KKhaemenes PrecalculusOpen-Age · Printable Analytical Lesson
Unit 08Course HomeTeacher Key

Unit 08 · Lesson 07 · Week 25 · Polar Coordinates & Complex Numbers

Roots of Complex Numbers & Geometric Structure

Interpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling. In this lesson, the central focus is roots of complex numbers & geometric structure. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.

Open-AgePrintableFoundation · Core · ExtendedMastery target 80%

Learning intentions

  • Interpret the defining quantities and notation used in roots of complex numbers & geometric structure.
  • Construct or analyze at least two representations related to roots of complex numbers & geometric structure.
  • Show symbolic work with correct domains, units, intervals, or angle measures.
  • Explain why the selected method is valid and check the result against another representation.

Essential question

How do polar coordinates and complex-number forms reveal rotation, magnitude, symmetry, and curves that Cartesian form can obscure?

Multiple-representation model

Complex visualInterpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling.Pθz=r cis θ

Interpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling.

z=a+bi=r(\cos\theta+i\sin\theta)z_1z_2=r_1r_2\operatorname{cis}(\theta_1+\theta_2)z^n=r^n\operatorname{cis}(n\theta)

Worked reasoning

Coordinate conversionFor (r,θ)=(6,2π/3), x=−3 and y=3√3.
RotationMultiplying by 2 cis(π/4) doubles magnitude and rotates every complex-plane point 45° counterclockwise.

Misconception check: When multiplying complex numbers in polar form, multiply magnitudes and add angles; do not multiply the angles.

Communication and reasoning

Write a complete response that identifies the representation used, shows the symbolic steps, interprets parameters or units, and verifies the conclusion with a second representation or reasonableness check.

Khaemenes Academy · Open-Age Precalculus

Roots of Complex Numbers & Geometric Structure

Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.

U08 · L07
Name
Date
1

Write 3+3i in polar form.

Complex visualInterpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling.Pθz=r cis θ
2

Multiply 2cis(π/6) by 3cis(π/4).

Complex visualInterpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling.Pθz=r cis θ
3

Divide 8cis(2π/3) by 2cis(π/6).

Complex visualInterpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling.Pθz=r cis θ
4

Use De Moivre to square 2cis(π/3).

Complex visualInterpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling.Pθz=r cis θ
5

How many distinct complex fourth roots does a nonzero number have?

Complex visualInterpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling.Pθz=r cis θ
6

What geometric action is multiplication by i?

Complex visualInterpret complex numbers geometrically and use polar form, multiplication, division, powers, and roots to model rotation and scaling.Pθz=r cis θ
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