Complex Numbers - Introduction to Complex Numbers

Figure \(1\). Argand diagram representing the complex number \(z\).

Once the Argand diagram is given, some essential vectorial data are readily available:

1. The real part \(x\) of \(z=x+yi\) is the horizontal component of the vector \(\overrightarrow{OZ}=\begin{pmatrix} x \\ y \end{pmatrix}\).
2. The imaginary part \(y\) of \(z=x+yi\) is the vertical component of the vector \(\overrightarrow{OZ}=\begin{pmatrix} x \\ y \end{pmatrix}\).
3. The length \(r\) of the vector \(\overrightarrow{OZ}=\begin{pmatrix} x \\ y \end{pmatrix}\) is \(|\overrightarrow{OZ}|=\sqrt{x^2+y^2}\).
4. The angle made by the vector \(\overrightarrow{OZ}=\begin{pmatrix} x \\ y \end{pmatrix}\) with the positive \(x\)-axis is labelled by \(\theta\).