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Unformatted text preview: N.C.R. Makins - University of Illinois at Urbana-Champaign 2004 Complex Numbers and the Zen of i Introduction Nature is filled with things which oscillate . Pendulums oscillate, the moon oscillates across the plane of the sky, and atomic electrons oscillate round-and-round the nuclei to which they are bound. In our study of electromagnetism, we have encountered yet more examples: currents and voltages oscillate in AC circuits, and the electromagnetic fields of a radiated plane wave oscillate around each other. At their core, these oscillations are sinusoidal in nature. Things which oscillate in time (like the position of a pendulum, or the angle of the pickup-coil in a motor) vary like sin( ! t ) or cos( ! t ) or sin( ! t + " ) things which oscillate in space (like the size of the electric field in a radio wave) vary like sin( kz ) or cos( kz ) or sin( kz + ! ) and so on, and so on. So far, we have used only the trigonometric functions sin and cos to describe this type of behavior. But there is a much better mathematical framework available! Continuing to use sin and cos to describe oscillations is like trying to complete your income tax forms using Roman numerals! The best mathematical technology for describing things which oscillate is complex numbers . These numbers are strange indeed, and they arise from the mystical contemplation of the following truly- bizarre question: This question has no rational answer or does it? To answer it, we must expand our minds , in the manner of the Zen scholars of ancient times Exercise 10.1: Eulers Theorem Have you ever heard of a Zen koan? Koans are mind-expanding riddles that Zen masters have used for centuries to teach their students about the limitations of the mind and its preconceptions. The riddles are non- sensical, in the sense that they have no solution that the rational mind can grasp. In contemplating these riddles, the mind realizes that it is limited and becomes quiet, opening the student to the possibility of greater truth. Here are a couple of famous Zen koans: What is the sound of one hand clapping? What is your original face before you were born? What is the square-root Physics 199HO 10.2 N.C.R. Makins, University of Illinois, 2004 10.2 Heres another one from Richard Feynman (who used it to torment a group of philosophy students during his grad-school days ): Show me the inside of a brick. And now here is a famous mathematical koan: What is the square-root of 1? That question has no mathematical answer within the world of real numbers. And so, just like the Zen students, we must expand our minds and our mathematics into another dimension!...
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- Spring '10