Complex numbers: polar and Euler form; De Moivre practice questions.

The principal argument (in radians) of z = 1 + i is

• 0
• π/4
• π/2
• π

Answer: π/4

z lies in the first quadrant with equal real and imaginary parts, so arg(z) = arctan(1/1) = π/4.

Determine the modulus of the complex number z = -2 + 2√3 i.

• 2√3
• 4
• 16
• 8

Answer: 4

|z| = √((-2)² + (2√3)²) = √(4 + 12) = √16 = 4. Choosing 16 means forgetting the final square root.

Write z = 3(cos(π/6) + i sin(π/6)) in Cartesian form x + iy.

• (3√3)/2 + (3/2)i
• 3/2 + ((3√3)/2)i
• 3√3 + 3i
• 3/2 − ((3√3)/2)i

Answer: (3√3)/2 + (3/2)i

cos(π/6) = √3/2 and sin(π/6) = 1/2, so z = 3(√3/2 + i/2) = (3√3)/2 + (3/2)i. Swapping cos and sin gives the classic wrong answer.

Calculate (√3 + i)⁶ using polar form.

• −64
• 64
• 64i
• −64i

Answer: −64

√3 + i has modulus 2 and argument π/6, so the sixth power is 2⁶ cis(6 × π/6) = 64 cis(π) = −64.

Given z = 2e^(iπ/3), determine z³.

• 8
• 8i
• 4
• −8

Answer: −8

z³ = 2³ e^(iπ) = 8(cos π + i sin π) = −8. The modulus is cubed and the argument is tripled.

Let z₁ = cis(3π/4) and z₂ = cis(π/2). Determine the principal argument of z₁z₂, where −π < arg z ≤ π.

• 5π/4
• 3π/4
• −3π/4
• π/4

Answer: −3π/4

Arguments add: 3π/4 + π/2 = 5π/4, which lies outside (−π, π], so subtract 2π to get the principal value −3π/4.