836 Professor Hiamitton on Conjugate Functions, 
always be determined, in thought and in expression, by two whole numbers, namely 
the sub-multipling number, called also the denominator, and the multipling number, 
called also the nwmerator, (of the fraction or fractional number,) which mark the two 
successive or component acts that make up the complex act of fractioning. Hence 
also the reciprocal number, or reciprocal of any proposed whole number, which marks 
the act of multiplication conceived to be equivalent to the act of sub-multipling by 
that whole number, coincides with the fractional number which has the same whole 
number for its denominator, and the number 1 for its numerator, because a step is not 
altered when it is multipled by positive one. And any whole number itself, consi- 
dered as the mark of any special act of multipling, may be changed to a fractional 
number with positive one for its denominator, and with the proposed whole number for 
its numerator ; since such a fractional number, considered as the mark of a special 
act of multiplication, is only the complex mark of a complex act of thought equi- 
‘valent to the simpler act of multipling by the numerator of the fraction ; because the 
other component act, of sub-multipling by positive one, produces no real alteration. 
Thus, the conceptions of whole numbers, and of reciprocal numbers, are included in 
the more comprehensive conception of fractional numbers; and a complete theory of 
the latter would contain all the properties of the former. 
17. To form now a notation of fractions, we may agree to denote a fractional num- 
ber by writing the numerator over the denominator, with a bar between ; that is, we 
may write 
ce = ~a, or more fully, ¢ = oe (121.) 
in mn 
when we wish to express that two commensurable steps, b and c, (which we shall, for 
the present, suppose to be both effective steps,) may be severally formed from some 
one common base or unit-step 2, by multipling that base by the two (positive or 
contra-positive) whole numbers « and v, so that 
bomwxa, C= vxKa. (122.) 
[We shall suppose throughout the whole of this and of the two next following arti- 
cles, that all the steps are effective, and that all the numerators and denominatcrs are 
positive or contra-positive, excluding for the present the consideration of null steps, 
and of null numerators or null denominators. | 
Under these conditions, the step ¢ is a fraction of b, and bears to that step b the 
fractional ratio ~, called also ‘the ratio of v to u;” and e may be deduced or gene- 
p 
rated as a product from b by a corresponding act of fractioning, namely, by the act of 
