The Geometry of C2

Geometrically, complex numbers can be thought of as the points in the (infinite, 2-dimensional) plane. Thus, whereas real numbers represent the real number line, complex numbers represent the complex plane, or C2. The real numbers are a subset of the complex numbers. This is indicated by the real number line being the horizontal axis of the complex plane.

Algebraically, a complex number is of the form X + i*Y, where X and Y are real numbers, and i is the square root of -1. (Since the square root of -1 does not exist as a real number, complex numbers were initially, and now somtimes archaically, called imaginery numbers.) Many, if not most, of the operators and functions defined for real numbers are also defined for complex numbers.

Complex analysis is an extremely important tool in mathematics, engineering and science because of its ability to represent two-dimensional phenomena in a form to which much of the machinery of algebra and calculus can be applied. Much of the need for trigonometry disappears when using complex numbers. Consider that the following two functions are equivalent:

f(X) = sin(X*pi)
f(X) = i*X*4
Complex numbers implicitly represent angular quantities and operations.

Maps in C2

Functions, particularly ones which operate on C2, are often called maps. Simple operators, such as addition, multiplication, and exponentiation are maps over C2 (C2 is both the domain and range of the function), as are combinations of them. One way to visualize such a map is sometimes called an object-space method. It assigns domain coordinates to all elements of a grid or image based on the spatial positions of elements, applies the function, and stretches the grid or image to the range coordinates. The Complex Map Visualizer applet is such a viewer. It allows one to view a grid in C2 mapped by a chain of functions.

An opposite approach employs a so-called image-space method. It divides the view space up into pixels, assigns domain coordinates to each based on their position in the view, applies the function, and uses range coordinates as positions in some pattern or image to obtain a color to assign to the view pixel. Such methods, used with forms of iteration, are the basis for for the creation of images widely knows as fractals, but more specifically as Julia or Mandelbrot sets.

Complex Analysis Web Resources