Argument
Given a complex number , the argument
is the signed angle the ray
makes with the positive real axis. (Note that this means the argument of the complex number 0 is undefined.)
Unfortunately, this means that is not a proper function but is instead a multi-valued function: for example, any positive real number has argument 0, but also has argument
. This means that the argument may be best considered as an equivalence class
. The advantages of this are several: most importantly, they make
into a continuous function. They also make some properties of the argument "look nicer." For example, under this interpretation, we can write
. The other common solution (restricting the range of
to some interval, usually
or
) forces us to state this equality modulo
.
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