Equality of complex numbers

Two complex numbers \(z = x + yi\) and \(w = u + vi\) are equal, denoted by \(z = w\), if

\[x = u \text{ and}\hspace{0.5em} y = v,\]

or equivalently, \(\mathrm{Re}(z) = \mathrm{Re}(w)\) and \(\mathrm{Im}(z) = \mathrm{Im}(w)\).

Remarks. The equality of complex numbers is the same as the equality of the corresponding representing \(2\)D vectors. That is, if \(z\) is identified with the position vector \(\begin{pmatrix} x \\ y \end{pmatrix}\) and \(w\) is identified with \(\begin{pmatrix} u \\ v \end{pmatrix}\), then we say that \(z = w\) if and only if

\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} u \\ v \end{pmatrix} \iff x = u \text{ and}\hspace{0.5em} y = v.\]