Complex Numbers Flashcards

If z = (3 + 4i)/(1 - 2i), what is the imaginary part of z?

Answer: 2

Multiply numerator and denominator by the conjugate of the denominator: (3 + 4i)(1 + 2i) / ((1 - 2i)(1 + 2i)). Numerator: 3 + 6i + 4i + 8i² = 3 + 10i - 8 = -5 + 10i. Denominator: 1 + 4 = 5. So z = (-5 + 10i)/5 = -1 + 2i. The imaginary part is 2.

The complex number z satisfies z² = -5 + 12i. Which of the following is a possible value of z?

Answer: 3 + 2i

Let z = a + bi. Then z² = a² - b² + 2abi = -5 + 12i. So a² - b² = -5 and 2ab = 12, giving ab = 6, so b = 6/a. Substituting: a² - 36/a² = -5 → a⁴ + 5a² - 36 = 0 → (a² + 9)(a² - 4) = 0 → a² = 4 → a = 2. Then b = 3. So z = 2 + 3i or z = -2 - 3i. Check: (3 + 2i)² = 9 + 12i - 4 = 5 + 12i ≠ -5 + 12i. Only 2 + 3i works: (2 + 3i)² = 4 + 12i - 9 = -5 + 12i. ✓

If ω = e^(2πi/3), which of the following equals 1 + ω + ω²?

Answer: 0

ω = e^(2πi/3) is a primitive cube root of unity, satisfying ω³ = 1. The three cube roots of unity (1, ω, ω²) are roots of x³ - 1 = (x - 1)(x² + x + 1) = 0. Since ω ≠ 1, it satisfies x² + x + 1 = 0, which means ω² + ω + 1 = 0. Therefore 1 + ω + ω² = 0.

In the complex plane, the set of all z such that |z - 2| = |z + 2i| represents a line. What is the y-intercept of that line?

Answer: 0

Let z = x + yi. Setting |z - 2|² = |z + 2i|²: (x-2)² + y² = x² + (y+2)². Expanding both sides: x² - 4x + 4 + y² = x² + y² + 4y + 4. The x², y², and constant terms cancel, leaving -4x = 4y, so y = -x. This line passes through the origin, giving a y-intercept of 0.

If P(x) = x⁴ + 4 is factored completely over the complex numbers, and two of its roots are 1 + i and 1 - i, what are the other two roots?

Answer: -1 + i and -1 - i

x⁴ + 4 = (x² + 2)² - (2x)² = (x² - 2x + 2)(x² + 2x + 2) using Sophie Germain identity. The roots of x² - 2x + 2 = 0 are x = (2 ± √(4-8))/2 = 1 ± i. The roots of x² + 2x + 2 = 0 are x = (-2 ± √(4-8))/2 = -1 ± i. So the other two roots are -1 + i and -1 - i.

If z = cos θ + i sin θ, which expression is equivalent to z^n + z^(-n)?

Answer: 2 cos(nθ)

By De Moivre's theorem, z^n = cos(nθ) + i sin(nθ) and z^(-n) = cos(-nθ) + i sin(-nθ) = cos(nθ) - i sin(nθ). Adding: z^n + z^(-n) = 2cos(nθ). This is a standard result used to express powers of cosine in terms of multiple angles.