Definition:Complex Number/Imaginary Part

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

The imaginary part of $z$ is the coefficient $b$ (note: not $i b$).

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

Polar Form

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

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

The imaginary part of $z$ is:

$\map \Im z = r \sin \theta$

Also denoted as

Variants of $\map \Im z$ for the imaginary part of a complex number $z$ are:

$\map {\mathrm {Im} } z$
$\mathrm {Im} \set z$
$\map {\mathscr I} z$
$\map {\mathrm {im} } z$
$\map {\mathfrak I} z$
$\map {\mathbf I} z$ or $\mathbf I z$

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 imaginary parts can be found here.