Any complex number can be written in the form where is the imaginary unit and and are real numbers. Then the real part of , usually denoted or , is just the value .
Geometrically, if a complex number is plotted in the complex plane, its real part is its -coordinate (abscissa).
A complex number is real exactly when .
The function can also be defined in terms of the complex conjugate of : . (Recall that if , ).
- . Note in particular that is not in general a multiplicative function, for arbitrary complex numbers .
Practice Problem 1
Find the conditions on and so that .