notice that the absolute value of a complex number is a non-negative real number, with
Most concepts in analysis can be extended from to , as long as they do not rely on the order in . Sometimes even works better, either because it is algebraically closed or because of Goursat's theorem. Even when the order in is important, often it is enough to order the absolute values of complex numbers. See ground field for some of the concepts whose precise definition may vary with the choice of or (or even other possibilities).
Geometry of complex numbers
The complex numbers form a plane, the complex plane . The standard real-valued coordinates on this plane are and , with the identity function on . Much of complex analysis can be understood through differential topology by identifying with , using either and or and . (For example, Cauchy's integral theorem is Green's/Stokes's theorem.)
It is often convenient to use the Alexandroff compactification of , the Riemann sphere (note that the bar here does not indicate complex conjugation). One may think of as ; functions valued in but containing ‘poles’ may be taken to be valued in , with whenever is a pole of .