Fry — Real and Complex Numbers as Adjectives or Operators. 23 



III. — Complex Numbers. 



To generalise still further our operations we take, instead of two units, 

 four units in cyclical order a, a', /3, /3'. Of these four, two a, |3, are a pair of 

 the sort we have considered up to this, so that they combine by addition, 

 7a + 9]3 = 2/3, and so on. The other pair </, j3' are similarly related, and form 

 any other pair. 



For instance, a, j3 might refer to money clue to or owed by a person, 

 a, /3' to distance moved along a line. Or in particular a, /3 might refer to 

 distances measured along a line, and a', )3' to distances measured along a 

 second line inclined at an angle to the first. A simpler representation of 

 the result of complex multiplication is obtained when the angle is taken to 

 be a right angle. Again a, /3 might be any pair of units, and a, /3' a pair 

 to measure quantities of the same kind, but we agree to keep the two 

 quantities distinct. 



Having made a selection of any such four, and arranged for them 

 a cyclical order a, a, /3, /3', we introduce a new symbol i such that ai 

 written before any of the quantities ba, ba', bj3, 5)3', means that b is to be 

 multiplied by a and the unit changed to the next in cyclical order. Of 

 course + a or a written in front of ba, &c, still means that b is to be 

 multiplied by a and the unit not changed, and - a written in front ba, 

 &c, still means that b is to be multiplied by a and the unit changed to /3, &c. 



Thus aiba = aba, aiba' = abfi, aibji = a&j3', aibfi' = aba. 



Hence, fa = ia = ft = - a, and so on, so that f is equivalent to the 

 operator - 1. Again, i 3 a = |3' = - ia, thus the operator - i that is 

 (- 1) (li) (but it is customary to omit the 1 in each), is equivalent to a 

 reversal of the order of substitution laid down for i. 



As Pa = - a, i z a = - ia, i'a = a, our most general quantity, 

 "a + ba' + c /3 + dfi' may now be written as xa + iya or (x + iy) a. 

 Thus our most general number is x + iy, and is a complex adjective 

 qualifying the noun a. 



Now, defining (cc + iy)za to mean xza + ii/za, where za is any complex 

 quantity, we get (x + iy) (x' + iy') za = \xx' - yy' + i {.xy + x'y)\za, hence x + iy 

 and / + iy' are commutative, and hence, denoting complex numbers by 

 z\, z 2 , S3, &c, in ZiZ&j&e.a, s,, z 2 , z 3 , &c, are commutative and associative. 



Now if (z, + :.) a = sa, as the symbols - and i operate on a in its 

 relation to j3/3'a in consecutive cyclical order in the same way as they 

 operate on a in its relation to a'(3j3', and so on, it follows that we may 

 replace a by a' or j3 or /3'. Also it follows by intuition that we may increase 



PKOC. H.I. A., VOL. XXXII., SECT, A. [4] 



