(i.) _Preliminary_
27. _Equality and Identity._--There is a certain difference between the
use of words referring to equality and identity in arithmetic and in
algebra respectively; what is an _equality_ in the former becoming an
_identity_ in the latter. Thus the statement that 4 times 3 is equal to
3 times 4, or, in abbreviated form, 4 X 3 = 3 X 4 (S 28), is a statement
not of identity but of equality; i.e. 4 X 3 and 3 X 4 mean different
things, but the operations which they denote produce the same result.
But in algebra a X b = b X a is called an identity, in the sense that it
is true whatever a and b may be; while n X X = A is called an equation,
as being true, when n and A are given, for one value only of X.
Similarly the numbers represented by 6/12 and 1/2 are not identical, but
are equal.
28. _Symbols of Operation._--The failure to observe the distinction
between an identity and an equality often leads to loose reasoning; and
in order to prevent this it is important that definite meanings should
be attached to all symbols of operation, and especially to those which
represent elementary operations. The symbols - and [:] mean respectively
that the first quantity mentioned is to be reduced or divided by the
second; but there is some vagueness about + and X. In the present
article a + b will mean that a is taken first, and b added to it; but a
X b will mean that b is taken first, and is then multiplied by a. In the
case of numbers the X may be replaced by a dot; thus 4.3 means 4 times
3. When it is necessary to write the multiplicand before the multiplier,
the symbol [X] will be used, so that b [X] a will mean the same as a X
b.
29. _Axioms._--There are certain statements that are sometimes regarded
as axiomatic; e.g. that if equals are added to equals the results are
equal, or that if A is greater than B then A + X is greater than B + X.
Such statements, however, are capable of logical proof, and are
generalizations of results obtained empirically at an elementary stage;
they therefore belong more properly to the laws of arithmetic (S 58).
(ii.) _Sums and Differences._
30. _Addition and Subtraction._--_Addition_ is the process of expressing
(in numeration or notation) a whole, the parts of which have already
been expressed; while, if a whole has been expressed and also a part or
parts, _subtraction_ is the process of expressing the remainder.
Except with very small numbers, addition and subtraction, on the
grouping system, involve analysis and rearrangement. Thus the sum of 8
and 7 cannot be expressed as ones; we can either form the whole, and
regroup it as 10 and 5, or we can split up the 7 into 2 and 5, and add
the 2 to the 8 to form 10, thus getting 8 + 7 = 8 + (2 + 5) = (8 + 2) +
5 = 10 + 5 = 15. For larger numbers the rearrangement is more extensive;
thus 24 + 31 = (20 + 4) + (30 + 1) = (20 + 30) + (4 + 1) = 50 + 5 = 55,
the process being still more complicated when the ones together make
more than ten. Similarly we cannot subtract 8 from 15, if 15 means 1 ten
+ 5 ones; we must either write 15 - 8 = (10 + 5) - 8 = (10 - 8) + 5 = 2
+ 5 = 7, or else resolve the 15 into an inexpressible number of ones,
and then subtract 8 of them, leaving 7.
Numerical quantities, to be added or subtracted, must be in the same
denomination; we cannot, for instance, add 55 shillings and 100 pence,
any more than we can add 3 yards and 2 metres.
31. _Relative Position in the Series._--The above method of dealing with
addition and subtraction is synthetic, and is appropriate to the
grouping method of dealing with number. We commence with processes, and
see what they lead to; and thus get an idea of sums and differences. If
we adopted the counting method, we should proceed in a different way,
our method being analytic.
One number is less or greater than another, according as the symbol (or
ordinal) of the former comes earlier or later than that of the latter in
the number-series. Thus (writing ordinals in light type, and cardinals
in heavy type) 9 comes after 4, and therefore 9 is greater than 4. To
find how much greater, we compare two series, in one of which we go up
to 9, while in the other we stop at 4 and then recommence our counting.
The series are shown below, the numbers being placed horizontally for
convenience of printing, instead of vertically (S 14):--
1 2 3 4 5 6 7 8 9
1 2 3 4 1 2 3 4 5
This exhibits 9 as the sum of 4 and 5; it being understood that the sum
of 4 and 5 means that we add 5 to 4. That this gives the same result as
adding 4 to 5 may be seen by reckoning the series backwards.