Complex Analysis- Lecture 5 - Geometric Representation of complex number 1

1.6 Geometric Representation of complex number:- `
A complex number Z=x+iy can be considered as an ordered pair (x,y), that is we can represent such numbers by points in an xy plane called complex PLAN. Some time we refer to the x and y axes as the real and imaginary axes respectively and the complex plan as the Z plan.


A complex number Z=x+iy can be considered as a vector (OP) ? whose initial point is the origin O and whose terminal (end) point P is the point (x,y) as in fig.
EX:- Graph each of the following:-
2+3i
-2+3i
-5-2i H.W
Sol:





According to the definition of the sum of two complex numbers, Z_1+Z_2 corresponds to the point (x_1+x_2,y_1+y_2 ). It also corresponds to a vector with those coordinates as its components . Hence Z_1+Z_2 is represented by vector whose initial point is the origin O and whose terminal point is the point (x_1+x_2,y_1+y_2 ) as shown in fig.

The difference Z_1-Z_2 is represented by vector from the point Z_2 to the point ? Z?_1. Addition of complex numbers corresponds to the parallelogram low for addition of vectors. Thus to add the complex numbers. Z_1 and Z_2 , we complete the parallelogram OABC . The diagonal OB which pass in the origin corresponds to Z_1+Z_2 , but the diagonal AC which don’t pass in the origin corresponds to Z_1-Z_2.