The sign of the quotient will be changed by changing the sign sither of the numerator or denominator, but will not be affected by changing the signs of both the terms. 70. We will add two propositions on the subject of fractions. I. If the same number be added to each of the terms of a proper fraction, the fraction resulting from these additions will be greater than the first; but if it be added to the terms of an improper fraction, the resulting fraction will be less than the first. Let the fraction be expressed by α b Let m represent the number to be added to each term: then the new fraction will be, a + m b + m In order to compare the two fractions, they must be reduced to the same denominator, which gives for Now, the denominators being the same, that fraction will be the greater which has the greater numerator. But the two numerators have a common part ab, and the part bm of the second is greater than the part am of the first, when b>a: hence ab + bm ab + am; that is, when the fraction is proper, the second fraction is greater than the first. If the given fraction is improper, that is, if a >b, it is plain that the numerator of the second fraction will be less than that of the first, since bm would then be less than am. II. If the same number be subtracted from euch term of a proper fraction, the value of the fraction will be diminished; but if it be subtracted from the terms of an improper fraction, the value of the fraction will be increased. Let the fraction be expressed by to be subtracted by m. Then, a and denote the number b' By reducing to the same denominator, we have, Now, if we suppose a <b, then am <bm; and if am < bm, then will ab am > ab that is, the new fraction will be less than the first. If a >b, that is, if the fraction is improper, then am >bm, and ab am < ab - bm, that is, the new fraction will be greater than the first. = + ebdxs bdfx6bdf6bdfx6 adfx2+bcfx+bde bdfx3 adfhx9 bcfhx8 bedhx bdfgx adfhx3 + bcfhx2 -bedhx 71. The symbol 0 is called zero, which signifies in ordinary language, nothing. In Algebra, it signifies no quantity: it is also used to expres a quantity less than any assignable quantity. The symbol is called the symbol for infinity; that is, it is used to represent a quantity greater than any assignable quantity. α b' If we take the fraction and suppose, whilst the value of a remains the same, that the value of b becomes greater and greater, it is evident that the value of the fraction will become less and less. When the value of b becomes very great, the value of the fraction becomes very small; and finally, when b becomes greater than any assignable quantity, or infinite, the value of the fraction becomes less than any assignable quantity, or zero. Hence, we say, that a finite quantity divided by infinity is equal to zero. a We may therefore regard and 0, as equivalent symbols. α If in the same fraction we suppose, whilst the value of a b remains the same, that the value of b becomes less and less, it is plain that the value of the fraction becomes greater and greater; and finally, when b becomes less than any assignable quantity, or zero, the value of the fraction becomes greater than any assignable quantity, or infinite. Hence, we say, that a finite quantity divided by zero is equal to infinity. a We may then regard and as equivalent symbols: Zero 0 and infinity are reciprocals of each other. 0 The expression is a symbol of indetermination; that is, it 0 is employed to designate a quantity which admits of an infinite number of values. The origin of the symbol will be explained in the next chapter. |