Pure Mathematics · 3rd secondary · First Term

Algebra - Complex numbers — The cubic roots of unity

About this lesson

This unit teaches the cubic roots of unity, specifically the roots of the equation Z³ = 1. It introduces the notation 1, ω, and ω² for these roots, where ω is a complex number. The unit covers the properties of these roots, including that their sum is zero, the product relationships (ω³ = 1, ω² = 1/ω), and their geometric representation as vertices of an equilateral triangle on the unit circle. It also includes examples and exercises involving algebraic manipulation with ω, such as simplifying expressions, proving identities, and forming quadratic equations with roots expressed in terms of ω. The unit uses De Moivre's theorem to derive the roots and emphasizes the conjugate of ω.

Main topics in this lesson

Cubic roots of unity
Properties of cubic roots of unity
Geometric representation of cubic roots
Algebraic manipulation with ω

Key terms and vocabulary

Cubic roots of unity
Root
Conjugate
De Moivre's theorem
Complex number
Equilateral triangle

Questions and answers on Algebra - Complex numbers — The cubic roots of unity - practice and revision

Multiple choice questions on this lesson in Pure Mathematics for 3rd secondary, First Term, with practice exercises for revision and exam preparation.

إذا كان ω هو الجذر التخيلي للوحدة، فإن قيمة ω⁹ + ω¹² تساوي:

1 2
2 1
3 صفر
4 ω

إذا كان ω هو الجذر التخيلي للوحدة، فإن قيمة (2 + 2ω + 2ω²) تساوي:

1 صفر
2 2
3 6
4 1

إذا كان ω هو الجذر التخيلي للوحدة، فإن قيمة ω + ω² + ω³ تساوي:

1 1
2 صفر
3 2
4 ω

إذا كان ω هو الجذر التخيلي للوحدة، فإن قيمة (1 + ω²)³ تساوي:

1 -1
2 1
3 ω
4 صفر

إذا كان ω هو الجذر التخيلي للوحدة، فإن قيمة (ω²)³ تساوي:

1 1
2 ω
3 ω²
4 صفر

إذا كان ω هو الجذر التخيلي للوحدة، فإن قيمة ω¹⁰ تساوي:

1 ω
2 ω²
3 1
4 -1

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