Complex Numbers and the Complex Exponential

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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+b2Figure 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 ...

  Complex, Arithmetic