## Representation of Complex Numbers:

Complex numbers are represented by points on a plane known as the ** complex plane** or the

**or the**

*Argand plane***. To represent z = x +**

*Gaussian Plane**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)]

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)]

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

mod z = | z | = OP = √(a^{2} + b^{2}) = √[{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 z_{1} = 1 + *i* and z_{2} = -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 z_{1} = tan^{-1} (1/1) = tan^{-1} (1) = π/4 and argument of z_{2} = tan^{-1} (-1/-1) = π/4. But the argument of z_{2} 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).

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)]

(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)]

**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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