Geometry of Complex Numbers
A complex number z = x+ζy can be represented by a point (x, y) on the plane which is known as the Argand plane.To represent z = x+ζygeometrically we take two mutually perpendicular straight lines X' OX and Y' OY called the Real axis and Imaginary axis respectively. Now, plot a point whose x and y coordinated are respectively the real and imaginary parts of z. This point P(x, y) represents the complex number z = x+ζy.
If a complex number is purely real, then its imaginary part is zero.Therefore, a purely real number is represented by a point on x-axis. A purely imaginary complex number is represented by a point on y-axis. This is why x-axis is known as the real axis and y-axis, as the imaginary axis.
Conversely, if P(x, y) is a point in the plane, then the point P(x, y) represents a complex number z = x+ζy. The complex number z = x=ζy is known as the affix of the point P.
There exists a one-one correspondence between the points of the plane and the members (elements) of the set C of all complex numbers. i.e., for every complex number z = x+ζy there exists uniquely a point (x, y) on the plane and for every point (x, y) of the plane there exists uniquely a complex number z = x+ζy
The plane in which we represent a complex number geometrically is known as the complex plane or Argand plane or the Gaussian plane. The point P, plotted on the Argand plane, is called the Agrand diagram.
The length of the line segment OP is called the modulus of z and is denoted by |z|.
Geometrical Representation of Addition
Let and be two complex numbers represented by points P(x,y)and Q (x,y)in the Argand plane. Join the origin O with P and Q and complete the parallelogram OPRQ by taking OP and OQ as two adjacent sides. Draw perpendiculars PK, QL and RM from P, Q and R respectively on x-axis. Also draw PN to RM. Since the diagonals of a parallelogram bisect each other, therefore coordinates of the mid point of PQ and also that of OR are So, the coordinates of R are .Hence R represents the complex number