Page 10 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY
P. 10

Unit 1: Complex Numbers




          Now, suppose z = (x, y) = x + y and w = (u, v) = u + v. Then we have                Notes
                                       zw = (x + y) (u + v)

                                              = xu + (xv + yu) ±  yv
                                                           2
          We need only see what   is:   = (0, 1) (0, 1) = (–1, 0), and we have agreed that we can safely
                                   2
                               2
          abbreviate (–1, 0) as –1. Thus,   = –1, and so
                                   2
                                     zw = (xu – yv) + (xv + yu)
          and we have reduced the fairly complicated definition of complex arithmetic simply to ordinary
          real arithmetic together with the fact that   = –1.
                                             2
                                                                z
          Let’s take a look at division–the inverse of multiplication. Thus,    stands for that complex
                                                                w
          number you must multiply w by in order to get z . An example:


                                      z  x   y  x   y u   v
                                                     .
                                     w    u   v    u   v u   v
                                         (xu   yv)   (yu xv)
                                                      
                                       
                                                2
                                               u  v 2
                                                     
                                         xu  yv  yu xv
                                               
                                          2
                                                   2
                                         u   v 2  u   v 2
                                              2
                                                 2
             Notes   This is just fine except when u  + v  = 0; that is, when u = v = 0. We may, thus,
             divide by any complex number except 0 = (0, 0).
          One final note in all this. Almost everyone in the world except an electrical engineer uses the
          letter i to denote the complex number we have called . We shall accordingly use i rather than
           to stand for the number (0, 1).

          1.1 Geometry

          We now have this collection of all ordered pairs of real numbers, and so there is an uncontrollable
          urge  to  plot  them on  the usual  coordinate axes.  We see  at once  that there  is a  one-to-one
          correspondence between the complex numbers and the points in the plane. In the usual way, we
          can think of the sum of two complex numbers, the point in the plane corresponding to z + w is
          the diagonal of the parallelogram having z and w as sides:

                                            Figure  1.1




















                                           LOVELY PROFESSIONAL UNIVERSITY                                    3
   5   6   7   8   9   10   11   12   13   14   15