T = x/ar^ + y2 > 0, ^ = tan"! — 



X 



The geometrical significance of r and ^ on the plot is shown in Figure 18. Here r is called 

 the modulus or absolute value of 2 and is denoted by \z\ or mod 2. The angle 6 is called the 

 amplitude, or sometimes the argument of 2; it is denoted by amp 2, or arg 2. 



The amplitude is multiple-valued, since if 6 is one value, another possible value is 

 ^j + 27177 where n is any positive or negative integer. A complex number is completely speci- 

 fied when its modulus and amplitude are given; for, in terms of r and 6, 



2 = X + iy = r (cos 6 + i sin d) 

 In referring to the amplitude, however, it is often necessary to specify which of its many 

 values is meant. The value of the amplitude such that 



is called its principal value; this value is often tacitly understood. It should be carefully 

 noted that no ambiguity attaches to the value of the complex number itself; the ambiguity 

 attaches only to its polar representation. 



Numbers for which r = 1 are represented on the diagram by points lying on the unit 

 circle, or a circle about the origin of radius unity. 



Two complex numbers are equal only when their real parts are equal and their imaginary 

 parts are also equal. For this reason every equation between complex numbers is equivalent 

 to two real equations, one containing the real parts, and the other the imaginary parts with i 

 omitted. Equal complex numbers have equal moduli, and their amplitudes can differ only by 

 an integer multiplied by 2??. 



In the diagram, the sum of two complex numbers is represented by the vector sum of 

 the vectors representing the two numbers, as in Figure 19. For, if 



2^ = aTj + zyj, 2^ = X^ + iy2» 



then 



2^ + 2^ = x^ + x^ +i {yy + 2/2) 

 It should be noted that the amplitude of the sum or difference of two numbers is not uniquely 

 fixed by an assignment of the amplitudes of the two numbers; it is partly arbitrary and must be 

 separately chosen if needed. 



The product, on the other hand, has nothing to do with the ordinary vector products of 

 the corresponding vectors. Multiplication and division can be done in cartesian form, thus; 



SyZ^^XyX^-y^y^ + iix^y^ + x^y^), 



2y ^ Xy + iy^ ^ XyX^ + y^y^ ^ x^y^ - x^y^ 

 ' ■ -- ; ^2 ^2 ■•" *2i'2 x^ + y^ x^ + y^ 



34 



