Question

What is a principal argument in complex numbers?

Answer 1

Principal argument:

The principal argument is the angle between the positive real axis and the line joining the origin and $$z$$.

The argument of a complex number $z=x+iy$ is,

$arg\left(z\right)={\tan }^{-1}\left(\frac{y}{x}\right)$ , when $x>0$

$arg\left(z\right)={\tan }^{-1}\left(\frac{y}{x}\right)+\mathrm{\pi }$ , when $x<0$

The principal value of an argument is denoted by $Arg\left(z\right)$.

The range of the value of an argument is:

$arg\left(z\right)\in \left\{Arg\left(z\right)+2\pi n|n\in Z\right\}$

Thus,

$Arg\left(z\right)=arg\left(z\right)–2\pi n$

Hence, the value of the principal argument of the complex numbers lies in the interval $\left(-\mathrm{\pi },\mathrm{\pi }\right]$.