| 20s. | × £153 = 153 × | 20s. | × £1 = 153 × 20s. = 3060s. |
| £1 | £1 |
Hence we can only regard £153 as being equal to 3060s. if we regard this converting factor as unity.
(ii) In the case of partition we can express the complete operation if we extend the meaning of division so as to enable us to divide 20 apples by 5 boys. We thus get (20 apples)/(5 boys) = (4 apples)/(1 boy), which means that the distribution can be effected by distributing at the rate of 4 apples per boy. The converting factor mentioned under (i) therefore represents a rate; and partition, applied to concrete cases, leads to a rate.
In reference to the use of the sign × with the converting factor, it should be observed that “(7 ℔)/(4 ℔) ×” symbolizes the replacing of so many times 4 ℔ by the same number of times 7 ℔, while “7⁄4 ×” symbolizes the replacing of 4 times something by 7 times that something.
X. Arithmetical Reasoning
91. Correspondence of Series of Numbers.—In §§ 33-42 we have dealt with the parallelism of the original number-series with a series consisting of the corresponding multiples of some unit, whether a number or a numerical quantity; and the relations arising out of multiplication, division, &c., have been exhibited by diagrams comprising pairs of corresponding terms of the two series. This, however, is only a particular case of the correspondence of two series. In considering addition, for instance, we have introduced two parallel series, each being the original number-series, but the two being placed in different positions. If we add 1, 2, 3, ... to 6, we obtain a series 7, 8, 9, ..., the terms of which correspond with those of the original series 1, 2, 3,...
Again, in §§ 61-75 and 84-88 we have considered various kinds of numbers other than those in the original number-series. In general, these have involved two of the original numbers, e.g. 53 involves 5 and 3, and log2 8 involves 2 and 8. In some cases, however, e.g. in the case of negative numbers and reciprocals, only one is involved; and there might be three or more, as in the case of a number expressed by (a + b)n. If all but one of these constituent elements are settled beforehand, e.g. if we take the numbers 5, 52, 53, ..., or the numbers 3√1, 3√2, 3√3, ... or log10 1.001, log10 1.002, log10 1.003 ... we obtain a series in which each term corresponds with a term of the original number-series.
This correspondence is usually shown by tabulation, i.e. by the formation of a table in which the original series is shown in one column, and each term of the second series is placed in a second column opposite the corresponding term of the first series, each column being headed by a description of its contents. It is sometimes convenient to begin the first series with 0, and even to give the series of negative numbers; in most cases, however, these latter are regarded as belonging to a different series, and they need not be considered here. The diagrams, A, B, C are simple forms of tables; A giving a sum-series, B a multiple-series, and C a series of square roots, calculated approximately.
92. Correspondence of Numerical Quantities.—Again, in § 89, we have considered cases of multiple-tables of numerical quantities, where each quantity in one series is equivalent to the corresponding quantity in the other series. We might extend this principle to cases in which the terms of two series, whether of numbers or of numerical quantities, merely correspond with each other, the correspondence being the result of some relation. The volume of a cube, for instance, bears a certain relation to the length of an edge of the cube. This relation is not one of proportion; but it may nevertheless be expressed by tabulation, as shown at D.