Complex Numbers

Definitions and Polar Coordinates

The basic definition of complex numbers is that we extend our real number line by adding a number not on that line, the square root of 1 and all multiples of that by a real number. This leads to two independent (orthogonal) real lines, equivalent to a plane. We write

and any complex number as z = x + iy where x and y are real numbers. These are represented on a plane as shown in the figure on the right. The complex number z is represented by the vector shown in the diagram.

Polar coordinates are very convenient when working with complex numbers. We define the length of the vector z and the angle it makes with the x axis as:

This gives the inverse relations

This gives the important representation of a complex number


University of MarylandPhysics DepartmentPhysics 374 Home

This page prepared by

Edward F. Redish
Department of Physics
University of Maryland
College Park, MD 20742
Phone: (301) 405-6120

Last revision 2. September, 2005.