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Grade 12 Math Lab: Analysis & Inference FREE
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Roots of Unity
Learning Objectives
- Find nth roots of unity
- Plot on complex plane
- Apply De Moivre's theorem
Key Concepts
Root of Unity
Complex number z where zⁿ = 1
De Moivre's Theorem
(cosθ + i·sinθ)ⁿ = cos(nθ) + i·sin(nθ)
Theory
**nth roots of unity:** zₖ = e^(2πik/n) = cos(2πk/n) + i·sin(2πk/n), k = 0, 1, …, n-1
**Properties:**
- Form a regular n-gon on the unit circle
- Sum of all nth roots = 0
- Product = (-1)^(n+1)
**De Moivre's Theorem:** (r·e^(iθ))ⁿ = rⁿ·e^(inθ)
**Cube roots of unity:** 1, e^(2πi/3), e^(4πi/3)
Examples
Find the 4th roots of unity
Solution: 1, i, -1, -i