Complex.swift

struct Complex: Equatable {
    let real: Double
    let imm: Double
}

extension Complex: CustomStringConvertible {
    var description: String { return "\(real) + \(imm)i" }
}

extension Complex: TwoDimensions {
    typealias ComponentVal = Double
    static func make2D(_ fst: Double, _ snd: Double) -> Complex {
        return Complex(real: fst, imm: snd)
    }
    var fst: Double { return real }
    var snd: Double { return imm }
}

extension Complex: Additive {}

extension Complex: Scalar { typealias ScalarVal = Double }

extension Complex: Magnitude {}

// Complex numbers have a multiplication such that
// (a + bi) * (c + di) = a * c + (a * d)i + (b * c)i + (b * d)i²
// since i² = -1 we have
// (a + bi) * (c + di) = a * c - b * d + (a * d + b * c)
extension Complex: Multiplicative {
    static var multId: Complex {
        return Complex(real: 1.0 , imm: 0.0)
    }
    static func * (_ lhs: Complex, _ rhs: Complex) -> Complex {
        return Complex(real: lhs.real * rhs.real - lhs.imm * rhs.imm,
                       imm:  lhs.real * rhs.imm  + lhs.imm * rhs.real)
    }
}