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
Lets take a look at divisionthe 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
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