78 G. FLEURI. 



The sum of two complex quantities is the diagonal of the 

 parallelogram constructed on those quantities ; a definition which 

 includes, as a particular case, the definition of the addition of two 

 directive quantities and is easily extended to any number of 

 complex quantities. 



Let AB = a a BC = bp (Fig. 6.) 

 c From our definition 



^i^h^.. First, the operation is uniform 



~~b ^ \ "x Second, Completing the parallelogram 



Fi £ 6 on AC we see that 



AB + BC = AD + DC 

 But AB = DC = a a BC = AD = bp 



therefore : 



a a + b P = h p + a a 

 the operation is commutative. 



Third, Considering another complex quantity CD = C y we 

 have : (Fig. 7.) 



AD = AB + BC + CD 

 and AD = AB + BD 



whence 



a a + b /3 + c y = a a + ( b f3 + c y) 

 the addition of complex quantities is 



'B' associative. 



Ei £- 7 - Fourth, a a + o = a a 



Subtraction. — It is the inverse operation of addition. 



a a — b f3 = a a + b p + ir 

 for, from definition of subtraction and associativity of addition 



a a + b f3 + 7r + b f3 = a a + ( b f3 + 7r + b /S) = a a 

 since 



bft + 7r + bo — o (definition of addition) 



therefore the subtraction of a complex quantity bo is the same as 



the addition of that quantity with the argument increased by ir. 



