Modulus of a complex number — A/L Combined Mathematics

The modulus of a complex number z = x + iy is defined as the non-negative real distance from the origin to z on the Argand plane: |z| = √(x² + y²) = √(z z̄). Key modulus properties include: (1) |z| ≥ 0, with |z| = 0 ⟺ z = 0; (2) |z| = |z̄| = |-z|; (3) |z₁ z₂| = |z₁| |z₂| and |z₁ / z₂| = |z₁| / |z₂|; (4) |zⁿ| = |z|^n for n ∈ ℤ; (5) The Triangle Inequality: |z₁ + z₂| ≤ |z₁| + |z₂|, with equality holding if and only if the origin and the points z₁, z₂ are collinear with z₁ and z₂ lying on the same ray; (6) Lower bound: ||z₁| - |z₂|| ≤ |z₁ - z₂|.

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

1. Triangle Inequality Proof: Proving |z₁ + z₂|² = (z₁ + z₂)(z̄₁ + z̄₂) = |z₁|² + |z₂|² + 2 Re(z₁ z̄₂) ≤ |z₁|² + |z₂|² + 2|z₁||z₂| = (|z₁| + |z₂|)².
2. Modulus Product Theorem: Applying |z₁ z₂| = |z₁||z₂| without expanding full algebraic products.
3. Maximum/Minimum Range Problems: Finding the maximum and minimum values of |z - z₀| using ||z| - |z₀|| ≤ |z - z₀| ≤ |z| + |z₀| when |z| is fixed.