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 .
22 +12 = 5. zlies in the first quadrant so its argument θis an angle between 0 and π/2. From tanθ= 1 2 we then conclude arg(2 + i) = θ= arctan 1 2. 3. Geometry of Arithmetic Since we can picture complex numbers as points in the complex plane, we can also try to visualize the arithmetic operations “addition” and “multiplication.” To ...
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