Simple Complex Dilation If $a&gt;0$ and $$w = f(z) = az \text{, where }(a,z)\in \mathbb{C}$$ how is it that, in general, $$|w| = a|z|, Arg(w) = Arg(z)$$   

It's written in my teacher's notes. Why do w and z have to like on the same ray? I thought multiplying complex numbers meant multiplying moduli and adding the args.
Multiplying complex numbers does scale the moduli and add the arguments. But the argument of a positive real number is $0$. Hence Multiplying any complex number by a positive real will only scale it, while keeping it on the same ray (i.e. argument).
Multiplying complex numbers does scale the moduli and add the arguments. But the argument of a positive real number is $0$. Hence Multiplying any complex number by a positive real will only scale it, while keeping it on the same ray (i.e. argument).