In this part, we introduce the number system of complex numbers. In the set of real numbers $\mathbb R,$ which has been defined as the set of all limits of rational Cauchy sequences, it is not possible to solve all polynomial equations. The simplest polynomial equation, for which there is no real number solving it is the equation $$x^2 + 1 = 0.$$
Complex numbers $\mathbb C$ allow to find the zeros of this polynomial and it turns out that the set of complex numbers is sufficient to ensure that all polynomial equations can be solved.
Definitions: 1
Parts: 2 3
Proofs: 4 5
Propositions: 6 7