Fool Proof Mathematics/CP1/Complex numbers

Consider a mathematics which only allows for positive numbers. This drastically restricts the solutions obtainable such as being unable to answer "What is ?". A similar problem arises with polynomials, as without expanding our system of numbers some polynomials have no solution. A complex number encapsulates all real numbers as well as introducing the imaginary unit, , such that a complex number has 2 components: where . is an imaginary number. Two functions are introduced to distinguish between the 2 componentsː . Addition and subtraction of complex numbers works the same way as algebraic and root manipulation, the only difference to bear in mind is thatː which is a real number, allowing further simplification. As previously alluded to, quadratics now have solutions for when the discriminant is less than 0ː means the quadratic has 2 distinct complex solutions. Worked Examples Solve the equation ː The complex conjugate of a complex number has the same real part but the inverse of its' imaginary part. This allows the us to utilize the difference of two squares identity, making the product of 2 complex numbers realːWe can use an already learnt trick of rationalizing the denominator to divide 2 complex numbers by multiplying the fraction by the denominators' complex conjugate, i.e: .