Argand diagram representing a complex number

By the way addition, subtraction and scalar multiplication of complex numbers are defined, it is apparent that these operations are just like those defined on \(2\)D vectors. Based on this consideration, it makes a lot of sense to represent all these operations, i.e., complex addition, complex subtraction and scalar multiplication in a \(2\)D diagram – called an Argand diagram.

An Argand diagram representation of a complex number \(z = x + yi\) is just the point \(Z\) whose position vector with respect to a given origin \(O\) given by
\[ \begin{pmatrix} x \\ y \end{pmatrix}. \]
In other words, \(z = x + yi\) is to be identified as the point \(Z\) whose coordinates are \((x, y)\).
The Argand diagram representing \(z = x + yi\) is given in Figure \(1\) below.

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

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