| Fig. 1. |
26. Representation of Geometrical Magnitude by Number.—The application of arithmetical methods to geometrical measurement presents some difficulty. In reality there is a transition from a cardinal to an ordinal system, but to an ordinal system which does not agree with the original ordinal system from which the cardinal system was derived. To see this, we may represent ordinal numbers by the ordinary numerals 1, 2, 3, ... and cardinal numbers by the Roman I, II, III, ... Then in the earliest stage each object counted is indivisible; either we are counting it as a whole, or we are not counting it at all. The symbols 1, 2, 3, ... then refer to the individual objects, as in fig. 1; this is the primary ordinal stage. Figs. 2 and 3 represent the cardinal stage; fig. 2 showing how the I, II, III, ... denote the successively larger groups of objects, while fig. 3 shows how the name II of the whole is determined by the name 2 of the last one counted.
| Fig. 2. | Fig. 3. |
When now we pass to geometrical measurement, each “one” is a thing which is itself divisible, and it cannot be said that at any moment we are counting it; it is only when one is completed that we can count it. The names 1, 2, 3, ... for the individual objects cease to have an intelligible meaning, and measurement is effected by the cardinal numbers I, II, III, ..., as in fig. 4. These cardinal numbers have now, however, come to denote individual points in the line of measurement, i.e. the points of separation of the individual units of length. The point III in fig. 4 does not include the point II in the same way that the number III includes the number II in fig. 2, and the points must therefore be denoted by the ordinal numbers 1, 2, 3, ... as in fig. 5, the zero 0 falling into its natural place immediately before the commencement of the first unit.
| Fig. 4. | Fig. 5. |
Thus, while arithmetical numbering refers to units, geometrical numbering does not refer to units but to the intervals between units.
III. Arithmetic of Integral Numbers
(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 × 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 × will be used, so that b × a will mean the same as a × b.