The Paradigm of Complex Probability, Numerical Analysis, and Chaos Theory

The set of imaginary numbers is taken into account by extending the probability system of five axioms of Andrey Nikolaevich Kolmogorov which was put forward in 1933. This is achieved by adding three new and supplementary axioms. Hence, any random experiment can thus be performed in the extended complex probability set C = R + M which is the sum of the real set R of real probabilities and the imaginary set M of imaginary probabilities. The objective here is to determine the complex probabilities by encompassing and considering additional new imaginary dimensions to the event that occurs in the “real” laboratory. The outcome of the stochastic phenomenon in C can be foretold perfectly whatever the probability distribution of the input random variable in R is since the corresponding probability in the whole set C is permanently and constantly equal to one. Thus, the consequence that follows indicates that randomness and chance in R is substituted now by absolute determinism in C. This is the result of the fact that the probability in C is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. This novel complex probability paradigm will be applied to numerical analysis and to chaos theory to prove henceforth that chaos vanishes totally and completely in the probability universe C = R + M.