BodhiAI

Nov. 11, 2019

**Geometry of Complex Numbers**

**Introduction**

**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.**

**Important**

**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 **