Question

What is the principal argument in complex numbers ?

Answer 1

In order to answer this question, first we will explain the definition or the exact meaning of principal argument of the complex numbers and then we will represent the argument algebraically and we will also find the range of the principal argument of the complex numbers.

Complete step by step answer:
In simple words, the principal or primary argument can be defined as the angle formed by the line OP with the +ve $x - axis$ by analysing the complex number represented by point $P\left( {Re\left( z \right),Im\left( z \right)} \right)$ in the argand plane. The principal argument's value is such that $- \pi < \theta = \pi$ . It is also the angle between the positive real axis and the line joining the origin and $z$ .

The value of the Principal argument is denoted by $Arg(z)$. We can write the argument of the complex number or their general form, $z = x + iy$ and algebraically we can represent the argument of the complex number as:
$\arg (z) = {\tan ^{ - 1}}(\dfrac{y}{x})\,,\,when\,x > 0 \\\ \Rightarrow \arg (z) = {\tan ^{ - 1}}(\dfrac{x}{y}) + \pi \,,\,when\,x < 0$
Now, we will find the range of the value of argument of the complex numbers:
$\arg (z) \in \\{ Arg(z) + 2\pi n\,|\,n \in Z\\}$
Thus, $Arg(z) = \arg (z) - 2\pi n$

Hence, the value of the principal argument of the complex numbers lies in the interval $( - \pi ,\pi )$.

Note: A complex number is expressed in polar form by the equation $r\left( {\cos \theta + isin\theta } \right)$ , where $\theta$ is the argument. $Arg\left( z \right)$ signifies the argument function, where $z$ signifies the complex number, i.e. $z = x + iy$ .