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Geometric Representation of Complex Numbers

  • Argand diagram: Used for graphically representing complex numbers.

  • Complex plane: Composed of a horizontal (real axis) and a vertical (imaginary axis).

  • Rectangular form: z = a + bi (where a = real part, b = imaginary part).

Polar Form of Complex Numbers

  • Polar form: z = r(cos(θ) + i sin(θ)) = r cis(θ)

  • Conversion: Rectangular form can be converted to polar form using r and θ calculated from the complex number.

Example Problems

  1. Express z = 5 + i in polar form.

  2. Convert z = 3 + 5 cis(π) into rectangular form.

Multiplying Complex Numbers

  • Formula: (r_1 cis(α))(r_2 cis(β)) = (r_1 r_2) cis(α + β)

  • Multiplication in rectangular form:

    • Use distribution to multiply two complex numbers.

    • Example: Multiply (3 + 5i)(3 - 5i).

Powers of Complex Numbers

  • Raising to the nth power: For z = a + bi, (z^n) = ? can be complex.

  • DeMoivre’s Theorem: If z = r cis(θ), then (z^n) = r^n cis(nθ).

Example Evaluation

  • Evaluate (1 - 3i)^4 using DeMoivre’s theorem and convert the answer to rectangular form.