346 Professor HamiLton on Conjugate Functions, 
whole number, but lies between two successive whole numbers, a next preceding and 
a next succeeding, in the general progression of numbers ; and _ these may be dis- 
covered by the process of approximate division above mentioned, while each of the 
two remainders of that approximate division is the numerator of a new fraction, 
which retains the proposed denominator, and must be added algebraically as a cor- 
rection to the corresponding approximate integer quotient, in order to express, by the 
help of it, the quotient of the accurate division. For example, 
82 e8 
Sa ae ee 
Or} oe 
AS) PS 
ap (S) Ue 
Or| 
trl bo 
® 
w 
5) 
In general, a fractional number may be called a mixed number, when it is thus ex- 
pressed as the algebraic sum of a whole number and a proper fraction, this last name 
being given to a fractional number which lies between zero and positive or contra- 
positive one. We may remark that an ordinal relation between two fractional 
numbers is not altered by dividing them both by one common positive divisor ;_ but if 
the divisor be contra-positive, it changes a relation of subsequence to one of pre- 
cedence, and conversely, without disturbing a relation of coincidence. 
20. In all the formulz of the three last articles, we have supposed that all the 
numerators and all the denominators of those formule are positive or contra-positive 
whole numbers, excluding the number zero. However, the general conception of a 
fraction as a multiple of a sub-multiple, permits us: to suppose that the multipling 
number or numerator is zero, and shows us that then the fractional step itself is null, 
if the denominator be different from zero; that is, 
o xb =0if wk 0. (159.) 
Thus, although we supposed, in the composition (149.) of successive fractional steps, 
(with positive or contra-positive numerators and denominators,) that the resultant 
step was effective, yet we might have removed this limitation, and have presented the 
formule (150.) for fractional sums as extending even to the case when the resultant 
step is null, if we had observed that in every such case the resultant numerator of the 
formula is zero, while the resultant denominator is different from zero, and therefore 
that the formula rightly expresses that the resultant fraction or sum is null. For 
cee : at Ov-. 
example, the addition of any two opposite fractional numbers, such as ~ ‘anid! |= an 
be 
which » and v are different from zero, conducts to a null sum, under the form 
sa , in which the numerator © v+v is zero, while the denominator is different 
from zero. 
