Complex number (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Complex number" in English language version.

Last modified: 11:39:09, 24 June 2026 (UTC)

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Campbell, George Ashley (April 1911). "Cisoidal oscillations" (PDF). Proceedings of the American Institute of Electrical Engineers. XXX (1–6). American Institute of Electrical Engineers: 789–824 [Fig. 13 on p. 810]. Bibcode:1911PAIEE..30d.789C. doi:10.1109/PAIEE.1911.6659711. S2CID 51647814. Retrieved 24 June 2023. p. 789: The use of i (or Greek ı) for the imaginary symbol is nearly universal in mathematical work, which is a very strong reason for retaining it in the applications of mathematics in electrical engineering. Aside, however, from the matter of established conventions and facility of reference to mathematical literature, the substitution of the symbol j is objectionable because of the vector terminology with which it has become associated in engineering literature, and also because of the confusion resulting from the divided practice of engineering writers, some using j for +i and others using j for −i.

ams.org (Global: 434^th place; English: 249^th place)

mathscinet.ams.org

Marker, David (1996). "Introduction to the Model Theory of Fields". In Marker, D.; Messmer, M.; Pillay, A. (eds.). Model theory of fields. Lecture Notes in Logic. Vol. 5. Berlin: Springer-Verlag. pp. 1–37. ISBN 978-3-540-60741-0. MR 1477154.
McCrimmon, Kevin (2004). A Taste of Jordan Algebras. Universitext. Springer. p. 64. ISBN 0-387-95447-3. MR 2014924

archive.org (Global: 6^th place; English: 6^th place)

"Complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales.", Penrose 2005, pp.72–73. Penrose, Roger (2005). The Road to Reality: A complete guide to the laws of the universe. Alfred A. Knopf. ISBN 978-0-679-45443-4.
Axler, Sheldon (2010). College algebra. Wiley. p. 262. ISBN 9780470470770.
Ahlfors 1979. Ahlfors, Lars (1979). Complex analysis (3rd ed.). McGraw-Hill. ISBN 978-0-07-000657-7.
Ahlfors 1979, p. 15. Ahlfors, Lars (1979). Complex analysis (3rd ed.). McGraw-Hill. ISBN 978-0-07-000657-7.
Ahlfors 1979, p. 16. Ahlfors, Lars (1979). Complex analysis (3rd ed.). McGraw-Hill. ISBN 978-0-07-000657-7.
Descartes, René (1954) [1637]. La Géométrie | The Geometry of René Descartes with a facsimile of the first edition. Dover Publications. ISBN 978-0-486-60068-0. Retrieved 20 April 2011. {{cite book}}: ISBN / Date incompatibility (help)
Mourey, C.V. (1861). La vraies théore des quantités négatives et des quantités prétendues imaginaires [The true theory of negative quantities and of alleged imaginary quantities] (in French). Paris, France: Mallet-Bachelier. 1861 reprint of 1828 original.
Warren, John (1828). A Treatise on the Geometrical Representation of the Square Roots of Negative Quantities. Cambridge, England: Cambridge University Press.
Cauchy, Augustin-Louis (1821). Cours d'analyse de l'École royale polytechnique (in French). Vol. 1. Paris, France: L'Imprimerie Royale. p. 183.
Ahlfors 1979, p. 46. Ahlfors, Lars (1979). Complex analysis (3rd ed.). McGraw-Hill. ISBN 978-0-07-000657-7.

books.google.com (Global: 3^rd place; English: 3^rd place)

Aufmann, Barker & Nation 2007, p. 66, Chapter P Aufmann, Richard N.; Barker, Vernon C.; Nation, Richard D. (2007). College Algebra and Trigonometry (6 ed.). Cengage Learning. ISBN 978-0-618-82515-8.
William Ford (2014). Numerical Linear Algebra with Applications: Using MATLAB and Octave (reprinted ed.). Academic Press. p. 570. ISBN 978-0-12-394784-0. Extract of page 570
Dennis Zill; Jacqueline Dewar (2011). Precalculus with Calculus Previews: Expanded Volume (revised ed.). Jones & Bartlett Learning. p. 37. ISBN 978-0-7637-6631-3. Extract of page 37
Kasana, H.S. (2005). "Chapter 1". Complex Variables: Theory And Applications (2nd ed.). PHI Learning Pvt. Ltd. p. 14. ISBN 978-81-203-2641-5.
Nilsson, James William; Riedel, Susan A. (2008). "Chapter 9". Electric circuits (8th ed.). Prentice Hall. p. 338. ISBN 978-0-13-198925-2.
Lloyd James Peter Kilford (2015). Modular Forms: A Classical And Computational Introduction (2nd ed.). World Scientific Publishing Company. p. 112. ISBN 978-1-78326-547-3. Extract of page 112
Cynthia Y. Young (2017). Trigonometry (4th ed.). John Wiley & Sons. p. 406. ISBN 978-1-119-44520-3. Extract of page 406
Joseph Mazur (2016). Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers (reprinted ed.). Princeton University Press. p. 138. ISBN 978-0-691-17337-5. Extract of page 138
Bryan Bunch (2012). Mathematical Fallacies and Paradoxes (reprinted, revised ed.). Courier Corporation. p. 32. ISBN 978-0-486-13793-3. Extract of page 32
Euler, Leonhard (1748). Introductio in Analysin Infinitorum [Introduction to the Analysis of the Infinite] (in Latin). Vol. 1. Lucerne, Switzerland: Marc Michel Bosquet & Co. p. 104.
Gauss, Carl Friedrich (1799) "Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse." [New proof of the theorem that any rational integral algebraic function of a single variable can be resolved into real factors of the first or second degree.] Ph.D. thesis, University of Helmstedt, (Germany). (in Latin)
Ewald, William B. (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vol. 1. Oxford University Press. p. 313. ISBN 9780198505358. Retrieved 18 March 2020.
Caparrini, Sandro (2000). "On the Common Origin of Some of the Works on the Geometrical Interpretation of Complex Numbers". In Kim Williams (ed.). Two Cultures. Birkhäuser. p. 139. ISBN 978-3-7643-7186-9.
Hankel, Hermann (1867). Vorlesungen über die complexen Zahlen und ihre Functionen [Lectures About the Complex Numbers and Their Functions] (in German). Vol. 1. Leipzig, [Germany]: Leopold Voss. p. 71. From p. 71: "Wir werden den Factor (cos φ + i sin φ) haüfig den Richtungscoefficienten nennen." (We will often call the factor (cos φ + i sin φ) the "coefficient of direction".)

doi.org (Global: 2^nd place; English: 2^nd place)

Campbell, George Ashley (April 1911). "Cisoidal oscillations" (PDF). Proceedings of the American Institute of Electrical Engineers. XXX (1–6). American Institute of Electrical Engineers: 789–824 [Fig. 13 on p. 810]. Bibcode:1911PAIEE..30d.789C. doi:10.1109/PAIEE.1911.6659711. S2CID 51647814. Retrieved 24 June 2023. p. 789: The use of i (or Greek ı) for the imaginary symbol is nearly universal in mathematical work, which is a very strong reason for retaining it in the applications of mathematics in electrical engineering. Aside, however, from the matter of established conventions and facility of reference to mathematical literature, the substitution of the symbol j is objectionable because of the vector terminology with which it has become associated in engineering literature, and also because of the confusion resulting from the divided practice of engineering writers, some using j for +i and others using j for −i.
Buée (1806). "Mémoire sur les quantités imaginaires" [Memoir on imaginary quantities]. Philosophical Transactions of the Royal Society of London (in French). 96: 23–88. doi:10.1098/rstl.1806.0003. S2CID 110394048.
Warren, John (1829). "Consideration of the objections raised against the geometrical representation of the square roots of negative quantities". Philosophical Transactions of the Royal Society of London. 119 (119): 241–254. doi:10.1098/rstl.1829.0022. S2CID 186211638.
Warren, John (1829). "On the geometrical representation of the powers of quantities, whose indices involve the square roots of negative numbers". Philosophical Transactions of the Royal Society of London. 119: 339–359. doi:10.1098/rstl.1829.0031. S2CID 125699726.
Lester, J.A. (1994). "Triangles I: Shapes". Aequationes Mathematicae. 52: 30–54. doi:10.1007/BF01818325. S2CID 121095307.
Kalman, Dan (2008a). "An Elementary Proof of Marden's Theorem". American Mathematical Monthly. 115 (4): 330–38. doi:10.1080/00029890.2008.11920532. ISSN 0002-9890. S2CID 13222698. Archived from the original on 8 March 2012. Retrieved 1 January 2012.

education.fr (Global: low place; English: low place)

bibnum.education.fr

Argand (1806). Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques [Essay on a way to represent complex quantities by geometric constructions] (in French). Paris, France: Madame Veuve Blanc.

encyclopediaofmath.org (Global: 7,049^th place; English: 5,456^th place)

Solomentsev 2001: "The plane $\mathbb {R} ^{2}$ whose points are identified with the elements of $\mathbb {C}$ is called the complex plane ... The complete geometric interpretation of complex numbers and operations on them appeared first in the work of C. Wessel (1799). The geometric representation of complex numbers, sometimes called the 'Argand diagram', came into use after the publication in 1806 and 1814 of papers by J.R. Argand, who rediscovered, largely independently, the findings of Wessel". Solomentsev, E.D. (2001) [1994], "Complex number", Encyclopedia of Mathematics, EMS Press

hathitrust.org (Global: 296^th place; English: 153^rd place)

babel.hathitrust.org

Argand 1814, p. 204 defines the modulus of a complex number but he doesn't name it:
"Dans ce qui suit, les accens, indifféremment placés, seront employés pour indiquer la grandeur absolue des quantités qu'ils affectent; ainsi, si $a=m+n{\sqrt {-1}}$ , $m$ et $n$ étant réels, on devra entendre que $a_{'}$ ou $a'={\sqrt {m^{2}+n^{2}}}$ ."
[In what follows, accent marks, wherever they're placed, will be used to indicate the absolute size of the quantities to which they're assigned; thus if $a=m+n{\sqrt {-1}}$ , $m$ and $n$ being real, one should understand that $a_{'}$ or $a'={\sqrt {m^{2}+n^{2}}}$ .]
Argand 1814, p. 208 defines and names the module and the direction factor of a complex number: "... $a={\sqrt {m^{2}+n^{2}}}$ pourrait être appelé le module de $a+b{\sqrt {-1}}$ , et représenterait la grandeur absolue de la ligne $a+b{\sqrt {-1}}$ , tandis que l'autre facteur, dont le module est l'unité, en représenterait la direction."
[... $a={\sqrt {m^{2}+n^{2}}}$ could be called the module of $a+b{\sqrt {-1}}$ and would represent the absolute size of the line $a+b{\sqrt {-1}}\,,$ (Argand represented complex numbers as vectors.) whereas the other factor [namely, ${\tfrac {a}{\sqrt {a^{2}+b^{2}}}}+{\tfrac {b}{\sqrt {a^{2}+b^{2}}}}{\sqrt {-1}}$ ], whose module is unity [1], would represent its direction.] Argand (1814). "Reflexions sur la nouvelle théorie des imaginaires, suives d'une application à la demonstration d'un theorème d'analise" [Reflections on the new theory of complex numbers, followed by an application to the proof of a theorem of analysis]. Annales de mathématiques pures et appliquées (in French). 5: 197–209. Argand (1814). "Reflexions sur la nouvelle théorie des imaginaires, suives d'une application à la demonstration d'un theorème d'analise" [Reflections on the new theory of complex numbers, followed by an application to the proof of a theorem of analysis]. Annales de mathématiques pures et appliquées (in French). 5: 197–209.
Hamilton, Wm. (1844). "On a new species of imaginary quantities connected with a theory of quaternions". Proceedings of the Royal Irish Academy. 2: 424–434.
Wessel, Caspar (1799). "Om Directionens analytiske Betegning, et Forsog, anvendt fornemmelig til plane og sphæriske Polygoners Oplosning" [On the analytic representation of direction, an effort applied in particular to the determination of plane and spherical polygons]. Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter [New Collection of the Writings of the Royal Danish Science Society] (in Danish). 5: 469–518.
Gauss 1831. Gauss, C. F. (1831). "Theoria residuorum biquadraticorum. Commentatio secunda" [Theory of biquadratic residues. Second memoir.]. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores (in Latin). 7: 89–148.
Français, J.F. (1813). "Nouveaux principes de géométrie de position, et interprétation géométrique des symboles imaginaires" [New principles of the geometry of position, and geometric interpretation of complex [number] symbols]. Annales des mathématiques pures et appliquées (in French). 4: 61–71.
Gauss 1831, p. 96 Gauss, C. F. (1831). "Theoria residuorum biquadraticorum. Commentatio secunda" [Theory of biquadratic residues. Second memoir.]. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores (in Latin). 7: 89–148.
Gauss 1831, p. 96 Gauss, C. F. (1831). "Theoria residuorum biquadraticorum. Commentatio secunda" [Theory of biquadratic residues. Second memoir.]. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores (in Latin). 7: 89–148.
Gauss 1831, p. 98 Gauss, C. F. (1831). "Theoria residuorum biquadraticorum. Commentatio secunda" [Theory of biquadratic residues. Second memoir.]. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores (in Latin). 7: 89–148.

jmilne.org (Global: low place; English: low place)

Milne, James S. (2022), Fields and Galois Theory (PDF), 5.10, J.S. Milne, p. 63
Milne, James S. (2022), Fields and Galois Theory (PDF), 5.10, J.S. Milne, p. 17

maa.org (Global: 4,358^th place; English: 2,907^th place)

mathdl.maa.org

Kalman, Dan (2008a). "An Elementary Proof of Marden's Theorem". American Mathematical Monthly. 115 (4): 330–38. doi:10.1080/00029890.2008.11920532. ISSN 0002-9890. S2CID 13222698. Archived from the original on 8 March 2012. Retrieved 1 January 2012.
Kalman, Dan (2008b). "The Most Marvelous Theorem in Mathematics". Journal of Online Mathematics and Its Applications. Archived from the original on 8 February 2012. Retrieved 1 January 2012.

mathforum.org (Global: low place; English: low place)

Nahin, Paul J. (2007). An Imaginary Tale: The Story of √−1. Princeton University Press. ISBN 978-0-691-12798-9. Archived from the original on 12 October 2012. Retrieved 20 April 2011.

mpg.de (Global: 1,722^nd place; English: 1,710^th place)

echo.mpiwg-berlin.mpg.de

Wallis, John (1685). A Treatise of Algebra, Both Historical and Practical ... London, England: printed by John Playford, for Richard Davis. pp. 264–273.

projecteuclid.org (Global: 4,781^st place; English: 3,117^th place)

Marker, David (1996). "Introduction to the Model Theory of Fields". In Marker, D.; Messmer, M.; Pillay, A. (eds.). Model theory of fields. Lecture Notes in Logic. Vol. 5. Berlin: Springer-Verlag. pp. 1–37. ISBN 978-3-540-60741-0. MR 1477154.

royalsocietypublishing.org (Global: 1,880^th place; English: 2,810^th place)

Buée (1806). "Mémoire sur les quantités imaginaires" [Memoir on imaginary quantities]. Philosophical Transactions of the Royal Society of London (in French). 96: 23–88. doi:10.1098/rstl.1806.0003. S2CID 110394048.

semanticscholar.org (Global: 15^th place; English: 8^th place)

api.semanticscholar.org

Campbell, George Ashley (April 1911). "Cisoidal oscillations" (PDF). Proceedings of the American Institute of Electrical Engineers. XXX (1–6). American Institute of Electrical Engineers: 789–824 [Fig. 13 on p. 810]. Bibcode:1911PAIEE..30d.789C. doi:10.1109/PAIEE.1911.6659711. S2CID 51647814. Retrieved 24 June 2023. p. 789: The use of i (or Greek ı) for the imaginary symbol is nearly universal in mathematical work, which is a very strong reason for retaining it in the applications of mathematics in electrical engineering. Aside, however, from the matter of established conventions and facility of reference to mathematical literature, the substitution of the symbol j is objectionable because of the vector terminology with which it has become associated in engineering literature, and also because of the confusion resulting from the divided practice of engineering writers, some using j for +i and others using j for −i.
Buée (1806). "Mémoire sur les quantités imaginaires" [Memoir on imaginary quantities]. Philosophical Transactions of the Royal Society of London (in French). 96: 23–88. doi:10.1098/rstl.1806.0003. S2CID 110394048.
Warren, John (1829). "Consideration of the objections raised against the geometrical representation of the square roots of negative quantities". Philosophical Transactions of the Royal Society of London. 119 (119): 241–254. doi:10.1098/rstl.1829.0022. S2CID 186211638.
Warren, John (1829). "On the geometrical representation of the powers of quantities, whose indices involve the square roots of negative numbers". Philosophical Transactions of the Royal Society of London. 119: 339–359. doi:10.1098/rstl.1829.0031. S2CID 125699726.
Lester, J.A. (1994). "Triangles I: Shapes". Aequationes Mathematicae. 52: 30–54. doi:10.1007/BF01818325. S2CID 121095307.
Kalman, Dan (2008a). "An Elementary Proof of Marden's Theorem". American Mathematical Monthly. 115 (4): 330–38. doi:10.1080/00029890.2008.11920532. ISSN 0002-9890. S2CID 13222698. Archived from the original on 8 March 2012. Retrieved 1 January 2012.

st-andrews.ac.uk (Global: 1,053^rd place; English: 1,274^th place)

mathshistory.st-andrews.ac.uk

"Adrien Quentin Buée (1745–1845): MacTutor".

us.archive.org (Global: 1,020^th place; English: 1,018^th place)

ia800708.us.archive.org

Campbell, George Ashley (April 1911). "Cisoidal oscillations" (PDF). Proceedings of the American Institute of Electrical Engineers. XXX (1–6). American Institute of Electrical Engineers: 789–824 [Fig. 13 on p. 810]. Bibcode:1911PAIEE..30d.789C. doi:10.1109/PAIEE.1911.6659711. S2CID 51647814. Retrieved 24 June 2023. p. 789: The use of i (or Greek ı) for the imaginary symbol is nearly universal in mathematical work, which is a very strong reason for retaining it in the applications of mathematics in electrical engineering. Aside, however, from the matter of established conventions and facility of reference to mathematical literature, the substitution of the symbol j is objectionable because of the vector terminology with which it has become associated in engineering literature, and also because of the confusion resulting from the divided practice of engineering writers, some using j for +i and others using j for −i.

web.archive.org (Global: 1^st place; English: 1^st place)

Nahin, Paul J. (2007). An Imaginary Tale: The Story of √−1. Princeton University Press. ISBN 978-0-691-12798-9. Archived from the original on 12 October 2012. Retrieved 20 April 2011.
Jeff Miller (21 September 1999). "MODULUS". Earliest Known Uses of Some of the Words of Mathematics (M). Archived from the original on 3 October 1999.
Kalman, Dan (2008a). "An Elementary Proof of Marden's Theorem". American Mathematical Monthly. 115 (4): 330–38. doi:10.1080/00029890.2008.11920532. ISSN 0002-9890. S2CID 13222698. Archived from the original on 8 March 2012. Retrieved 1 January 2012.
Kalman, Dan (2008b). "The Most Marvelous Theorem in Mathematics". Journal of Online Mathematics and Its Applications. Archived from the original on 8 February 2012. Retrieved 1 January 2012.

wolfram.com (Global: 1,091^st place; English: 1,169^th place)

mathworld.wolfram.com

Weisstein, Eric W. "Complex Number". mathworld.wolfram.com. Retrieved 12 August 2020.

worldcat.org (Global: 4^th place; English: 4^th place)

search.worldcat.org

Jurij., Kovič. Tristan Needham, Visual Complex Analysis, Oxford University Press Inc., New York, 1998, 592 strani. OCLC 1080410598.
Kalman, Dan (2008a). "An Elementary Proof of Marden's Theorem". American Mathematical Monthly. 115 (4): 330–38. doi:10.1080/00029890.2008.11920532. ISSN 0002-9890. S2CID 13222698. Archived from the original on 8 March 2012. Retrieved 1 January 2012.

Complex number (English Wikipedia)

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