The argument of a non-zero complex number $z$ is a multi-valued function defined by $$\arg z = \theta + 2\pi n, \, \, \, n \in \mathbb{Z}$$
where $\theta \equiv \text{Arg }z $, the principal value of the argument, which by convention is taken to be $$-\pi < \theta \leq \pi$$
But this is just by convention, you could equally well have taken the principal value to be $$0 < \theta \leq 2\pi$$ and all your work with complex numbers would be just as valid.
We now have $$\arg z = \text{Arg }z + 2\pi n, \, \, \, \, n \in \mathbb{Z}$$ we can now define $$\text{Arg }z = \arg z + 2\pi \left\lfloor\frac{1}{2} - \frac{\arg z}{2\pi}\right\rfloor $$