The imaginary part of a complex number with , denoted in this work, is the real number multiplying i, so . In terms of itself,

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 .

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).

See also

Absolute Square, Complex Argument, Complex Conjugate, Complex Modulus, Complex Plane, Imaginary Number, Purely Imaginary Number, Real Part, Sign

Related Wolfram sites

https://functions.wolfram.com/ComplexComponents/Im/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 16, 1972.International Organization for Standardization. Quantities and units--Part 2: Mathematics. ISO 80000-2:2019(E), 2nd ed., corrected version 2021-11. Geneva, Switzerland: ISO, pp. 1 and 18, 2019. https://www.iso.org/standard/64973.html.Knuth, D. E. plain.tex. https://mirrors.ctan.org/systems/knuth/dist/lib/plain.tex.Knuth, D. E. symbol.mf. https://mirrors.ctan.org/systems/knuth/dist/cm/symbol.mf.Krantz, S. G. Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 2, 1999.National Institute of Standards and Technology. "Calculus of a Complex Variable." §1.9(i) in Digital Library of Mathematical Functions. https://dlmf.nist.gov/1.9#i.Suba, A. From My Pen, On Life. 2012. Quoted on Goodreads at https://www.goodreads.com/quotes/530740-life-is-a-complex-with-real-and-imaginary-parts.Unicode Consortium. "Letterlike Symbols." In The Unicode Standard, Version 17.0. https://www.unicode.org/Public/17.0.0/charts/PDF/U2100.pdf.Wolfram Research. "TraditionalForm." https://reference.wolfram.com/language/ref/TraditionalForm.html.

Referenced on Wolfram|Alpha

Imaginary Part

Cite this as:

Weisstein, Eric W. "Imaginary Part." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ImaginaryPart.html

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