ART. 133. To reduce a complex fraction to a simple one. This is merely a case of division, in which the dividend and divisor are either fractions or mixed quantities. b at c Thus M n b is the same as to divide a+ by m ( a +2 ) ÷ ( m—? ) — ac+b ̧ mr—n_ac+bx. (+)÷( = с g n mr-n — acr+br cmr-cn Let the following examples be solved in the same manner. 3x ART. 134. Resolution of fractions into series. DEF.-An infinite series consists of an unlimited number of terms which observe the same law. The law of a series is a relation existing between its terms, so that, when some of them are known, the succeeding terms may be easily obtained. be found by multiplying the preceding term by -1. Any proper algebraic fraction, whose denominator is a polynomial, may, by division, be resolved into an infinite series; for the numerator is a dividend, and the denominator a divisor, so related to each other that the division can never terminate, and the quotient will therefore be an infinite series. After finding a few terms of the series, the law of continuation is, in general, easily seen, and the succeeding terms may be found without continuing the division. In a similar manner, let the fractions in each of the following examples be resolved into an infinite series. are finite determinate b When the two terms of a fraction quantities, the fraction has necessarily a finite determinate value, that is, the quotient of a divided by b. Let us now examine the cases where the numerator or denominator, or both, reduce to zero. ART. 135. To prove that 5 =0. While the denominator b is a constant number, if the numerator a diminishes, the value of the fraction diminishes. Thus, in.. the fractions 7, §, §, and 1, each is less than the preceding. Hence, as the numerator a diminishes, and approaches to zero, the value of diminishes and approaches to zero; and finally, b when a=0, the expression reduces to zero. Or thus: Since the product of zero, by any number, is zero, therefore the quotient of zero, divided by any number, is zero. That is, since 0xb=0, therefore 0 ART. 136. To prove that =0. If the numerator a, of a fraction, remains constant, and the denominator diminishes, the value of the fraction increases. a Thus: 1st. Suppose the denominator 1; then a. From this it is evident, that if the denominator is less than any assignable quantity, that is 0, the value of the fraction is greater than any assignable quantity, that is infinitely great, or infinity. This is designated by the sign ; that is When both numerator and denominator are zero, the fraction α becomes Now since the divisor zero, multiplied by any b number whatever, produces the dividend zero; therefore the quotient of zero, divided by zero, may be taken any number whatever; that 0 is, the fraction is indeterminate. 0 0 It is important, however, for the pupil to know, that the form is often the result of a particular supposition, when both terms but if we cancel the common factor, a-b, and then make b=a, we have x=2a. Similarly, the fraction x= a2-1 a2+a-2 0 becomes when a=1; but if we divide both terms by their common factor, a-1, we have These examples show, that if the value of any quantity is, before we decide that it is really indeterminate, we must see that the apparent indetermination has not arisen from the existence of a factor, which, by a particular supposition, became equal to zero. ART. 138. THEOREM.. - If the same quantity be added to both terms of a proper fraction, the new fraction resulting will be greater than the first; but if the same quantity be added to both terms of an improper fraction, the new fraction resulting will be less than the first. α Let be a proper fraction, a being less than b. Let m represent the quantity to be added to each term, then a+m the resulting fraction is b+m er, we must reduce them to a common denominator; Since the denominators are the same, that fraction is the greatest which has the greatest numerator. α When is a proper fraction, a is less than b; therefore am is less than bm, that is, the resulting fraction is greater than the first. But if a b is an improper fraction, it is evident that ab+am>ab+bm; that is, the resulting fraction is less than the first. ART. 139. THEOREM.- If the same quantity be subtracted from both terms of a proper fraction, the new fraction resulting will be less than the first; but if the same quantity be subtracted from both terms of an improper fraction, the new fraction resulting will be greater than the first. a Let be a proper fraction, a being less than b. Let m repreb sent the quantity to be subtracted from each term, then the resulting fraction is am b-m To determine which fraction is the greater, we reduce them to a common denominator, and compare |