The imaginary part of a complex number 
with 
, denoted ![Im[z]](/images/equations/ImaginaryPart/Inline3.svg)
in this work, is the real number
multiplying i, so ![Im[x+iy]=y](/images/equations/ImaginaryPart/Inline4.svg)
. In terms of 
itself,
![Im[z]=(z-z^_)/(2i),](/images/equations/ImaginaryPart/NumberedEquation1.svg) |
where 
is the complex conjugate of 
.
Geometrically, the imaginary part is the signed coordinate of the orthogonal projection of 
onto the imaginary axis
in the complex plane. The projected point
is ![iIm[z]](/images/equations/ImaginaryPart/Inline9.svg)
.
For the operator symbols, two typographic conventions are common. The upright two-letter abbreviation 
is used by ISO 80000-2 and the NIST Digital Library of Mathematical Functions. By
contrast, the traditional plain TeX control sequences \Im and \Re
render as the black-letter symbols identified by Unicode as U+2111 BLACK-LETTER CAPITAL
I and U+211C BLACK-LETTER CAPITAL R. The Computer Modern source names these glyphs
"Uppercase Fraktur I" and "Uppercase Fraktur R." Unicode separately
encodes the script or calligraphic capitals U+2110 SCRIPT CAPITAL I and U+211B SCRIPT
CAPITAL R.
Separately, two argument-delimiter conventions are common. ISO normally places a function argument
in parentheses, but permits the parentheses
to be omitted after a multi-letter function name when
the argument is simple and unambiguous. This accounts
for both 
and 
. Some mathematical publishing styles use square
brackets as alternative delimiters, especially around an expression containing
parentheses. ISO instead recommends parentheses
for grouping because square brackets can have other
meanings.
The imaginary part is implemented in the Wolfram Language as Im[z]
and displayed in traditional form formatting as 
.
Angela Suba uses the terminology in the literary metaphor "Life is a complex with real and imaginary parts" (Suba 2012, quoted on Goodreads).
