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After completing this section, students should be able to do the following.

• Explain why complex numbers are important in circuits and electromagnetics.
• Sketch a complex number in rectangular and polar coordinates, and label magnitude, phase, real and imaginary parts.
• Derive the magnitude and phase from the real and imaginary parts of a complex number.
• Derive the real and imaginary parts of a complex number from the magnitude and phase.
• Explain how Euler’s formula relates to sinusoidal signals and complex numbers.
• Describe which coordinate system to use when adding/subtracting and which one when multiplying/dividing two complex numbers.
• Apply complex numbers to solve a circuit element’s impedance if the phasor of current through and voltage on it are known.
• Apply complex numbers to solve for voltage on a circuit element if phasor of current and impedance are known.
• Apply complex conjugate operation to a complex number in rectangular and polar coordinates.
• Derive magnitude of a complex number from a complex number and complex conjugate of the same number.
• Visualize the position of purely imaginary and purely real complex numbers on a unit circle.
• Convert visually purely imaginary and purely real complex numbers from rectangular to polar coordinates and vice versa.
• Prove that the magnitude of a complex number is a square root of the product of the number and its complex conjugate.