# 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, √(x^{2} + y^{2}) 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*nπ* 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*<*θ*≤*π*.

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 2*π*

**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*nπ*

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*,*z*1,*z*2 be non-zero complex number and *m * be any integer, then

1. arg(*z*_{1}.*z*_{2})=arg(*z*_{1})+arg(*z*_{2})

2. arg(*z*_{1}* /* *z*_{2})=arg(*z*_{1})–arg(*z*_{2})

3. arg(*z** ^{m}*)=m arg(z)