Roots of unity (HSC SSCE Mathematics Extension 2): Flashcards
📚Flashcards
Practise the cards
10 cards from this deck
ShowHide
Practise the cards
10 cards from this deck
What are complex roots of unity?
What are complex roots of unity?
Numbers solving z^n = 1 in the complex plane.
Where do roots of unity lie geometrically?
Where do roots of unity lie geometrically?
On the unit circle in the complex plane.
What is the angle between roots of unity?
What is the angle between roots of unity?
Equal angles of 2π/n on the unit circle.
How do you express z^n - 1?
How do you express z^n - 1?
(z - 1)(z^{n-1} + ... + z + 1)
What are the solutions to z^4 - 1?
What are the solutions to z^4 - 1?
z = 1, -1, i, -i.
What is the symmetry property of roots of unity?
What is the symmetry property of roots of unity?
They appear in conjugate pairs for even n.
What is the formula for the n-th roots of unity?
What is the formula for the n-th roots of unity?
ω_k = e^{2πik/n}, for k = 0, 1, ..., n-1.
What role does ω play in polynomial root simplification?
What role does ω play in polynomial root simplification?
ω represents complex roots, simplifying calculations.
What is the significance of roots of unity?
What is the significance of roots of unity?
They assist in polynomial factorisation.
How do roots of unity help in polynomial division?
How do roots of unity help in polynomial division?
They speed up finding roots and zeroes.