Representations of Complex Numbers
Geometric representation
The complex number z = x +iy can be uniquely represented by the point P(x, y) in the coordinate plane and conversely corresponding to the point P(x, y) in the plane there exists a unique complex number z = x +iy. The plane is called the complex plane and the representation of complex numbers as points in the plane is called Argand diagram.
Notice that length OP = x² +y² = z
Also note that every real number x = x +0i is represented by point (x, 0)
lying on xaxis, and every purely imaginary number iy is represented by point
(0, y) lying on yaxis. Consequently, xaxis is called the real axis and
yaxis is called the imaginary axis.
 If z_{1} = x_{1} +iy_{1} and z_{2}
= x_{2} +iy_{2} are two complex numbers, then the distance
between two corresponding points P_{1}(x_{1}, y_{1})
and P_{2}(x_{2}, y _{2}) is
P_{1}P_{2} = [(x_{1} x_{2})² +(y_{1} y_{2})²] = z_{1} z_{2} = z_{2} z_{1}.  If P_{1}(x_{1}, y_{1}) and P_{2}(x
_{2}, y_{2}) are two points, then the point Q dividing [P_{1}P_{2}]
in the ratio m : n is given
Putting m = n = 1, the midpoint of z_{1} and z_{2} =
Trigonometric representation
z = x +iy = r cos +i r sin
= r (cos +i sin )
This form of z is called trigonometric form or polar form. Thus if modulus of
z is r and amp(z) = , then z = r (cos
+i sin ) = r cis .
Notice that if x = 0, then = /2;
if y > 0 and =  /2 if y < 0.
If x 0, then tan =
r sin /r cos = y/x, so
that
amp (z) = = tan^{1(}y/x)
The unique value of q such that  <
is called
principal value of amplitude or argument.
For example, let z = 1 +i. Then
r = [(1)² + (1)²] = 2
and = tan^{1} 1 = /4.
As another example, let r = 2 ,
=  /4
Then z = r (cos +i sin )
Remark
Frequently we have to convert the complex number z = x +i y to its polar
form z = r cis . Do the calculations as follows:
r = x² +y²
To find :
If x = 0 i.e. if z is purely imaginary, then
= /2 if y> 0,
= /2 if y < 0.
If y = 0 i.e. if z is purely real, then
= 0 if x > 0, =
if x < 0.
Otherwise, let
be such that 0 < < /2
Then =
if x > 0, y > 0 (i.e. z is in first quadrant)
=

if x < 0, y > 0 (i.e. z is in second quadrant)
=
+
if x < 0, y < 0 (i.e. z is in third quadrant)
= 
if x>0, y<0 (i.e. z is in fourth quadrant).
Conversely, given z = r cis , convert it to standard form
z = x +iy by using x = r cos , y = r sin .
Illustrative Examples
Example
If z is any complex number, show that zRe(z)z. When do the equality signs hold?
Solution
Let z = x +iy = r cis = r (cos +i
sin ) where r = z 0 and
= amp(z).
We know that 1 cos
1 for all
=> r r cos r
(as r0)
=> z Re(z) z.
(because Re(z) = r cos )
Now z = Re(z) <=> [x²+y²] = x
<=> y = 0 and x 0.
Also Re(z) = z <=> x = [x²+y²] <=> y
= 0 and x 0.
Hence z = Re(z) = z <=> y = 0 and x = 0 => z = 0.
Example
Show that the area of the triangle on the Argand plane formed by the complex numbers z, iz and z +iz is z²/2
Solution
Note that the points O(0), P(z), R(z +iz) and Q(iz) form parallelogram OPRQ.
Also OP = z = iz = OQ and POQ = 90°
Thus 0, z, z +iz, iz form a square of side z.
Hence area of triangle with vertices z, iz and z +iz
= (1/2)z.z = (1/2)z²
Exercise
 Represent the following complex numbers in polar form
(i) 3 +i
(ii)
(iii) sin 90° +cos 90°
(iv) tan i
(v) 1 +sin +i cos
(vi) 1 i  Represent the following complex numbers in standard form x +iy
(i) cis 2/3
(ii)  If z is a complex number, represent z and iz in complex plane.
 Using distance formula, prove that the points 1, (1 +3i)/2 and (1 3i)/2 form the vertices of an equilateral triangle in complex plane.
 Using distance formula or otherwise, prove that the points 2 +3i, 1 +2i and 2 +5i are collinear.
[Hint. Let the points A, B, C represent the complex numbers 2 +3i,1 +2i and 2 +5i respectively. Show that AB +BC = AC.]  If z_{1}, z_{2}, z_{3}, z_{4} are complex
numbers, show that they are vertices of a parallelogram in the Argand diagram
if and only if z_{1} +z_{3} = z_{2} +z_{4}.
[Hint. What is midpoint of AC? BD?]
Answers
1. (i) 2 cos/6 (ii) cis 0 (iii) cis 0(iv) (v)
(vi) 2 cis 3/4
2. (i)(1/2) + [(3)/2] i (ii) 1 i