Complex Variables for Strategy: From Bombelli's Imaginary Number, Steinmetz's Phasor Domain to Sunzi's Art of War

摘要

Rafael Bombelli was born in Bologna, Italy, on 1526. He was trained and worked as an engineer-architect until died on 1573 without much recognition until hundred years later by the math society as a great Italian mathematician, and there is a lunar crater named Bombelli for his contributions to imaginary and complex numbers. Bombelli is the author of a treatise on algebra and is a central figure in the understanding of imaginary numbers. His mathematical achievement was never fully appreciated during his life time, but his failure to repair the Ponte Santa Maria 1561 attempt, a bridge in Rome, was probably fully recognized by his comtemporaries, until hundred years later the great mathematician Leibniz (1646-1716) recognized his great contribution to the finding and operation of complex numbers and hailed him an "outstanding master of the analytical art!" In Algebra 1569, Bombelli solved equations, using the method of del Ferro/Tartaglia, he introduced +i and -i and described how they both worked in Algebra. But it's interesting to note that even Bombelli himself had never realized his own great contribution to math, since he still referred to complex numbers as useless, not to mention his contemporaries. Nonetheless, Bombelli's Algebra was finally widely read and respected by the math giants such as Leibniz used it to study cubic equations and Leonhard Euler quoted from it in his text, Algebra. Bombelli's contribution to math, especially his understanding about real numbers can result from the operations of complex numbers, led to the breakthrough in calculation of alternating current and voltage with R, C & L (Resistance, Capacitance & Inductance) as the three variables to represent the operation of any electrical devices or power lines. Looking back at the development of modern societies supported by high-tech electronic devices and energies supplied from alternating current based powerlines, we should even give higher praise to Bombelli's finding and development of imaginary numbers and operation of complex variables in the first place. With understanding of the invention and finding of imaginary numbers and operation of complex variables, the authors would like to discuss how the strategies outlined by the greatest strategist Sunzi, who lived about 2500 years ago in China, who wrote the classical military text, "Art of War", can be in line of the concept and operation of complex variables. In this paper, the authors will combine the understanding of imaginary numbers by an Italian mathematician and the studies of a Chinese engineer and Sunzi strategy researcher, to show that Sunzi's military strategy based on I-Ching's philosophy with Yin & Yang as variables, can related to the Real & Imaginary numbers, and their operations and transitions from real to imaginary field in strategical thinking and business planning in daily life as well.