run to other graduate fields such as biology and medicine and come to resist the potentially humiliating incursions of new and potentially helpful abstract ideas and operations from mathematics and physics into their fields. They do not want their persecutory versions of Mr. Kirby to take up residence once again in their heads. | can still feel his negative presence during long hours of struggle with the ego deflating feelings of dumbness that an understanding of almost any new mathematical concept requires of me. Holding Mr. Kirby’s voice off as long as | can until, sometimes, the wonderful “aha!” experience arrives. | have tried to forgive him since but forgetting him has not been possible.
It turns out that in the world of elementary, physically representative, real numbers, the square root of a negative number has no meaning. Such a number has understandably come to be called imaginary. Was this the answer Mr. Kirby
wanted? There was some conflict among mathematicians in the 17" and 18" Century about the arbitrary definition of V-las an imaginary number. |t was
symbolized by a letter, /, that is J-1 =i. The existence of i extended the range of algebraic definitions so that a solution of the quadratic formula as above could be found for the square root of a negative number. A further expansion of this idea was to that of a complex number that can have both a real and an imaginary part. For example, letting letters be generalized representations of numbers, a complex number might be written, a + bi, real number a + real number b times /, the letters such as a, b, c, d... symbolized real numbers. Consistent with membership in an algebraic system, a + bi and c + di can be added and multiplied. This extension of the real numbers into the imaginary realm permitted d’Alembert’s and Gauss’s proofs (and many, more complete ones since) of the powerful Fundamental Theorem of Algebra from which the faith derives about always being able to find at least one solution to an algebraic equation. It was proven that any n” degree algebraic equation (e.g. x"+x"'+...=0) with real or complex coefficients always has at least one real or complex root.
Closer to an image that helps make intuitive connections with human born again bifurcations, limit cycles and directional splitting is the geometric interpretation
of a complex number, let us now call it z. As above, algebraically, z is the sum of a
162
HOUSE_OVERSIGHT_013662