Definition:Complex Number/Real Part

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Let $z = a + i b$ be a complex number.

The real part of $z$ is the coefficient $a$.

The real part of a complex number $z$ is usually denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ by $\map \Re z$ or $\mathop \Re z$.

Polar Form

Let $z$ be a complex number expressed in polar form:

$z = \polar {r, \theta}$

The real part of $z$ is:

$\map \Re z = r \cos \theta$

Also denoted as

Variants of $\map \Re z$ for the real part of a complex number $z$ are:

$\map {\mathrm {Re} } z$
$\mathrm {Re} \set z$
$\map {\mathscr R} z$
$\map {\mathrm {re} } z$
$\map {\mathfrak R} z$
$\map {\mathbf R} z$ or $\mathbf R z$

While the fraktur font is falling out of fashion, because of its cumbersome appearance and difficulty to render in longhand, its use for this application is conveniently unambiguous.

Also see

  • Results about real parts can be found here.