real part a, plus b times the imaginary part, b/’; that is, z = a + bi. We can then set up a geometric space to represent z by imagining a two dimensional plane with the horizontal real axis extending from left to right, the usual x axis, and the vertical dimension, called the imaginary axis, extending from bottom to top like the standard y axis. These two axes, going from negative values to positive ones, left to right and bottom to top, cross at the shared value of 0. Thus a and 6 can be visualized as the rectangular coordinates of a point in the plane and the point locates the complex number, z = a + bi. Since real parts and imaginary parts are like apples and pears and for addition, like must be added to like, if two complex numbers, a + bi and c + di are equal, then a = c and b = d and their sum is written (a+ c)+(b+d)i. Now that we’ve set up a point z on the plane, located with a complex number at z = a + bi, we can then draw an arrow, called a vector, from the intersection of the imaginary and real axis at 0 to this point z. Its length from 0 to z, 0z, we'll call that length p, is the size or amplitude-like modulus of the complex number, z = a +i. The angle this 0z vector makes with the real, Oa-axis, lets call this angle ¢, is called the argument of complex number z = a +bi. p is a length that can grow or shrink, ¢ is an angle that can rotate. We imagine vectorial movement like that of a variable length hand of a clock. This geometric explication of complex numbers prepares us to visualize complex numbered eigenvalue solutions to matrices representing the relevant equations that bifurcate to limit cycles and directional good and evil splitting. represents the dilatable clock’s radial amplitude of circular motion and ¢, the angle of vectorial turning from the 0a-axis. The complex conjugate of the complex number, a + b/ is the complex number a — bi in which the sign of the imaginary part is reversed. Geometrically, this means that a pair of complex conjugate