2.0 Algebraic operations

In performing operations with complex numbers we can proceed as in the algebra of real numbers and replacing ${i^2}$ by $-1$ and using various results of iota, as explained in the previous section, when it occurs.

Addition: $$(a + ib) + (c + id) = (a + c) + i(b + d)$$

Subtraction: $$(a + ib) - (c + id) = (a - c) + i(b - d)$$

Multiplication: $$(a + ib)(c + id) = (ac - bd) + i(ad + bc)$$

Division: $$\begin{equation} \begin{aligned} \frac{{a + ib}}{{c + id}} = \frac{{a + ib}}{{c + id}}.\frac{{c - id}}{{c - id}} = \frac{{ac - iad + ibc - {i^2}bd}}{{{c^2} - {i^2}{d^2}}} \\ {\text{ = }}\frac{{ac + bd + (bc - ad)i}}{{{c^2} + {d^2}}} = \frac{{ac + bd}}{{{c^2} + {d^2}}} + \frac{{bc - ad}}{{{c^2} + {d^2}}}i \\\end{aligned} \end{equation} $$