numbers such that the group of things represented by b can not be counted out of the group of things represented by a, then the symbol is a fraction. I. example, a is called the numerator and b the denominator of the fraction

a

5

The numerator and the denominator of the fraction are called the terms of the fraction.

The symbol() is always subject to the relation

(1)

(1)

b = በ.

[872]

An integral expression may be regarded as a fraction whose denominator is unity: thus, a+b is the same as a+b by the definition of quotient, ("+"). 1 = a + b.

is a rational fraction when a and b are integers.

5

8

5 8

The denominator of the fraction declares that the number of things in a certain group is 8, and the numerator 5 declares that of this group of 8 things 5 are taken; and in the use of the fraction the unit group is a group of 8 things. Thus, if one has a quantity of vinegar to measure, and finds that a gallon measure, which contains 8 pints, can be filled 13 times and that besides a pint measure can be filled 5 times, the measure of the quantity of vinegar is 13 gallons. A gallon is the unit, and the denominator 8 declares that the unit gallon is divided into 8 parts, and that 5 of these parts, or pints, are taken in the fraction Here gal. means one of the eight equal parts of a gallon (a pint), or briefly, 1 eighth of a gallon; and gal. means 5 of the eight equal parts of a gallon (or

8

5

8

8

5 pints) or briefly, 5 eighths of a gallon; and similarly for any other fraction.

127. The rules of division are purely the formal consequences of the fundamental laws of the multiplication of numbers, III, IV, V, definition IX (61), theorem XI (863), and the corresponding laws of addition and subtraction.

The rules of division, or, what is the same thing, the rules of the operation of fractions, can be deduced in the same way (872) as the rules of subtraction (38, 1-5). They follow without regard to the meaning of the symbols a, b, c, =, +, —, ab, ab, (872).

a

128. The rule governing the dependence of signs of a fraction upon the signs of its terms is deduced from the rules of the signs of products when the factors have different signs (241, 6 and 8), thus:

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Proofs: According to the rules used in establishing the preceding rule, we have:

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The sign written before the fraction is called the sign of the fraction.

Thus, if the sign of both numerator and denominator are changed, the sign of the fraction is not changed; but if the sign of either one is changed, the sign before the fraction is changed.

In case the numerator or denominator is a polynomial, we must be careful, in changing the signs, to change the sign of each of its b ቢ terms ( 41, 3, 4, 5). Thus, the fraction can be written, by changing the signs of both numerator and denominator, in the

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с

d

129. It follows from 41, 6, 8, that if the terms of a fraction are the indicated products of two or more parentheses, the sign of the fraction will remain the same, if the signs of an even number of the parentheses be changed, but the sign of the fraction will be changed if the signs of an odd number of parentheses be changed.

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If the integer in the numerator of a fraction is less than its denominator the fraction is said to be a proper, or pure fraction, and if greater, an improper fraction.

REDUCTION OF FRACTIONS

130. The Reduction of Fractions to their Lowest Terms.

Let the line AB be divided into seven equal parts, at D, E, F, G, II, I.

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Now let each of these parts be subdivided into 3 equal parts.

Then AB contains 21 of these subdivisions and AG contains 12

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That is, the value of the fraction is not altered by multiplying

both its terms by 3, and the value of the fraction altered by dividing both of its terms by 3.

12

is not

21

131. The result of the previous section is a particular case of the following:

Theorem I. It does not alter the value of a fraction to multiply or divide both of its terms by the same quantity.

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On multiplying both members of the equation by e, it becomes

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Hence, it follows from (1) and (2), to reduce a fraction to lower terms, divide both numerator and denominator by any factor common to both.

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