What does this inequality mean?

Question

I was reading complex numbers when I came across this inequality:

$- \pi ≤ arg(z) ≤ \pi$

Where $z$ is a complex number, I want to know what this inequality mean? Can't it simply be that,

$0 ≤ arg(z) ≤ 2 \pi$

Answer 1

It is called Argument.

The argument of $z$ is the angle between the positive real axis and the line joining the point to the origin, the argument is a multi-valued function operating on the nonzero complex numbers.

$$z=x+iy$$ $$arg(z)=\tan^{-1}(\frac{y}{x})$$

The $\tan^{-1}$ function usually returns a value in the range (−π, π].

When a well-defined function is required than the usual choice, known as the principal value, is the value in the open-closed interval $(−π,π]$, that is from −π to π radians, excluding −π rad itself.

Some authors define the range of the principal value as being in the closed-open interval [0, 2π).

Answer 2

You probably want strict inequality at one end of the interval, otherwise the argument is not unambiguously defined.

Any interval of length $2\pi$ which includes one endpoint but not the other will work as the set of possible values for the argument of any nonzero complex number. That's because it represents exactly one traversal of the unit circle.