Chapter IX: Units

89. _Change of Denomination_ of a numerical quantity is usually called _reduction_, so that this term covers, e.g., the expression of L153, 7s. 4d. as shillings and pence and also the expression of 3067s. 4d. as L, s. and d.

The usual statement is that to express L153, 7s. as shillings we multiply 153 by 20 and add 7. This, as already explained (S 37), is incorrect. L153 denotes 153 units, each of which is L1 or 20s.; and therefore we must multiply 20s. by 153 and add 7s., i.e. multiply 20 by 153 (the unit being now 1s.) and add 7. This is the expression of the process on the grouping method. On the counting method we have a scale with every 20th shilling marked as a L; there are 153 of these 20's, and 7 over.

A
+-------------+---------+
| 1s. | 12d. |
+-------------+-------------+---------+
|L1 | 20s. | |
+-------------+-------------+---------+
|L153, 7s. 4d.| 3067s. 4d. | 36808d. |
+-------------+-------------+---------+

B
+-----------+---------------+
| 20s. | L1 |
+---------+-----------+---------------+
| 12d. | 1s. | |
+---------+-----------+---------------+
| 36808d. | 3067s. 4d.| L153, 7s. 4d. |
+---------+-----------+---------------+

The simplest case, in which the quantity can be expressed as an integral number of the largest units involved, has already been considered (SS 37, 42). The same method can be applied in other cases by regarding a quantity expressed in several denominations as a fractional number of units of the largest denomination mentioned; thus 7s. 4d. is to be taken as meaning 7(4/12)s., but L0, 7s. 4d. as L0 (7(4/12))/20 (S 17). The reduction of L153, 7s. 4d. to pence, and of 36808d. to L, s. d., on this principle, is shown in diagrams A and B above.

For reduction of pounds to shillings, or shillings to pounds, we must consider that we have a multiple-table (S 36) in which the multiples of L1 and of 20s. are arranged in parallel columns; and similarly for shillings and pence.

90. _Change of Unit._--The statement "L153 = 3060s." is not a statement of _equality_ of the same kind as the statement "153 X 20 = 3060," but only a statement of _equivalence_ for certain purposes; in other words, it does not convey an absolute truth. It is therefore of interest to see whether we cannot replace it by an absolute truth.

To do this, consider what the ordinary processes of multiplication and division mean in reference to concrete objects. If we want to give, to 5 boys, 4 apples each, we are said to multiply 4 apples by 5. We cannot multiply 4 apples by 5 boys, for then we should get 20 "boy-apples," an expression which has no meaning. Or, again, to distribute 20 apples amongst 5 boys, we are not regarded as dividing 20 apples by 5 boys, but as dividing 20 apples by the number 5. The multiplication or division here involves the omission of the unit "boy," and the operation is incomplete. The complete operation, in each case, is as follows.

(i) In the case of multiplication we commence with the conception of the number "5" and the unit "boy"; and we then convert this unit into 4 apples, and thus obtain the result, 20 apples. The conversion of the unit may be represented as multiplication by a factor (4 apples)/(1 boy), so that the operation is [(4 apples)/(1 boy)] X (5 boys) = 5 X [(4 apples)/(1 boy)] X (1 boy) = 5 X 4 apples = 20 apples. Similarly, to convert L153 into shillings we must multiply it by a factor 20s./L1, so that we get

(20s./L1) X L153 = 153 X (20s./L1) X L1 = 153 X 20s. = 3060s.

Hence we can only regard L153 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 X with the converting factor, it should be observed that "(7 lb.)/(4 lb.) X" symbolizes the replacing of so many times 4 lb. by the same number of times 7 lb., while "(7/4) X" symbolizes the replacing of 4 times something by 7 times that something.