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.