Complex Numbers 35
i + V3 (1-i)^2
(c) (l+i)2 (d) J3-i
- Given z = (1 +cos()+ i sin B)/(1 +cos()' + i sin()'), find:
(a) lzl (b) Rez (c) Imz
7. If z = x+iy =f. O, and tis a real parameter, show that 2+cost+isint =
3/ z implies that x^2 + y^2 - 4x + 3 = 0.
1.10 GEOMETRIC REPRESENTATION OF THE
OPERATIONS WITH COMPLEX NUMBERS
We wish to show how the arithmetical operations with complex numbers
can be carried out by means of simple geometrical constructions. For this
purpose it is more convenient to represent the complex numbers by position
vectors rather than by points in the complex plane.
(a) Addition. Let z 1 = (a, b) and z 2 = ( c, d) be any two complex num-
bers, and let OP and OQ be their corresponding position vectors (Fig. 1.4).
We show that the position vector OR of the sum (a+ c, b + d) of two given
numbers is simply the geometrical sum of the vectors OP and OQ, namely,
the vector with origin 0 and end point at the point R obtained by parallel
translation of OQ in the direction and magnitude of OP, or alternatively,
by a parallel displacement of OP in the direction and magnitude of OQ.
If OP and OQ are not collinear, OR coincides in magnitude and direction
with the diagonal through 0 of the parallelogram with OP and OQ as a
pair of adjacent sides (this diagonal being considered as a directed segment
with the origin at 0).
To see this, it suffices to note that in a parallel translation of the vector
OQ in the direction and magnitude of OP, the vector OQ assumes the
position PR, while the triangle OAQ moves to the position P DR (in either
of the two cases shown in Fig. 1.4). Hence, denoting the signed measure of
y
0
- /I
I I
P' --~D i
I I
y
B C X 0
Fig. 1.4
