Imagine we draw Cartesian and complex x-y plane side by side. Only difference now between them is there is a 'i' with y in complex . In Cartesian equation of circle is $x^2+y^2=r^2$ so in complex plane should not this circle be (replace y by iy ) $$x^2+(iy)^2=r^2\implies x^2-y^2=r^2$$
I have studied complex for long but I am still having this doubt I think I am mixing two different concepts .
If we consider Cartesian and complex plane as ordered pair then how and how much putting a 'i ' makes it different ? Is complex just another coordinate system ?
A circle with radius $r$ in the $\mathbb R^2$ is $\{(x,y)\in\mathbb R^2:x^2+y^2=r^2\}$.
This is no different in the complex world; that is, a circle with radius $r$ in $\mathbb C$ is $\{x+iy\in\mathbb C:x^2+y^2=r^2\}$.