Representation of a Complex Number

Early in the 19^th century, Karl Friedrich Gauss and William Hamilton independently and almost simultaneously proposed idea of redefining complex number as ordered pair of real numbers, i.e., a+ib = (a,b).

To each complex number there corresponds one and are point in plane, and conversely to each point in the plane there corresponds one and only are complex number. Because of this we often refer to the complex number z as the point z.

(a) Cartesian form (Geometric Representation):

Every complex number z = x+iy can be represented by a point on the Cartisiar plane known as complex plane (Argand plane) by the ordered pair (x, y).

                        
Length OP is called modules of the complex number which is denoted by |z| and 0 is the argument or amplitude.

        |z| = √(x² + y²) and tanθ = (y/x) (angle made by positive x-axis).

Note: (i) Argument of a complex number is a many values function. If θ  is the argument of a complex number then αnΠ + θ; n ε I will also be the argument of that complex number. Any two arguments of a complex number differ by 2nΠ .

(ii) The unique value of θ such that -Π<Q≤Π  is called the principal value argument.

(iii) By specifying the modules and argument a complex number is defined completely. For the complex number 0+0i, the argument is not defined as this is the only complex number which is only given by its modules.

(b) Trigonometric / Polar Representation:

        z = r(cosθ + i sin θ) where |z| = r, arg z = θ I  = r(cosθ - i sin θ).

 Note: cosθ + i sinθ is also written as cisθ.