The complex number system — A/L Combined Mathematics

The complex number system extends real numbers to provide solutions to polynomial equations with negative discriminants, introducing the imaginary unit i defined by i² = -1 (or i = √(-1)). A complex number is uniquely expressed in Cartesian (standard) form as z = x + iy, where x, y ∈ ℝ. The real part of z is denoted by Re(z) = x and the imaginary part by Im(z) = y. If y = 0, z is purely real; if x = 0 and y ≠ 0, z is purely imaginary. Two complex numbers z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ are equal if and only if their real parts and imaginary parts are independently equal: x₁ = x₂ and y₁ = y₂.

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

1. Equality Equating Technique: Equating real and imaginary parts separately from an equation like f(z) = 0 to form two independent real equations in x and y.
2. Powers of i Reduction: Reducing high integer powers of i using the cyclic property: i¹ = i, i² = -1, i³ = -i, i⁴ = 1.
3. Purely Real/Imaginary Conditions: Enforcing Im(z) = 0 for purely real conditions and Re(z) = 0 for purely imaginary conditions.