Complex Numbers Flashcards

What is the sum of all cube roots of unity?

Answer: 0

The cube roots of unity 1, ω, ω² are roots of z³ − 1 = 0, and by Vieta's formulas their sum equals 0.

Using De Moivre's theorem, what is (cos(π/4) + i·sin(π/4))^8?

Answer: 1

By De Moivre's theorem: cos(8·π/4) + i·sin(8·π/4) = cos(2π) + i·sin(2π) = 1.

How many distinct solutions does z⁴ = 1 have in the complex numbers?

Answer: 4

A degree-4 polynomial has exactly 4 complex roots; the four 4th roots of unity are 1, i, −1, −i.

If z = 2(cos(π/3) + i·sin(π/3)), what is |z|?

Answer: 2

In polar form z = r·(cos θ + i·sin θ), the modulus is r = 2.

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

Answer: 2π/3

The point −1 + i√3 lies in the second quadrant; tan⁻¹(√3/1) = π/3, so the argument is π − π/3 = 2π/3.

What is i^100?

Answer: 1

Since 100 ≡ 0 (mod 4), i^100 = (i⁴)^25 = 1^25 = 1.

What is (1 + i)² / (1 − i)?

Answer: −1 + i

(1+i)² = 2i; then 2i/(1−i) · (1+i)/(1+i) = 2i(1+i)/2 = i + i² = −1 + i.