(ii.) To continue the division we may take as our new unit a submultiple of Q, such as Q/r, where r is an integer, and repeat the process.
Submultiples.-The relation of a unit to its successive multiples as shown in a multiple-table is expressed by saying that it is a submultiple of the multiples, the successive submultiples being one-half, one-third, one-fourth,..
The relation of submultiple is the converse of that of multiple; thus if a is I of b, then b is 5 times a.
The determination of a submultiple is therefore equivalent to completion of the diagram E or E' of § 35 by entry of the unit, when the number of times it is taken, and the product, are given.
Dividing 12 by 2, we get a submultiple 6, which again has a prime 2 as a factor.
A fraction of a quantity is a submultiple, or a multiple of a submultiple, of that quantity.
Subdivision of Submultiple.-By 7 of A we mean 5 times the unit, 7 times which is A.
The relation expressed by proportion includes the relations expressed by multiple, submultiple, fraction and ratio.
(iii) Finally, there are the cases of linear measurement, where it is theoretically possible to find, by geometrical methods, an exact submultiple of a given unit, but both the unit and the submultiple are not really concrete objects, but are spatial relations embodied in objects.
The former would be an exact submultiple of the 30-day month, but the exact relation of seven days to the month is not very clear.