26. A Continued Fraction is a fraction whose numerator is an integer, and whose denominator is an integer plus (or minus) another fraction whose numerator is an integer, and whose denominator is an integer plus (or minus) a third fraction, etc. Observe that in this reduction the work proceeds from below 3. Ex. 3 2+1+1=2+2+1-1 (+1)-1 x2+2+ = = 28. By Art. 5, a fraction is a number which, multiplied by the denominator, gives the numerator. Therefore the fraction ¦ is a number which, multiplied by 0, gives 0. But by Ch. III., § 3, Art. 16, any number, multiplied by 0, gives 0. Therefore the fraction & may denote any number whatever. For this reason, it is called an Indeterminate Fraction. Some Principles of Fractions. 29. The following principles will be of use in subsequent work : then each of these fractions is equal to the fraction Let the common value of the given fractions be v. Then from Multiplying these equations by a, b, c, etc., respectively, and adding corresponding members of the resulting equations, we have (iii.) If the fraction be in its lowest terms, then d positive integer, is in its lowest terms. For by Ch. VIII., § 2, Art. 13 (vii.), no and dr are prime to each other when n and d are prime to each other. n N (iv.) If two fractions, and whose terms are positive integers, be n d D' equal, and if be in its lowest terms, then N= kn, D = kd, wherein k is d Since N is an integer, this equation shows that nD is exactly divisible by d; that is, that it contains d as a factor. Also, since is in its lowest d n terms, n and d are prime to each other. Therefore, by Ch. VIII., § 2, Art. 13 (iv.), d is a factor of D; that is, D = kd, wherein k is a positive integer. Substituting kd for D in (1), we have n N D' (v.) If two fractions and whose terms are positive integers, be d equal, and each be in its lowest terms, then N: =n and D= d. This principle follows directly from (iv.). |