AMPLITUDE AND PHASE BY COMPLEX NUMBERS 39 



significant to write them in their exponential forms: 



Zl - ttiC 



22 = a2e^^^. 



Then, just as 

 so also 



hie^^'-ho^^'^^ = hih2e"'^^'+^^\ 



Z1Z2 = aia2<'''^^'+^-\ (4.9) 



To multiply two complex numbers, multiply their amplitudes and add 

 their phases. 

 Furthermore, 



^ = ^e^(«i-e2), (410) 



Z2 0-2 



To divide two complex numbers, divide their amplitudes and subtract 

 their phases. 



The above elementary properties of complex numbers suffice for the 

 purposes of an elementary theory of phase microscopy. The reader to 

 whom complex numbers are new should not attempt to attach any 

 significance to a complex number beyond the definition of a complex 

 number, together with its amplitude and phase, and beyond the rules of 

 addition and multiplication of two complex numbers. It is natural to 

 use complex numbers in treating the interference between two waves or 

 the passage of a wave through a diffraction plate for the following reasons : 



1. The amplitude and phase of a wave are exactly analogous to the 

 amplitude and phase of a complex number. 



2. The interference between two waves of different amplitude and 

 phase is exactly analogous to the addition of two complex numbers. 



3. The alteration of the amplitude and phase of a wave upon passing 

 through a medium, for example the diffraction plate, is exactly analogous 

 to the multiplication of two complex numbers one of w^hich represents 

 the amplitude and phase of the incident wave and the other of which 

 represents the amplitude and phase transmission of the medium. 



The first analogy is self-evident. With respect to the second analogy, 

 let the amplitude and phase of the two interfering waves be denoted by 

 the complex numbers zi = Uic'^^ and 22 = 0,2^'^'' and let the vectors 

 representing zi and 22 be drawn upon the XY plane as in Fig. 11.11. 

 The parallelogram construction for finding the amplitude a and the 

 phase angle 6 of the complex number 2 which is equal to the sum of the 

 complex numbers 21 and 22 will be recognized as identical with the 



