Arg(x/y) = Arg(x) - Arg(y)

 

It is easy to see that \( \mathrm{arg} \left ( \frac{a}{b} \right ) = \mathrm{arg} \left ( a \right )  - \mathrm{arg} \left ( b \right )  \) for any complex number \( a \) and \( b \). But, this is not the case for the principle value that is \( \mathrm{Arg} \) does not always equal to \( \mathrm{Arg} \left ( a \right ) - \mathrm{Arg} \left ( b \right )  \).

For example, the principle value argument of \( \frac {-i}{i} \) is \( \mathrm{Arg} \left ( -1 \right ) = \pi \) but \( \mathrm{Arg} \left ( -i \right )  - \mathrm{Arg} \left ( i \right ) = -\frac {\pi }{2} - \frac  {\pi }{2} = -\pi \neq \pi = \mathrm{Arg} \left ( \frac {-i}{i} \right )\).

In order \( \mathrm{Arg} = \mathrm{Arg} \left ( a \right ) - \mathrm{Arg} \left ( b \right )  \) to be true, the necessary condition is \(  -\pi  < \mathrm{Arg} \left ( a \right ) - \mathrm{Arg} \left ( b \right ) \leq \pi \). In other words, \( \mathrm{Arg} \left ( a \right )  - \mathrm{Arg} \left ( b \right ) \) must be in the range of the function \( \mathrm{Arg} \) that is \( \left ( -\pi , \pi \right ] \).