A collation of representations
A collation of representations
IB Year 6 | Grade 12
We now revisit the tabular collation of the representations of a complex number:
| Vectorial data | New name in Argand diagram | Notation | Formula |
|---|---|---|---|
| Position vector of \(Z\) | Complex number | \(z = x + yi\) | N. A. |
| Horizontal component of \(\overrightarrow{OZ} = \begin{pmatrix} x \\ y \end{pmatrix}\) | Real part | \(\mathrm{Re}(z)\) | \(x\) |
| Vertical component of \(\overrightarrow{OZ} = \begin{pmatrix} x \\ y \end{pmatrix}\) | Imaginary part | \(\mathrm{Im}(z)\) | \(y\) |
| Length of \(\overrightarrow{OZ}\) | Modulus | \(|z| = r\) | \(\sqrt{x^2 + y^2}\) |
| Angle made with positive \(x\)-axis | Argument | \(\arg(z) = \theta\) | Not so simple |
Since \(x = r \cos \theta\) and \(y = r \sin \theta\), where \(r \neq 0\), it follows that
\[z = x + iy = (r \cos \theta) + i(r \sin \theta) = r(\cos \theta + i \sin \theta).\]
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