Transcription of 1 Complex algebra and the complex plane - MIT Mathematics
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1 Complex algebra and the complex plane - MIT Mathematics
Topic 1 NotesJeremy Orloff1 Complex algebra and the Complex planeWe will start with a review of the basic algebra and geometry of Complex numbers. Mostlikely you have encountered this previously in or MotivationThe equationx2= 1 has no real solutions, yet we know that this equation arises naturallyand we want to use its roots. So we make up a new symbol for the roots and call it acomplex symbols iwill stand for the solutions to the equationx2= 1. We willcall these new numbers Complex numbers. We will also write 1 = iNote: Engineers typically usejwhile mathematicians and physicists usei.
ei = cos( ) + isin( ): (1) There are many ways to approach Euler’s formula. Our approach is to simply take Equation 1 as the de nition of complex exponentials. This is legal, but does not show that it’s a good de nition. To do that we need to show …
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