Complex numbers are used to solve quadratic equations that have no real solutions, such as x2 + 1 = 0. Euler introduced the symbol i to represent the square root of -1, allowing numbers of the form a + bi, where a is the real part and bi is the imaginary part. Complex numbers can be represented graphically on a plane with real numbers on the x-axis and imaginary numbers on the y-axis. They can also be expressed in polar form as r(cosθ + i sinθ) or in exponential form as reiθ. Operations like addition, subtraction, multiplication and division can be performed with complex numbers.

1. Complex Number
 Consider a simple quadratic equation x2
+ 2 = 0
 Since the square of any non-zero real number is positive, there are no
real solutions for the equations so there was a need to find a system
which could answer to this problem.
 Euler used the symbol 'i' to denote to solve the above equation.
 A symbol of the form a+ib where a is the real part, b is the imaginary
part, 'i' indicates the imaginary part with i = √-1
 Square root of a negative number is known as an imaginary number.
Graphical representation of Complex numbers
A complex number z = a + ib is shown as a point
z in an Argand diagram.
The image points of the real numbers 'a' are on
the x-axis. Therefore we say that the x-axis is
the real axis.
The image points of the 'pure imaginary numbers' 'bi' are on the y-axis.
Therefore we say that the y-axis is the imaginary axis.
The complex number z = a + ib,
POLAR FORM OF COMPLEX NUMBER
MODULUS-ARGUMENT FORM OF COMPLEX NUMBER
A complex number, x + iy is shown in an Argand
diagram.
The complex number z = x + iy can be
expressed in polar form as,

2. x = r cos θ
y = r sin θ
z = x + iy = r (cos θ + i sin θ)
Argument of z by Argand Diagram
arg(z) = θ = x
y1
tan −
(depends on quadrant.)
Take note: Principle value of θ must be
between –π and π.
Modulus of z
|z| = 22
yxr +=
Principal Values of Complex numbers
The value θ of the argument which satisfied the inequality –π< θ ≤ π is called
the principal value of the argument.
Case (i) when a>0 and b>0 then arg z= θ =tan-1
(b/a)
Case (ii) when a<0 and b>0 then arg z= θ = π- tan-1
(b/IaI)
Case (iii) when a<0 and b<0 then arg z= θ = - π + tan-1
(b/a)
Case (iv) when a>0 and b<0 then arg z= θ = - tan-1
(IbI/a)
Some Important Result on Argument
• Argument of the complex number ‘0’ is not defined.
• If z1=z2 Iz1I=Iz2I and arg z1 =arg z2
• I z1+z2I=I z1I + I z2I arg z1 = arg z2
• I z1+z2I= I z1-z2I arg z1 - arg z2 = π/2
• If agr z=π/2 or - π/2, z is purely imaginary

3. • If arg z=0 or π, z is purely real.
• arg z1z2 = arg z1 + arg z2
• arg (z1/z2) = arg z1 - arg z2
• If arg z*= - arg z
Conjugate Complex Number
Complex number z= (a,b)=a+ib and z*=(a,-b)=a-ib, where a and b are real
numbers, i = √-1 and b≠0 are said to be complex conjugate of each other.
De Moivre's Theorem
For any real number 21 ,θθ , )sin()cos()sin)(cossin(cos 21212211 θθθθθθθθ +++=++ iii
In particular, if 21 θθθ == , we have θθθθ 2sin2cos)sin(cos 2
ii +=+ .
For any positive integer n , by induction on n , the result may be generalized
as
θθθθ nini n
sincos)sin(cos +=+
Above equation is also true for negative integer.
De Moivre's Theorem for Rational Index
Let n be a positive integer and θ be a real number. Then
n
k
i
n
k
i n
θπθπ
θθ
+
+
+
=+
2
sin
2
cos)sin(cos
1
, where 1,,2,1,0 −= nk 
Exponential FORM OF COMPLEX NUMBER
If z = x + iy = r (cos θ + i sin θ) then exponential form is
z = reiθ
= r (cos θ + i sin θ)
z1z2 = (reiθ1
) (reiθ2
) = rei (θ1 +θ2)
=r1 r2 [cos (θ1 + θ2) + sin (θ1 + θ2)]
Some Important Result on Modulus

4. If z1 and z2 be any two complex numbers, then
If z1 and z2 are two complex numbers, then the distance between z1 and z2 are
I z1- z2I.
Segment joining points A(z1) and B(z2) is divided by point P(z) in the ratio
m1:m2
then z=(m1z2 + m2z1)/ (m1 + m2)
m1 and m2 are real.
Equation of a Circle
Equation of circle I z - z0I =r
where z0 centre of circle
r radius of circle
zz*+ a z* +a*z + k = 0 represent a circle with centre –a and radius √(IaI2
-k)

5. (z- z1)(z*- z2*)+( z- z2)( z*- z1*)=0 represent a circle with diameter AB where
A(z1) and B(z2).
Equation of a Ellipse
If I z-z1I + I z-z2I = 2a where 2a> I z1-z2I then point z
describes an ellipse having foci at z1 and z2 and a € R+
Equation of Hyperbola
If I z-z1I - I z-z2I = 2a where 2a< I z1-z2I then point z describes an hyperbola
having foci at z1 and z2 and a € R+

6. (z- z1)(z*- z2*)+( z- z2)( z*- z1*)=0 represent a circle with diameter AB where
A(z1) and B(z2).
Equation of a Ellipse
If I z-z1I + I z-z2I = 2a where 2a> I z1-z2I then point z
describes an ellipse having foci at z1 and z2 and a € R+
Equation of Hyperbola
If I z-z1I - I z-z2I = 2a where 2a< I z1-z2I then point z describes an hyperbola
having foci at z1 and z2 and a € R+