Algebra of complex numbers — A/L Combined Mathematics

Fundamental algebraic operations on complex numbers follow standard field axioms with i² = -1: (1) Addition: (x₁ + iy₁) + (x₂ + iy₂) = (x₁ + x₂) + i(y₁ + y₂); (2) Subtraction: (x₁ + iy₁) - (x₂ + iy₂) = (x₁ - x₂) + i(y₁ - y₂); (3) Multiplication: (x₁ + iy₁)(x₂ + iy₂) = (x₁x₂ - y₁y₂) + i(x₁y₂ + x₂y₁); (4) Division: The quotient z₁ / z₂ (for z₂ ≠ 0) is evaluated by rationalizing the denominator, multiplying numerator and denominator by the complex conjugate z̄₂ = x₂ - iy₂: z₁/z₂ = [(x₁ + iy₁)(x₂ - iy₂)] / (x₂² + y₂²).

Check before you answer (A/L Pure Maths Paper 1)

1. Denominator Realization: Multiplying both numerator and denominator by the complex conjugate of the denominator when simplifying fractional complex numbers.
2. Cartesian Form Representation: Expressing final answers strictly in the standard form a + ib, clearly separating real and imaginary terms.
3. Square Roots of a Complex Number: Setting (u + iv)² = x + iy and solving the resulting real system u² - v² = x and 2uv = y to find both square roots.