Fry — Real and Complex Numbers as Adjectives or Operators. 19 



get c'.'a, such a quantity is .ra + ?//3 = .*a - ya. The introduction of the 

 units a and /3 has made the operation of subtraction always possible. 



The addition and multiplication of real numbers is a distinct forward 

 intellectual step, which will be made immediately, but is not required for 

 the consideration of simple and simultaneous equations, which seem to me 

 to be simplified by retaining the unit «. 



As an example — solve the equation 



t(*-5) = i(2*-.6) + l. 



In this form the question has no meaning until we shall have advanced 

 to the consideration of the addition and multiplication of real numbers, 

 but in the following simpler form it has a definite meaning — 



Find a real number x to satisfy 



I (xa + 5/3) = | (2xa + 6/3) + a, 

 or j (.''« - 5a) = 3-(2.ra - 6a) + a. 



Multiply across by 20. We can do so, because if two quantities are equal, 

 it is intuitively true that they are still equal if we change the unit to o.a, 

 and in addition we may alter the unit a to /3 on each side, as the operations 

 we use have just the same effect on /3 in its connexion with a as they have 

 on a in its connexion with /3. Thus we can multiply both sides of any 

 equation by any real number. 



So we get 



5.ra + 25/3 = S.ca + 24/3 + 20a, 

 add to each side 8.r/3 + 25a, and as 25a + 25/3 = and 5.ra + 8.r/3 = 3,*-/3, etc. ; 

 .-. 3.r/3 = 24/3 + 20a + 25a = 21a ; 

 xf5 = 7a ; 

 x = - 7. 



Observe we can reverse every step and proceed backwards from x = - 7 to 

 the equation. 



Simple problem. — How far should I walk in a forward or backward 

 direction along a road, so that should I walk four times as far in the same 

 direction, and then walk 31 miles in a forward direction, I should be 11 miles 

 in front of the point from which I started ? 



In this problem, denoting a mile walked in a forward direction by a, and 

 one in a backward direction by /3, we want to find a real number x to satisfy 



Axa + 31a = 11a. 



Adding 31/3 to each side, ixa = 11a + 31/3 = 20/3, .\ x = - 5, or I walk 

 backward one mile. 



