Complex Numbers Flashcards

If z = 3 + 4i, what is |z|, the modulus of z?

Answer: 5

The modulus is √(3² + 4²) = √(9 + 16) = √25 = 5.

What is the product (2 + 3i)(1 − 2i)?

Answer: 8 − i

Expanding: 2 − 4i + 3i − 6i² = 2 − i + 6 = 8 − i, since i² = −1.

What is i^23?

Answer: −i

Since i repeats with period 4 and 23 ≡ 3 (mod 4), i^23 = i^3 = −i.

If z = 1 + i, what is z^4?

Answer: −4

z² = (1+i)² = 2i, so z⁴ = (2i)² = 4i² = −4.

What is the real part of (3 + 4i) / (1 + 2i)?

Answer: 11/5

Multiply numerator and denominator by the conjugate (1−2i): (3+4i)(1−2i)/5 = (11−2i)/5, giving real part 11/5.

If z₁ = 1 + 2i and z₂ = 2 − i, what is z₁ + z₂?

Answer: 3 + i

Add real and imaginary parts separately: (1+2) + (2−1)i = 3 + i.

If z = cos(π/6) + i·sin(π/6), what is z + z̄ (z plus its complex conjugate)?

Answer: √3

z = √3/2 + i/2 and z̄ = √3/2 − i/2, so z + z̄ = √3 (the imaginary parts cancel).