1.2 Review of arithmetic in the complex numbers
I assume that you know what the complex numbers are and how to add, subtract and multiply then. But, just in case, here’s a review.
Suppose \(x_1, y_1, x_2, y_2\in \mathbb {R}\), and, for \(j = 1, 2\), set
\begin{equation} \label{2cx} z_j = x_j + iy_j\in \mathbb {C}. \end{equation}
1.1
Then, by definition,
\begin{align} z_1 + z_2 & = (x_1 + x_2) + i(y_1 + y_2)\\ z_1 - z_2 & = (x_1 - x_2) + i(y_1 - y_2)\\ z_1z_2 & = (x_1x_2 - y_1y_2) + i(x_1 y_2 + x_2 y_1). \end{align}
We also write \(0 = 0 + i0\), \(1 = 1 + i0\) and \(i = 0 + i1\). A consequence of this is then that \(i^2 = -1\). Here are a few more consequences.
Examples
1.5
We have \((1+i)^2 = (1+i)(1+i) = (1-1) + i + i = 2i\).
We have \((1+i) + (2+3i) = 3 + 4i\).
We have \((1+i) - (2+3i) = -1 -2i\).