SCALES OF NOTATION 41

99. If we consider only positiv or zero integers, the theorems
of §§ 96, 97 give the following.

Th. If a is any positiv or zero integer, 1, 13, 73, * * + an infinit
series of integers all greater than 1, if g, where k = 1, is the
exact or lower approximate quotient obtaind by dividing its prede-
cessor in the series a, qi, @2, qs, *** by ri, if also ar—1 s the re-
mainder corresponding to the quotient qi; then a quotient gnia,
where n = 0, will be arrived at whose value is zero, this being
the first quotient in case a = 0; if we stop the process of division
as soon as we obtain a zero quotient, there is only one set of
remainders, all positiv or zero and each less than its divisor;

if a = 0, there is only one remainder, which is 0;

if a —= 0, the last remainder a, is equal to q, and therfor not
2ero;

finally, if for convenience we set ro = 1,

n l
a = Z {alH (rm)}
=0 m=0
= qgro + a1ror1 + Qe orirs - asrorirars + < o o - anrorirers e - - 7a(Y)

Example.

 

 

 

 

467 =1+4+2:-24+1:2X3+42X3X44+3-2X3X4X5

() G. Chrystal, l.c., p. 163.