Argument of Complex Numbers

Argand Plane

The pair of numbers (x, y) can be represented on the XY plane, where x is called abscissa and y is called the ordinate. Similarly, we can represent complex numbers also on a plane called the Argand plane or complex plane. Similar to the X-axis and Y-axis in two-dimensional geometry, there are two axes in the Argand plane.

  • The axis which is horizontal is called the real axis
  • The axis which is vertical is called the imaginary axis

The complex number x+iy which corresponds to the ordered pair(x, y)is represented geometrically as the unique point (x, y) in the XY-plane.

For example,

The complex number, 5+3i corresponds to the ordered pair (5, 3) geometrically.

Similarly, -6+2i corresponds to the ordered pair (-6, 2).

  • Complex numbers in the form 0+ai, where “a” is any real number will lie on the imaginary axis.
  • Complex numbers in the form a+0i, where “a” is any real number will lie on the real axis.

It is obvious that the modulus of complex number x+iy, √(x2 + y2) is the distance between the origin (0, 0) and the point (x, y).

  • The conjugate of z = x+iy is z = x-iy which is represented as (x, -y) in the Argand plane. Point (x, -y) is the mirror image of the point (x, y) across the real axis in the Argand plane.

Argument of Complex Numbers

In polar form, a complex number is represented by the equation r(cos θ + i sin θ), here, θ is the argument. The argument function is denoted by arg(z), where z denotes the complex number, i.e. z = x + iy.

arg (z) = arg (x+iy) = tan-1(y/x)

Therefore, the argument θ is represented as:

θ = tan-1 (y/x)

Principal Argument of a Complex Number

The argument of a complex number z=a+ib is the angle θ of its polar representation. This angle is multi-valued. If θ is the argument of a complex number z,then θ+2 will also be argument of that complex number, where n is an integer.
While, principal argument of a complex number is the unique value of
θ such that π<θπ.

So, the principal argument of a complex number is always a unique data point, while argument of a complex number has multiple data points due to its integral multiple of 2π

For example:
z=i=P(0,1)  which lie on the positive imaginary axis; hence argument of z is  $\frac{\text{ π}}{2}$ π2  +2
But its Principal argument will be only  $\frac{\text{ π}}{2}$ π2   

Properties of Argument of Complex Numbers

The properties of argument of a complex number are as follows:
If
z,z1,z2 be non-zero complex number and m  be any integer, then

1. arg(z1.z2)=arg(z1)+arg(z2)
2. arg(z1 / z2)=arg(z1)–arg(z2)
3. arg(zm)=m arg(z)