Can any complex equation be seperated into two equations?

Question

Considering that √-1 is i, but negative numbers are actually positive numbers that indicate an opposite direction or quantity, can any complex equation be seperated into two positive equations? If so, is there a relationship between the answers of the two equations?

Answer 1

Yes, it can. Any complex equation $$A=B$$ where $A$ and $B$ are arbitrary complex expressions is equivalent to the pair of equations

$$\mathrm{Re}(A)=\mathrm{Re}(B)\\ \mathrm{Im}(A) = \mathrm{Im}(B)$$ where $\mathrm{Re}(z)$ describes the real part of $z$ and $\mathrm{Im}(z)$ is the imaginary part of $z$.

The equation is equivalent to the pair of equations in the sense that if the equation is true, then the pair of equations is true, and if the pair of equations is true, then the equation is also true.