<p>We now revisit the tabular collation of the representations of a complex number:</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start">Vectorial data</th><th style="text-align: left; --cell-justify: flex-start">New name in Argand diagram</th><th style="text-align: left; --cell-justify: flex-start">Notation</th><th style="text-align: left; --cell-justify: flex-start">Formula</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">Position vector of \(Z\)</td><td style="text-align: left; --cell-justify: flex-start">Complex number</td><td style="text-align: left; --cell-justify: flex-start">\(z = x + yi\)</td><td style="text-align: left; --cell-justify: flex-start">N. A.</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Horizontal component of \(\overrightarrow{OZ} = \begin{pmatrix} x \\ y \end{pmatrix}\)</td><td style="text-align: left; --cell-justify: flex-start">Real part</td><td style="text-align: left; --cell-justify: flex-start">\(\mathrm{Re}(z)\)</td><td style="text-align: left; --cell-justify: flex-start">\(x\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Vertical component of \(\overrightarrow{OZ} = \begin{pmatrix} x \\ y \end{pmatrix}\)</td><td style="text-align: left; --cell-justify: flex-start">Imaginary part</td><td style="text-align: left; --cell-justify: flex-start">\(\mathrm{Im}(z)\)</td><td style="text-align: left; --cell-justify: flex-start">\(y\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Length of \(\overrightarrow{OZ}\)</td><td style="text-align: left; --cell-justify: flex-start"><strong>Modulus</strong></td><td style="text-align: left; --cell-justify: flex-start">\(|z| = r\)</td><td style="text-align: left; --cell-justify: flex-start">\(\sqrt{x^2 + y^2}\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">Angle made with positive \(x\)-axis</td><td style="text-align: left; --cell-justify: flex-start"><strong>Argument</strong></td><td style="text-align: left; --cell-justify: flex-start">\(\arg(z) = \theta\)</td><td style="text-align: left; --cell-justify: flex-start">Not so simple</td></tr></tbody></table></div></div>
<p>Since&nbsp;\(x = r \cos \theta\) and \(y = r \sin \theta\), where&nbsp;\(r \neq 0\), it follows that</p>
<p style="text-align: center">\[z = x + iy = (r \cos \theta) + i(r \sin \theta) = r(\cos \theta + i \sin \theta).\]</p>

We now revisit the tabular collation of the representations of a complex number:

Since \(x = r \cos \theta\) and \(y = r \sin \theta\), where \(r \neq 0\), it follows that

\[z = x + iy = (r \cos \theta) + i(r \sin \theta) = r(\cos \theta + i \sin \theta).\]

A collation of representations

A collation of representations