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. Theplane in which one plot these Complex Numbers is called the Complex plane, or +bia=Re(z)b=Im(z) = argzr=|z|= a2+b2 Figure Complex can add, multiply and divide Complex Numbers . Here s how:To add (subtract)z=a+biandw=c+diz+w= (a+bi) + (c+di) = (a+c) + (b+d)i,z w= (a+bi) (c+di) = (a c) + (b d) multiplyzandwproceed as follows:zw= (a+bi)(c+di)=a(c+di) +bi(c+di)=ac+adi+bci+bdi2= (ac bd) + (ad+bc)iwhere we have use the defining propertyi2= 1 to get rid divide two Complex Numbers one always uses the following +bic+di=a+bic+di c dic di=(a+bi)(c di)(c+di)(c di)Now(c+di)(c di) =c2 (di)2=c2 d2i2=c2+d2,soa+bic+di=(ac+bd) + (bc ad)ic2+d2=ac+bdc2+d2+bc adc2+d2iObviously you do not want to memorize this formula: instead you remember the trick, to dividec+diintoa+biyou multiply numerator and denominator withc any com
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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