For polar forms z₁ = r₁(cosθ₁ + i sinθ₁) and z₂ = r₂(cosθ₂ + i sinθ₂),
z₁·z₂ = r₁r₂{cos(θ₁+θ₂) + i sin(θ₁+θ₂)} (Product: absolute values are multiplied and arguments are added)
z₁/z₂ = (r₁/r₂){cos(θ₁−θ₂) + i sin(θ₁−θ₂)} (Quotient: absolute values are divided and arguments are subtracted)
Geometrically, multiplying complex numbers represents scaling and rotation.
Example ① Calculate (1+i)(√3+i) as a product in polar form.
Solution:√2(π/4) · 2(π/6) = 2√2(cos(5π/12) + i sin(5π/12))
Solution:√2(π/4) · 2(π/6) = 2√2(cos(5π/12) + i sin(5π/12))
Key Points
- “Multiplication causes rotation” is the geometric meaning of complex numbers.
- Multiplying by i rotates a point by π/2.
Practice Problems
- z = √2(cos(π/6)+i sin(π/6)) and w = √3(cos(π/3)+i sin(π/3)). Find zw.
Solution and Explanation
- Answer:√6(cos(π/2)+i sin(π/2)) = √6·i