# Doubt on the $n$ distinct $n$th roots of a complex number

I know that, given $n\in\mathbb{N}$, for every $z\in\mathbb{C}$, $z\neq0$, there are exactly $n$ distinct $n$th roots of $z$.

To prove this: given $z\neq0$, a $n$th root of $z$ is a complex number $w$ such that $w^n=z$. By posing $$w = r(\cos\theta+i\sin\theta),$$ we have $w^n=r^n(\cos n\theta+i\sin n\theta)$. So, in order to have $w^n=z$, it must be $$r = |z|,\qquad n\theta=\phi+2k\pi,\quad k\in\mathbb{Z},$$ where $\phi$ is the argument of $z$, because two complex numbers are equal if and only if their modulus coincide and their arguments are the same up to a multiple of $2\pi$. Is it right?

My question: Why in some book, in order to have $w^n=z$, is $n\theta=\operatorname{Arg}z+2k\pi$, where $\operatorname{Arg}z$ is the principal value of the argument of $z$? Should $\phi$ be the argument (and not the principal value of the argument) of $z$?

Thank You