Complex Arithmetic

Why

We want to add and multiply complex numbers.¹

Definition

Let $z_1, z_2 \in \C $ with $z_1 = (x_1, y_1)$ and $z_2 = (x_2, y_2)$. The complex product of $z_1$ and $z_2$ is the complex number $(x_1x_2 - y_1y_2, x_1y_2 + y_1x_2)$.

Properties

Relaton to $\R ^2$

Addition in $\C $ corresponds to the usual addition of the corresponding vectors in the plane $\R ^2$. In other words, it corresponds to element-wise addition. However multiplication in $\C $ is not componentwise multiplication in $\R ^2$.

1. Future editions will expand in the genetic account for introducing complex numbers. ↩︎