Complex Numbers: One-Page Formula Sheet

Complex Numbers — formula sheet

Formulas

• z = r(cosθ + i sinθ) = re^(iθ)
• De Moivre: (cosθ + i sinθ)ⁿ = cos nθ + i sin nθ
• Triangle inequality: |z₁ + z₂| ≤ |z₁| + |z₂|
• |z − z₁| = |z − z₂| is the perpendicular bisector

Key ideas

• |z|² = z z̄
• arg(z₁z₂) = arg z₁ + arg z₂
• Cube roots of unity: 1 + ω + ω² = 0, ω³ = 1
• |z − z₀| = r is a circle

Traps to avoid

• Principal argument range (−π, π]
• Forgetting the quadrant when finding arg
• Treating i² as +1 in long expansions

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