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III. 



EEAL AND COMPLEX NUMBERS CONSIDERED AS ADJECTIVES 



OE OPEEATOES. 



By M. W. J. FEY, M.A. 



Read February 23. Published August IT, 1914. 



I. — Introductory. 



The object of this paper is to define the symbols +, -, and I, so that the 

 rules to be followed in using them may be obvious, and that what are called 

 negative and imaginary solutions of problems may have as real and precise a 

 meaning as those called positive. 



In this paper, the properties of ordinary or rational numbers, which 

 include, of course, integers and fractions, are assumed as proved, and 

 also the extension of these properties to irrational numbers. Such an 

 extension is purely arithmetical, in other words, quite distinct from the 

 extension of number by the introduction of algebraic symbols. 



Assuming, then, that this extension has been made, we can proceed to 

 reason as if we could always obtain rational numbers to measure quantities 

 of the same kind by means of a unit. By a unit we mean any arbitrary 

 quantity which we select as a standard in terms of which to measure other 

 quantities of the same kind. Here we note the fundamental difference 

 between the terms " number " and " quantity." A quantity is a noun, and 

 requires for its statement two elements. One is the unit we have selected, 

 in terms of which the magnitude of the quantity is to be expressed. The 

 other is a qualifying adjective, the number, which expresses the number of 

 times the unit is to be taken. Similarly, generalised numbers and complex 

 numbers, which will be introduced subsequently, are adjectives ; in fact, any 

 complex formula in algebra is an adjective which may be used to qualify any 

 unit. 



Now, when we proceed to make calculations involving the quantities 

 which occur in nature, we find that those of the same kind can further be 

 subdivided into two groups. For instance, distances may be measured 

 forwards or backwards along a line; money may be received or paid out; 



PEOC. K.I. A., VOL. XXXII., SECT. A. [3] 



