Property of Vulgar Fractions. 327 



If - =£ or — , =-^-7, we merely have the case of m = or n=0 

 v q v q 



respectively. 



It is evident from the preceding considerations that^>, q, 

 the numerator and denominator of the inferior convergent of 



j, are the remainders when u and v are divided respectively 



by a and b ; and that^/, q f , the numerator and denominator of 



the superior convergent of ■=, are the remainders when v! and 

 v' are divided by a and b. 



It was rather more convenient to define the convergents of 



7 as the fractions nearest to it, on each side, having denomi- 

 nators less than b : but they might equally well be defined as 

 the fractions nearest to -7, on each side, having numerators not 

 exceeding a\ for, the single case of a = l excepted, if - be a 

 convergent of j, then p> or <a, according as q> or <b. 



§ 5. The following ten consecutive fractions, which be- 

 long to a series in which the greatest admissible numerator or 

 denominator is 1000, will serve to illustrate the remarks in the 

 last section: — 



220' 817 ? 597' 974 J 377' 911 ? 534* 69l' 848' 157* 



12 



Consider the fraction ^p= ; we have 



31 + 29 60 5.12 



974 + 911 1885 "5. 377' 



so that the factor 5 divides out from both numerator and de- 

 nominator. To see how this is brought about by means of the 



lemma, we observe (i) that the convergents of -^-r are — — 



and ^y, and that the convergents of -^ are- 377 and ^0 ; 



29 12 17 



(ii) that the convergents of ^- are g^ and pj, and that 



17 12 5 



the convergents of ^ are ^ and ^'; (iii) that the con- 

 vergents of ^ are ^q and j^. Thus, considering the nu- 



