Representation of Complex Numbers

Representation of Complex Numbers:

Complex numbers are represented by points on a plane known as the complex plane or the Argand plane or the Gaussian Plane. To represent z = x + i y geometrically, two mutually perpendicular axes are taken, which are denoted by X and Y axes respectively, the X-axis being called the real axis and the Y-axis, the imaginary axis.

Thus, the complex number, z = x + i y, denotes the point P(x, y) on the Argand plane. The abscissa of the point denotes the real part and the ordinate, the imaginary part of the complex number. [Fig. (i)]

Argand Plane

As all points are alike, in order to differentiate one complex number from another, two terms namely modulus and argument or amplitude have been introduced.

The modulus of a complex number denotes the length of the line segment joining the origin and the point representing the complex number on the complex plane. [Fig. (ii)]

modulus and amplitude of z

Thus, if P denotes the complex number z = a + i b, then its modulus is given by-

mod z = | z | = OP = √(a2 + b2) = √[{Re (z)}2 + {Im (z)}2]

The argument or the amplitude of the complex number gives the angle which the line segment denoting the modulus of the complex number makes with the positive direction of the X-axis (the real axis) in the anticlockwise sense. If θ be the amplitude of z = a + i b, then-

tan θ = b/a = Im (z)/Re (z)
⇒ θ = tan-1 (b/a)

The angle θ may assume infinitely many values differing by multiples of 2π. However, the unique value of θ satisfying, -π < θ ≤ π, is known as the principal argument or the principal value of the amplitude and is denoted by Arg (z) for the complex number z.

The formula for finding out the argument of a complex number, namely tan θ = b/a has a drawback. Let z1 = 1 + i and z2 = -1 – i be two complex numbers. These two numbers are distinct and they must have different arguments. But application of the above formula gives argument of z1 = tan-1 (1/1) = tan-1 (1) = π/4 and argument of z2 = tan-1 (-1/-1) = π/4. But the argument of z2 is clearly (π + π/4) Fig. (iii) i.e., 5π/4 > π and the principal argument in this case are (π/4 – π) i.e., -3π/4 (as the principal argument satisfies -π < θ ≤ π).

argument of z2

The principal argument of the complex number z = a + i b for different signs of a and b may be determined following the conventions given below.

(1) If a > 0, b > 0, then the principal argument θ = α, where α is the acute angle given by tan α = | b/a | and we write Arg (z) = α. [Fig (iv)]

[Arg (z) ⇒ principal argument of z]

(2) If a < 0 and b > 0, then the principal argument θ is given by θ = π – α (where tan α = | b/a |) ⇒ Arg (z) = π – α. [Fig (v)]

principal argument of the complex number

(3) If a < 0 and b < 0, then θ = -(π – α) = α – π . (tan α = | b/a |) ⇒ Arg (z) = α – π. [Fig (vi)]

(4) If a > 0 but b < 0, then θ = -α, where α = tan-1 | b/a | ⇒ Arg (z) = -a. [Fig (vii)]

principal argument of the complex numbers

Note: In each of cases (3) and (4), θ is negative (measured in the clockwise direction starting from the zero line) as the principal argument θ must satisfy -π ≤ θ ≤ π.

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