258 Methods for the Solution of Special Problems [OH. vm 



Representation of complex quantities. 



309. If we write 



z = x + iy 



so that z is a complex quantity, we can suppose the 

 position of the point P indicated by the value of 

 the single complex variable z. If z is expressed in 

 Demoivre's form 



z = re io = r (cos 6 + i sin 6), 



ft/* and = tan" 1 - . The 



then we find that r = 



x 



FIG. 83. 



quantity r is known as the modulus of z and is denoted by \z\, while is 

 known as the argument of z and is denoted by arg z. The representation of 

 a complex quantity in a plane in this way is known as an Argand diagram. 



310. Addition of complex quantities. Let P be z=x + iy, and let P' be 

 z' x 4- iy'. The value of z + z is (# 4- x r ) + i (y + ?/'), so that if Q represents 

 the value z + z' it is clear that OPQP' will be a parallelogram. Thus to 

 add together the complex quantities z and z 1 we complete the parallelogram 

 OPP' , and the fourth point of this parallelogram will represent z + z' . 



The matter may be put more simply by supposing the complex quantity 

 z x-\-iy represented by the direction and length of a line, such that its 

 projections on two rectangular axes are x, y. For instance in fig. 83, the 

 value of z will be represented equally by either OP or P'Q. We now have 

 the following rule for the addition of complex quantities. 



To find z + /, describe a path from the origin representing z in magnitude 

 and direction, and from the extremity of this describe a path representing z'. 

 The line joining the origin to the extremity of this second path will repre- 

 sent z + z . 



311. Multiplication of complex quantities. If 



z = x 4- iy = r (cos 6 + i sin 6 ), 



and z' x' + iy' = r' (cos & + i sin 6'), 



then, by multiplication 



zz = rr' {cos (6 + 0') + i sin (6 + 6% 

 so that zz' - rr z\ z' , 



arg (zz) = + 6' = arg z + arg z 1 , 



and clearly we can extend this result to any number of factors. Thus we 

 have the important rules : 



The modulus of a product is the product of the moduli of the factors. 

 The argument of a product is the sum of the arguments of the factors. 



