(i.) _Exact Calculation._
95. _Working from Left._--It is desirable, wherever possible, to perform
operations on numbers or numerical quantities from the left, rather than
from the right. There are several reasons for this. In the first place,
an operation then corresponds more closely, at an elementary stage, with
the concrete process which it represents. If, for instance, we had one
sum of L3, 15s. 9d. and another of L2, 6s. 5d., we should add them by
putting the coins of each denomination together and commencing the
addition with the L. In the second place, this method fixes the
attention at once on the larger, and therefore more important, parts of
the quantities concerned, and thus prevents arithmetical processes from
becoming too abstract in character. In the third place, it is a better
preparation for dealing with approximate calculations. Finally,
experience shows that certain operations in which the result is written
down at once--e.g. addition or subtraction of two numbers or quantities,
and multiplication by some small numbers--are with a little practice
performed more quickly and more accurately from left to right.
96. _Addition._--There is no difference in principle between addition
(or subtraction) of numbers and addition (or subtraction) of numerical
quantities. In each case the grouping system involves rearrangement,
which implies the commutative law, while the counting system requires
the expression of a quantity in different denominations to be regarded
as a notation in a varying scale (SS 17, 32). We need therefore consider
numerical quantities only, our results being applicable to numbers by
regarding the digits as representing multiples of units in different
denominations.
When the result of addition in one denomination can be partly expressed
in another denomination, the process is technically called _carrying_.
The name is a bad one, since it does not correspond with any ordinary
meaning of the verb. It would be better described as _exchanging_, by
analogy with the "changing" of subtraction. When, e.g., we find that the
sum of 17s. and 18s. is 35s., we take out 20 of the 35 shillings, and
exchange them for L1.
To add from the left, we have to look ahead to see whether the next
addition will require an exchange. Thus, in adding L3, 17s. 0d. to L2,
18s. 0d., we write down the sum of L3 and L2 as L6, not as L5, and the
sum of 17s. and 18s. as 15s., not as 35s.
When three or more numbers or quantities are added together, the result
should always be checked by adding both upwards and downwards. It is
also useful to look out for pairs of numbers or quantities which make 1
of the next denomination, e.g. 7 and 3, or 8d. and 4d.
97. _Subtraction._--To subtract L3, 5s. 4d. from L9, 7s. 8d., on the
grouping system, we split up each quantity into its denominations,
perform the subtractions independently, and then regroup the results as
the "remainder" L6, 2s. 4d. On the counting system we can count either
forwards or backwards, and we can work either from the left or from the
right. If we count forwards we find that to convert L3, 5s. 4d. into L9,
7s. 8d. we must successively add L6, 2s. and 4d. if we work from the
left, or 4d., 2s. and L6 if we work from the right. The intermediate
values obtained by the successive additions are different according as
we work from the left or from the right, being L9, 5s. 4d. and L9, 7s.
4d. in the one case, and L3, 5s. 8d. and L3, 7s. 8d. in the other. If we
count backwards, the intermediate values are L3, 7s. 8d. and L3, 5s. 8d.
in the one case, and L9, 7s. 4d. and L9, 5s. 4d. in the other.
The determination of each element in the remainder involves reference to
an addition-table. Thus to subtract 5s. from 7s. we refer to an
addition-table giving the sum of any two quantities, each of which is
one of the series 0s., 1s., ... 19s.
Subtraction by counting forward is called _complementary addition_.
To subtract L3, 5s. 8d. from L9, 10s. 4d., on the grouping system, we
must _change_ 1s. out of the 10s. into 12d., so that we subtract L3, 5s.
8d. from L9, 9s. 16d. On the counting system it will be found that, in
determining the number of shillings in the remainder, we subtract 5s.
from 9s. if we count forwards, working from the left, or backwards,
working from the right; while, if we count backwards, working from the
left, or forwards, working from the right, the subtraction is of 6s.
from 10s. In the first two cases the successive values (in direct or
reverse order) are L3, 5s. 8d., L9, 5s. 8d., L9, 9s. 8d. and L9, 10s.
4d.; while in the last two cases they are L9, 10s. 4d., L3, 10s. 4d.,
L3, 6s. 4d. and L3, 5s. 8d.
In subtracting from the left, we look ahead to see whether a 1 in any
denomination must be reserved for changing; thus in subtracting 274 from
637 we should put down 2 from 6 as 3, not as 4, and 7 from 3 as 6.
98. _Multiplication-Table._--For multiplication and division we use a
_multiplication-table_, which is a multiple-table, arranged as explained
in S 36, and giving the successive multiples, up to 9 times or further,
of the numbers from 1 (or better, from 0) to 10, 12 or 20. The column
(vertical) headed 3 will give the multiples of 3, while the row
(horizontal) commencing with 3 will give the values of 3 X 1, 3 X 2,...
To multiply by 3 we use the row. To divide by 3, in the sense of
partition, we also use the row; but to divide by 3 as a unit we use the
column.
99. _Multiplication by a Small Number._--The idea of a large multiple
of a small number is simpler than that of a small multiple of a large
number, but the calculation of the latter is easier. It is therefore
convenient, in finding the product of two numbers, to take the smaller
as the multiplier.
To find 3 times 427, we apply the distributive law (S 58 (vi)) that
3.427 = 3(400 + 20 + 7) = 3.400 + 3.20 + 3.7. This, if we regard 3.427
as 427 + 427 + 427, is a direct consequence of the commutative law for
addition (S 58 (iii)), which enables us to add separately the hundreds,
the tens and the ones. To find 3.400, we treat 100 as the unit (as in
addition), so that 3.400 = 3.4.100 = 12.100 = 1200; and similarly for
3.20. These are examples of the associative law for multiplication (S 58
(iv)).