Complex Numbers: Cartesian, Polar, and Exponential Forms
A complex number z = a + bi consists of a real part a and an imaginary part b, where i = √(−1). In Cartesian form, complex arithmetic follows standard algebra rules with the substitution i² = −1. Multiplication: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. Division requires multiplying numerator and denominator by the complex conjugate of the denominator: (a + bi)/(c + di) × (c − di)/(c − di), which eliminates the imaginary denominator. The complex conjugate z* = a − bi satisfies z × z* = a² + b² = |z|², the square of the modulus. In polar form, z = r(cos θ + i sin θ), where r = |z| = √(a² + b²) is the modulus and θ = arg(z) = arctan(b/a) is the argument (taking care to place θ in the correct quadrant using the signs of a and b). Multiplication in polar form is elegant: multiply moduli and add arguments — |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg(z₁) + arg(z₂). This geometric interpretation reveals that multiplying by a complex number of modulus 1 is a pure rotation. De Moivre's Theorem states (r(cos θ + i sin θ))ⁿ = rⁿ(cos nθ + i sin nθ), allowing rapid computation of powers and roots of complex numbers — a key Further Pure topic tested in AQA FP1 and Edexcel Further Maths. The exponential form z = re^(iθ) (Euler's formula) unifies all three representations and is the foundation for A-Level Further Maths university entrance questions.