Sometimes I want to compute a line integral over some circle $|z|=r$, where I have $|dz|$ instead of $dz$ given to me.
Reparametrizing with $z=re^{it}$, it follows that $dz=rie^{it}dt=izdt$. But I always read that $ |dz|=|iz|dt=|z|dt=|z|\frac{dz}{zi} $ so $|dz|=-ir\frac{dz}{z}$. In the first equality, why does $|dz|=|iz|dt$ instead of $|iz||dt|$? Why doesn't the absolute value extend to the $dt$ as well?