(iii.) The general statement of the laws of operation of fractions is perhaps best deferred until we come to fractional numbers, when letters can be used to express the laws of multiplication and division of such numbers.
The idea of (-5) as a number with which we can perform such operations as multiplication comes later (ï¿½ 49)ï¿½ (ii.) On the other hand, the conception of a fractional number follows directly from the use of fractions, involving the subdivision of a unit.
We find that fractions follow certain laws corresponding exactly with those of integral multipliers, and we are therefore able to deal with the fractional numbers as if they were integers.
Thus the concrete fact required to enable us to pass arithmetically from the conception of a fractional number to the conception of a surd is the fact of performing calculations by means of logarithms.