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 × 3 = 3 × 4 (§ 28), is a statement not of identity but of equality; i.e. 4 × 3 and 3 × 4 mean different things, but the operations which they denote produce the same result. But in algebra a × b = b × a is called an identity, in the sense that it is true whatever a and b may be; while n × 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 ½ 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 ×. In the present article a + b will mean that a is taken first, and b added to it; but a × b will mean that b is taken first, and is then multiplied by a. In the case of numbers the × 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 × 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 (§ 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 (§ 14):—
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.
It is convenient to introduce the zero; thus
indicates that after getting to 4 we make a fresh start from 4 as our zero.
To subtract, we may proceed in either of two ways. The subtraction of 4 from 9 may mean either “What has to be added to 4 in order to make up a total of 9,” or “To what has 4 to be added in order to make up a total of 9.” For the former meaning we count forwards, till we get to 4, and then make a new count, parallel with the continuation of the old series, and see at what number we arrive when we get to 9. This corresponds to the concrete method, in which we have 9 objects, take away 4 of them, and recount the remainder. The alternative method is to retrace the steps of addition, i.e. to count backwards, treating 9 of one (the standard) series as corresponding with 4 of the other, and finding which number of the former corresponds with 0 of the latter. This is a more advanced method, which leads easily to the idea of negative quantities, if the subtraction is such that we have to go behind the 0 of the standard series.
32. Mixed Quantities.—The application of the above principles, and of similar principles with regard to multiplication and division, to numerical quantities expressed in any of the diverse British denominations, presents no theoretical difficulty if the successive denominations are regarded as constituting a varying scale of notation (§17). Thus the expression 2 ft. 3 in. implies that in counting inches we use 0 to eleven instead of 0 to 9 as our first repeating series, so that we put down 1 for the next denomination when we get to twelve instead of when we get to ten. Similarly 3 yds. 2 ft. means
The practical difficulty, of course, is that the addition of two numbers produces different results according to the scale in which we are for the moment proceeding; thus the sum of 9 and 8 is 17, 15, 13 or 11 according as we are dealing with shillings, pence, pounds (avoirdupois) or ounces. The difficulty may be minimized by using the notation explained in § 17.
(iii.) Multiples, Submultiples and Quotients.
33. Multiplication and Division are the names given to certain numerical processes which have to be performed in order to find the result of certain arithmetical operations. Each process may arise out of either of two distinct operations; but the terminology is based on the processes, not on the operations to which they belong, and the latter are not always clearly understood.
34. Repetition and Subdivision.—Multiplication occurs when a certain number or numerical quantity is treated as a unit (§ 11), and is taken a certain number of times. It therefore arises in one or other of two ways, according as the unit or the number exists first in consciousness. If pennies are arranged in groups of five, the total amounts arranged are successively once 5d., twice 5d., three times 5d., ... ; which are written 1 × 5d., 2 × 5d., 3 × 5d., ... (§ 28). This process is repetition, and the quantities 1 × 5d., 2 × 5d., 3 × 5d., ... are the successive multiples of 5d. If, on the other hand, we have a sum of 5s., and treat a shilling as being equivalent to twelve pence, the 5s. is equivalent to 5 × 12d.; here the multiplication arises out of a subdivision of the original unit 1s. into 12d.
Although multiplication may arise in either of these two ways, the actual process in each case is performed by commencing with the unit and taking it the necessary number of times. In the above case of subdivision, for instance, each of the 5 shillings is separately converted into pence, so that we do in fact find in succession once 12d., twice 12d., ...; i.e. we find the multiples of 12d. up to 5 times.
The result of the multiplication is called the product of the unit by the number of times it is taken.