Complex conjugate — A/L Combined Mathematics

The complex conjugate of a complex number z = x + iy is defined as z̄ = x - iy, formed by reflecting z across the real axis on the Argand plane. Fundamental properties include: (1) z̿ = z; (2) z + z̄ = 2 Re(z) and z - z̄ = 2i Im(z); (3) z is purely real ⟺ z̄ = z, and z is purely imaginary ⟺ z̄ = -z; (4) z z̄ = x² + y² = |z|²; (5) Conjugate of sum and product: (z₁ ± z₂)⁻ = z̄₁ ± z̄₂, (z₁ z₂)⁻ = z̄₁ z̄₂, and (z₁ / z₂)⁻ = z̄₁ / z̄₂. If a real polynomial P(z) = 0 has a complex root α, its conjugate ᾱ is also a root (Conjugate Root Theorem).

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

1. Proof of Distributive Properties: Proving algebraically that (z₁ z₂)⁻ = z̄₁ z̄₂ by expanding both left-hand and right-hand sides in terms of coordinates.
2. Purely Real Criterion Application: Proving an algebraic expression w is real by establishing that w̄ = w.
3. Conjugate Roots of Real Polynomials: Utilizing the fact that non-real roots of polynomials with real coefficients occur in complex conjugate pairs.