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328 B. EUSSELL : absorb, into an apparently numerical expression, some pro- perties which the unit does not share with all thinkable contents. In order that a fraction with any given denom- inator may be applicable to a collection of objects, these objects must be capable of division into parts equal in num- ber to the denominator. If they are not so divisible, the fraction cannot be applied to these objects. For example, a fractional number of pounds sterling has a meaning if the denominator is a factor of 960, which is the number of farthings in a pound. If the denominator is not a factor of 960, the fraction neglecting foreign currencies cannot be taken with arithmetical strictness, but only to the nearest farthing. Thus a fraction whose denominator is n only has meaning when applied to a unit which is itself divisible into n smaller units. 1 From this we derive the important result, that the graduated infinite series of fractions, called the number continuum, has meaning only when applied to a matter divisible ad lib. No fraction has meaning, for ex- ample, when applied to human beings. The same is true of negative numbers. The minus sign does not apply to the number, but to the nature of the unit, which is regarded as the opposite of some unit previously fixed, i.e., as neutralising the previous unit when synthesised with it. Thus a debt and an asset neutralise one another : hence a debt may be represented as a negative asset, and vice versa. But a negative number only has meaning in relation to a matter which is capable of having an opposite, and thus implies always a certain property of the unit. Imaginary numbers may be similarly explained. Like the symbol of negative numbers, the symbol of imaginary numbers, l - 1, is not to be regarded as itself a number, nor yet a quantity, but simply as expressing a qualitative relation of our new unit to some unit previously fixed. This relation is such that, when A is so related to B, and B to C, then A is the opposite of C in the sense explained in dealing with negative numbers. In this case, B is said to be / - 1 x A. Thus if our original unit be a foot towards the East, -J - 1 multi- plied by our original unit will give a foot towards the North, which is a new and qualitatively different unit. Hence the so-called imaginary numbers are capable of application, but they agree with fractions and negative numbers in that they are not applicable, like pure number, to any thinkable con- 1 Cf. Ehrenfels, Zur Philosophic der Mathematik ; Vierteljahrsschrift fiir wiss. Phil., xv., pp. 308-9.