The equations of the form x² + a² = 0 where a  R, can be solvable using the square root of -1. Square root of -1 is an imaginary unit denoted by 'i',   = i, integral powers of i are given by i =    =>  i² =  -1,  i³ =  i². i. = -i. Similarly
             and so on.
Note:  
                       
       If a, b are any two real numbers, then any complex number is represented by a + ib, if z = a + ib then 'a' is called real part and b is called imaginary part of the complex number.
       Any complex number z is said to be purely real if its imaginary part is zero; Im (z) = 0 and the number is said to be purely imaginary if its real part is zero, Re (z) = 0
       We know that set of complex numbers is denoted by 'C'. Every real number a can be written as z = a + 0i therefore every real number, i.e., the set of real numbers is a subset of complex numbers, R  C.

Note:    i.  Multiplication of complex numbers is commutative.
             ii.  Multiplication of complex numbers is associative.
            iii.  Identity element of complex numbers is 1.
                 The multiplicative inverse of the complex number a + ib is given by
                         

          iv. Multiplication is distributive over addition.
4.  The division of any two complex numbers is the method of multiplying are complex number with the multiplicative inverse of the other complex number.
             
5.  If z = a + ib is a complex number then  = a - ib is called the conjugate of complex number.
Properties:
       
         
6. The modulus of a complex number z is denoted by  z  where 
      
7.  The square root of a complex number is given by
         
         

Graphical Representation of complex number
The plane in which the complex numbers are represented is called complex plane or Argand plane. The x - axis is called real axis and the y - axis is called imaginary axis.
    The modulus of any complex number is equal to the distance of the point from the origin.
                 
            

 is called amplitude or argument of complex number. Where θ  (- ,  ) is called principal value.
The amplitude of    is - θ .
¤  The amplitude is positive when the point lies in the first and second quadrants and is negative in the third and fourth quadrants.
                    We know that z = a + ib = r (cos θ + i sin θ) is known a  
                     Mod Amplitude form of z.   z = r (cos θ + i sin θ) for r = 1
                                        z = cos θ + i sin θ
                   cosθ  + i sin θ can be written as cis θ = e ^iθ
Note:       If z₁ = r₁ cis(θ₁) and z₂ = r₂cis (θ₂) then     z1. z₂ = r₁r₂ cis (θ₁ + θ₂),