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Mathematics · 11. Complex numbers and De Moivre formula

Complex numbers, polar form, Euler identity and De Moivre theorem

Explore complex field C, Argand plane geometry, trigonometric polar form, Euler’s formula and De Moivre theorem.

40 minutesContent version: 2.1
Start the interactive pathwayOpen the virtual experiment

Driving question

Why does an imaginary unit i² = −1 unlock the entire number system, turning algebraic multiplication into geometric rotation?

Curriculum coverage

Vietnam 2018 Mathematics 12 (Complex numbers); Cambridge A-Level Mathematics 9709 P3; AP Calculus BC.

Model scope and limitations

Covers complex algebra, polar representations, De Moivre powers and roots; omits complex analysis contour integration.

Learning objectives

  • Perform addition, subtraction, multiplication, division, modulus and conjugate operations in C.
  • Convert fluently between rectangular form z = a + bi and polar/Euler form z = r(cos θ + i sin θ) = r e^(iθ).
  • Apply De Moivre’s theorem z^n = r^n(cos nθ + i sin nθ) to high powers and roots of unity.

Prerequisite knowledge

  • Trigonometry on the unit circle and angle addition formulas.
  • Cartesian coordinates Oxy and Euclidean distance.

Core knowledge

Imaginary unit and Argand plane

The imaginary unit satisfies i² = −1. Each number z = a + bi corresponds uniquely to point M(a, b) on the Argand plane; its modulus |z| = √(a² + b²) is distance from the origin.

Polar form and geometric multiplication

Writing z = r(cos θ + i sin θ) where r is modulus and θ is argument: multiplying numbers multiplies their moduli and adds their arguments (scaling and rotation).

De Moivre theorem and Euler formula

For any integer n, [r(cos θ + i sin θ)]^n = r^n(cos nθ + i sin nθ). Euler’s identity e^(iθ) = cos θ + i sin θ gives the famous relation e^(iπ) + 1 = 0.

Worked example

Compute z¹⁰ for z = 1 + i. Modulus r = √2, argument θ = π/4. By De Moivre: z¹⁰ = (√2)¹⁰ [cos(5π/2) + i sin(5π/2)] = 32[0 + i] = 32i.

Misconceptions to avoid

  • Multiplying any complex number by i geometrically represents a pure counterclockwise 90° rotation about the origin.
  • Complex numbers cannot be ordered by inequality (z1 > z2 is undefined in C; only their scalar moduli |z1| and |z2| can be compared).

Virtual experiment procedure: Complex numbers, Argand diagram and De Moivre powers

  1. Open the Argand plane simulator, dragging complex point z to observe dynamic rectangular (a, b) and polar (r, θ) readouts.
  2. Set integer power n on the slider and observe powers z², z³, ..., zⁿ spiraling outward or inward.
  3. Activate roots-of-unity mode to observe the n roots forming vertices of a regular polygon inscribed in the unit circle.

Safety and cautions

  • Pure mathematical computation is inherently safe; beware of floating-point rounding when evaluating irrational trigonometric ratios.

Evidence to collect

  • Conversion table between Cartesian (a, b) coordinates and polar (r, θ) representation.
  • Geometric diagram illustrating the n-th roots of unity arranged as a regular polygon on the Argand plane.
Open the virtual experiment
📋 Embed this simulation in online lectures (For Teachers & LMS)

Copy the iframe code below to embed this interactive simulation into Canvas, Moodle, Google Sites or educational blogs:

<iframe src="https://visuallab.win/math-lab/?module=complex_numbers_moivre" width="100%" height="560" frameborder="0" allow="fullscreen" loading="lazy"></iframe>

Diagnostic check

1. What is the modulus of complex number z = 3 − 4i?

  1. 5
  2. 7
  3. −1
Show answer and explanation

Correct answer: 5

|z| = √(3² + (−4)²) = √(9 + 16) = 5.

2. What is the complex conjugate of z = 2 + 5i?

  1. 2 − 5i
  2. −2 + 5i
  3. −2 − 5i
Show answer and explanation

Correct answer: 2 − 5i

The complex conjugate of a + bi is a − bi (negating the imaginary part).

Application practice

1. What is the principal argument θ of z = −1 + i√3?

  1. 2π/3 (120°)
  2. π/3 (60°)
  3. 5π/6 (150°)
Show answer and explanation

Correct answer: 2π/3 (120°)

r = 2; with cos θ = −1/2 and sin θ = √3/2, point lies in quadrant II, giving θ = 2π/3.

2. According to De Moivre, what is (cos(π/6) + i sin(π/6))⁶?

  1. −1
  2. 1
  3. i
Show answer and explanation

Correct answer: −1

De Moivre yields cos(6×π/6) + i sin(6×π/6) = cos π + i sin π = −1.

3. The 3 cube roots of unity form which shape on the Argand plane?

  1. Equilateral triangle
  2. Right isosceles triangle
  3. Line segment
Show answer and explanation

Correct answer: Equilateral triangle

The three roots have unit modulus and angle separation 120°, forming vertices of an equilateral triangle inscribed in unit circle.

Academic references

  1. Cambridge IGCSE Mathematics 0580 — official syllabus and resources
  2. Cambridge AS & A Level Mathematics 9709 — official syllabus
  3. OpenStax Precalculus 2e — functions, geometry and probability

This lesson is maintained against the cited sources below. Simulations are learning models and do not replace supervised physical-laboratory safety procedures.

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WebLab STEM
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Đức Tiến — Control & Automation Engineer