Fry — Real and Complex Numbers as Adjectives or Operators. 17 



Again, we may first calculate any group, say 9/3, 11a, 12/3 in 7a, 9/3, lie, 

 12/8, 3a, and denoting the result by (9/3, 11a, 12/3), we can prove 



7a, 9/3, 11a, 12/3, 3a = 7a, (9/3, 11a, 12/3), 3a. 



For bring the group into the initial position in the calculation by using the 

 commutative law, then associate its members together, or, in other words, 

 replace the group by a single quantity, then alter by the commutative law, 

 so that the other quantities are in their original order, thus 



7a, 9/3, 11a, 12/3, 3a = 9/3, 11a, 12/3, 7a, 3a = (9/3, 11a, 12/3), 7a, 3a 

 = 7a, (9/3, 11a, 12/3), 3a. 



The quantities are then said to be associative. Thus the calculation obeys 

 the commutative and the associative laws. These laws are dealt with at 

 some length, because the same method of proof will apply to prove that the 

 multiplication of generalised and complex numbers follows both the associative 

 and commutative laws. 



At this stage it would be possible and instructive to do some simple and 

 simultaneous equations ; but theoretically it is better to postpone doing so 

 until we shall have made the next step. 



II. — EEAL NUMBERS. 



Denoting ordinary numbers by the letters a, h, c, a u b ly c u &c, we can 

 multiply any quantity aa by b, and denoting the result by baa or b.a.a, 

 it follows by the commutative law assumed iu this paper as proved for 

 ordinary numbers that baa = aba. This is quite intelligible. Observe in 

 multiplication for the purpose of exposition it is more convenient to take 

 the numbers in order from right to left, than in order from left to right, 

 as we did in addition. Now not only can we multiply «a by b, but we 

 may also change the unit a to (3, or if the unit were /3, alter /3 to a. To 

 obtain brief methods of expressing these operations we define (+ b) aa to 

 mean — multiply aa by b and do not alter the unit to the other unit, so 

 that (+ b)aa = baa = + (ba) a; and we define (- b)aa to mean — multiply 

 aa by b and alter the unit to the other one, so that (- &)«■« = bafi, 

 (- b) «/3 = baa. If we perform any operation (- b) on aa, and then the 

 operation (- c) on the result, as the unit has been twice changed it is 

 unaltered, so that (- c) (- b) aa = cbaa = (cb) aa. Also as 



(- c) (+ b) aa = cbafi = - (cb)aa, and (+ c) (+ b) aa = cbaa = (cb)a, 



and (+ c) (- b) a a = cbafi = - (cb) aa, 



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