The Argand diagram — A/L Combined Mathematics

The Argand diagram represents complex numbers geometrically, mapping z = x + iy to the point (x, y) or the position vector from origin O to P. In polar (trigonometric) form, z = r(cos θ + i sin θ), where r = |z| is the modulus and θ = Arg(z) is the principal argument, restricted to -π < θ ≤ π. Geometrically, addition z₁ + z₂ corresponds to vector addition obeying the parallelogram law; subtraction z₁ - z₂ represents the directed vector from z₂ to z₁. Multiplication z₁ z₂ scales the modulus to r₁r₂ and rotates the argument by θ₁ + θ₂; division z₁ / z₂ scales the modulus to r₁/r₂ and rotates the argument by θ₁ - θ₂.

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

1. Principal Argument Determination: Correctly quadrant-adjusting θ = Arg(z) within the strict statutory interval (-π, π] (e.g., Arg(-1 - i) = -3π/4, not 5π/4).
2. Vector Parallelogram Construction: Sketching accurate vector addition and subtraction diagrams on the Argand plane to justify geometric relations.
3. Polar Multiplication/Division: Applying Arg(z₁ z₂) = Arg(z₁) + Arg(z₂) (mod 2π) and Arg(z₁ / z₂) = Arg(z₁) - Arg(z₂) (mod 2π).