Transcription of Complex Numbers and the Complex Exponential
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Complex Numbers and the Complex Exponential
Complex Numbers and the Complex Exponential1. Complex numbersThe equationx2+ 1 = 0 has no solutions, because for any real numberxthe squarex2is nonnegative, and sox2+ 1 can never be less than 1. In spite of this it turns out tobe very useful toassumethat there is a numberifor which one has(1)i2= numberis then an expression of the forma+bi, whereaandbare old-fashioned real Numbers . The numberais called thereal partofa+bi, andbis called itsimaginary the letterszandware used to stand for Complex any Complex number is specified by two real Numbers one can visualize themby plotting a point with coordinates (a,b) in the plane for a Complex numbera+bi.
Complex Numbers and the Complex Exponential 1. Complex numbers The equation x2 + 1 = 0 has no solutions, because for any real number xthe square x 2is nonnegative, and so x + 1 can never be less than 1.In spite of this it turns out to be very useful to assume that there is a number ifor which one has
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