Natural number (English Wikipedia)

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

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17books.google.com

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10web.archive.org

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8archive.org

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1encyclopediaofmath.org

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ams.org (Global: 451^st place; English: 277^th place)

mathscinet.ams.org

Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4 (1): 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856.
Baratella, Stefano; Ferro, Ruggero (1993). "A theory of sets with the negation of the axiom of infinity". Mathematical Logic Quarterly. 39 (3): 338–352. doi:10.1002/malq.19930390138. MR 1270381.

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

Rudin, W. (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. p. 25. ISBN 978-0-07-054235-8.
Peano, Giuseppe (1901). Formulaire des mathematiques (in French). Paris, Gauthier-Villars. p. 39.
Emerson, William (1763). The method of increments. p. 113.
Arithmetices principia: nova methodo (in Latin). Fratres Bocca. 1889. p. 12.
Peano, Giuseppe (1901). Formulaire des mathematiques (in French). Paris, Gauthier-Villars. p. 39.
Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4 (1): 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856.
Shields, Paul (1997). "3. Peirce's Axiomatization of Arithmetic". In Houser, Nathan; Roberts, Don D.; Van Evra, James (eds.). Studies in the Logic of Charles Sanders Peirce. Indiana University Press. pp. 43–52. ISBN 0-253-33020-3.
Was sind und was sollen die Zahlen? (in German). F. Vieweg. 1893. 71–73.

astronomicalheritage.net (Global: low place; English: low place)

www2.astronomicalheritage.net

"The Ishango Bone, Democratic Republic of the Congo". UNESCO's Portal to the Heritage of Astronomy. Archived from the original on 10 November 2014., on permanent display at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium.

bnf.fr (Global: 124^th place; English: 544^th place)

gallica.bnf.fr

Chuquet, Nicolas (1881) [1484]. Le Triparty en la science des nombres (in French).
Fontenelle, Bernard de (1727). Eléments de la géométrie de l'infini (in French). p. 3.

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

"Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (Eves 1990, p. 606) Eves, Howard (1990). An Introduction to the History of Mathematics (6th ed.). Thomson. ISBN 978-0-03-029558-4 – via Google Books.
Hamilton (1988, pp. 117 ff) calls them "Peano's Postulates" and begins with "1. 0 is a natural number."
Halmos (1974, p. 46) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I) $0 \in ω$ (where, of course, $0 = \emptyset$ " ( $ω$ is the set of all natural numbers).
Morash (1991) gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1: An Axiomatization for the System of Positive Integers) Hamilton, A.G. (1988). Logic for Mathematicians (Revised ed.). Cambridge University Press. ISBN 978-0-521-36865-0 – via Google Books. Halmos, Paul (1974). Naive Set Theory. Springer Science & Business Media. ISBN 978-0-387-90092-6 – via Google Books. Morash, Ronald P. (1991). Bridge to Abstract Mathematics: Mathematical proof and structures (Second ed.). Mcgraw-Hill College. ISBN 978-0-07-043043-3 – via Google Books.
Mendelson (2008, p. x) says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers." Mendelson, Elliott (2008) [1973]. Number Systems and the Foundations of Analysis. Dover Publications. ISBN 978-0-486-45792-5 – via Google Books.
Bluman (2010, p. 1): "Numbers make up the foundation of mathematics." Bluman, Allan (2010). Pre-Algebra DeMYSTiFieD (Second ed.). McGraw-Hill Professional. ISBN 978-0-07-174251-1 – via Google Books.
Ganssle, Jack G. & Barr, Michael (2003). "integer". Embedded Systems Dictionary. Taylor & Francis. pp. 138 (integer), 247 (signed integer), & 276 (unsigned integer). ISBN 978-1-57820-120-4. Archived from the original on 29 March 2017. Retrieved 28 March 2017 – via Google Books.
Stewart, Ian; Tall, David (12 March 2015). The Foundations of Mathematics. OUP Oxford. p. 160. ISBN 978-0-19-101648-6. Retrieved 30 July 2025.
Fokas, Athanassios; Kaxiras, Efthimios (12 December 2022). Modern Mathematical Methods For Scientists And Engineers: A Street-smart Introduction. World Scientific. p. 4. ISBN 978-1-80061-182-5. Retrieved 30 July 2025.
Mann, Charles C. (2005). 1491: New Revelations of the Americas before Columbus. Knopf. p. 19. ISBN 978-1-4000-4006-3. Archived from the original on 14 May 2015. Retrieved 3 February 2015 – via Google Books.
Evans, Brian (2014). "Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations". The Development of Mathematics Throughout the Centuries: A brief history in a cultural context. John Wiley & Sons. ISBN 978-1-118-85397-9 – via Google Books.
Fine, Henry Burchard (1904). A College Algebra. Ginn. p. 6.
Advanced Algebra: A Study Guide to be Used with USAFI Course MC 166 Or CC166. United States Armed Forces Institute. 1958. p. 12.
Gray, Jeremy (2008). Plato's Ghost: The modernist transformation of mathematics. Princeton University Press. p. 153. ISBN 978-1-4008-2904-0. Archived from the original on 29 March 2017 – via Google Books.
Eves 1990, Chapter 15 Eves, Howard (1990). An Introduction to the History of Mathematics (6th ed.). Thomson. ISBN 978-0-03-029558-4 – via Google Books.
Shields, Paul (1997). "3. Peirce's Axiomatization of Arithmetic". In Houser, Nathan; Roberts, Don D.; Van Evra, James (eds.). Studies in the Logic of Charles Sanders Peirce. Indiana University Press. pp. 43–52. ISBN 0-253-33020-3.
Fletcher, Harold; Howell, Arnold A. (9 May 2014). Mathematics with Understanding. Elsevier. p. 116. ISBN 978-1-4832-8079-0. ...the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication
Davisson, Schuyler Colfax (1910). College Algebra. Macmillian Company. p. 2. Addition of natural numbers is associative.
Brandon, Bertha (M.); Brown, Kenneth E.; Gundlach, Bernard H.; Cooke, Ralph J. (1962). Laidlaw mathematics series. Vol. 8. Laidlaw Bros. p. 25.

clarku.edu (Global: 6,602^nd place; English: 6,109^th place)

aleph0.clarku.edu

Euclid. "Book VII, definitions 1 and 2". In Joyce, D. (ed.). Elements. Clark University.
Euclid. "Book VII, definition 22". In Joyce, D. (ed.). Elements. Clark University. A perfect number is that which is equal to the sum of its own parts. In definition VII.3 a "part" was defined as a number, but here 1 is considered to be a part, so that for example $6 = 1 + 2 + 3$ is a perfect number.

digizeitschriften.de (Global: 5,626^th place; English: 8,880^th place)

Weber, Heinrich L. (1891–1892). "Kronecker". Jahresbericht der Deutschen Mathematiker-Vereinigung [Annual report of the German Mathematicians Association]. pp. 2:5–23. (The quote is on p. 19). Archived from the original on 9 August 2018; "access to Jahresbericht der Deutschen Mathematiker-Vereinigung". Archived from the original on 20 August 2017.

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

Woodin, Greg; Winter, Bodo (2024). "Numbers in Context: Cardinals, Ordinals, and Nominals in American English". Cognitive Science. 48 (6) e13471. doi:10.1111/cogs.13471. PMC 11475258. PMID 38895756.
Benacerraf, Paul (January 1965). "What Numbers Could not Be". The Philosophical Review. 74 (1): 47. doi:10.2307/2183530.
Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4 (1): 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856.
Baratella, Stefano; Ferro, Ruggero (1993). "A theory of sets with the negation of the axiom of infinity". Mathematical Logic Quarterly. 39 (3): 338–352. doi:10.1002/malq.19930390138. MR 1270381.
Kirby, Laurie; Paris, Jeff (1982). "Accessible Independence Results for Peano Arithmetic". Bulletin of the London Mathematical Society. 14 (4). Wiley: 285–293. doi:10.1112/blms/14.4.285. ISSN 0024-6093.

encyclopediaofmath.org (Global: 3,863^rd place; English: 2,637^th place)

Mints, G.E. (ed.). "Peano axioms". Encyclopedia of Mathematics. Springer, in cooperation with the European Mathematical Society. Archived from the original on 13 October 2014. Retrieved 8 October 2014.

google.com (Global: 163^rd place; English: 185^th place)

Cooke, Heather (26 October 2000). Primary Mathematics. SAGE. p. 14. ISBN 978-1-84787-949-3.
Zegarelli, Mark (28 January 2014). Basic Math and Pre-Algebra For Dummies. John Wiley & Sons. p. 21. ISBN 978-1-118-79199-8. Counting numbers (also called natural numbers): The set of numbers beginning 1, 2, 3, 4, ... and going on infinitely.
Rice, Harris (1922). "Errors in computations and the rounded number". The Mathematics Teacher. National Council of Teachers of Mathematics. p. 393. A counting number is the number given in answer to the question "How many?" In this class of numbers belongs zero and positive integers/

iso.org (Global: 629^th place; English: 610^th place)

"Standard number sets and intervals" (PDF). ISO 80000-2:2019 Quantities and units Part 2: Mathematics. International Organization for Standardization. 24 June 2025.

iteh.ai (Global: low place; English: low place)

cdn.standards.iteh.ai

"Standard number sets and intervals" (PDF). ISO 80000-2:2019 Quantities and units Part 2: Mathematics. International Organization for Standardization. 24 June 2025.

jstor.org (Global: 26^th place; English: 20^th place)

Benacerraf, Paul (January 1965). "What Numbers Could not Be". The Philosophical Review. 74 (1): 47. doi:10.2307/2183530.
Peirce, C. S. (1881). "On the Logic of Number". American Journal of Mathematics. 4 (1): 85–95. doi:10.2307/2369151. JSTOR 2369151. MR 1507856.

msu.edu (Global: 1,844^th place; English: 1,231^st place)

archive.lib.msu.edu

"Natural Number". archive.lib.msu.edu.

msu.ru (Global: 1,129^th place; English: 6,103^rd place)

hbar.phys.msu.ru

Deckers, Michael (25 August 2003). "Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius". Hbar.phys.msu.ru. Archived from the original on 15 January 2019. Retrieved 13 February 2012.

naturalsciences.be (Global: low place; English: low place)

naturalsciences.be

"Introduction". Ishango bone. Brussels, Belgium: Royal Belgian Institute of Natural Sciences. Archived from the original on 4 March 2016.

ishango.naturalsciences.be

"Flash presentation". Ishango bone. Brussels, Belgium: Royal Belgian Institute of Natural Sciences. Archived from the original on 27 May 2016.

nih.gov (Global: 4^th place; English: 4^th place)

ncbi.nlm.nih.gov

Woodin, Greg; Winter, Bodo (2024). "Numbers in Context: Cardinals, Ordinals, and Nominals in American English". Cognitive Science. 48 (6) e13471. doi:10.1111/cogs.13471. PMC 11475258. PMID 38895756.

pubmed.ncbi.nlm.nih.gov

Woodin, Greg; Winter, Bodo (2024). "Numbers in Context: Cardinals, Ordinals, and Nominals in American English". Cognitive Science. 48 (6) e13471. doi:10.1111/cogs.13471. PMC 11475258. PMID 38895756.

st-and.ac.uk (Global: 3,503^rd place; English: 3,612^th place)

www-history.mcs.st-and.ac.uk

"A history of Zero". MacTutor History of Mathematics. Archived from the original on 19 January 2013. Retrieved 23 January 2013.

st-andrews.ac.uk (Global: 1,547^th place; English: 1,410^th place)

mathshistory.st-andrews.ac.uk

"Earliest Known Uses of Some of the Words of Mathematics (N)". Maths History.

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

Ganssle, Jack G. & Barr, Michael (2003). "integer". Embedded Systems Dictionary. Taylor & Francis. pp. 138 (integer), 247 (signed integer), & 276 (unsigned integer). ISBN 978-1-57820-120-4. Archived from the original on 29 March 2017. Retrieved 28 March 2017 – via Google Books.
"Introduction". Ishango bone. Brussels, Belgium: Royal Belgian Institute of Natural Sciences. Archived from the original on 4 March 2016.
"Flash presentation". Ishango bone. Brussels, Belgium: Royal Belgian Institute of Natural Sciences. Archived from the original on 27 May 2016.
"The Ishango Bone, Democratic Republic of the Congo". UNESCO's Portal to the Heritage of Astronomy. Archived from the original on 10 November 2014., on permanent display at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium.
"A history of Zero". MacTutor History of Mathematics. Archived from the original on 19 January 2013. Retrieved 23 January 2013.
Mann, Charles C. (2005). 1491: New Revelations of the Americas before Columbus. Knopf. p. 19. ISBN 978-1-4000-4006-3. Archived from the original on 14 May 2015. Retrieved 3 February 2015 – via Google Books.
Deckers, Michael (25 August 2003). "Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius". Hbar.phys.msu.ru. Archived from the original on 15 January 2019. Retrieved 13 February 2012.
Gray, Jeremy (2008). Plato's Ghost: The modernist transformation of mathematics. Princeton University Press. p. 153. ISBN 978-1-4008-2904-0. Archived from the original on 29 March 2017 – via Google Books.
Weber, Heinrich L. (1891–1892). "Kronecker". Jahresbericht der Deutschen Mathematiker-Vereinigung [Annual report of the German Mathematicians Association]. pp. 2:5–23. (The quote is on p. 19). Archived from the original on 9 August 2018; "access to Jahresbericht der Deutschen Mathematiker-Vereinigung". Archived from the original on 20 August 2017.
Mints, G.E. (ed.). "Peano axioms". Encyclopedia of Mathematics. Springer, in cooperation with the European Mathematical Society. Archived from the original on 13 October 2014. Retrieved 8 October 2014.

wikisource.org (Global: 27^th place; English: 51^st place)

en.wikisource.org

Poincaré, Henri (1905) [1902]. "On the nature of mathematical reasoning". La Science et l'hypothèse [Science and Hypothesis]. Translated by Greenstreet, William John. VI.

wolfram.com (Global: 513^th place; English: 537^th place)

mathworld.wolfram.com

Weisstein, Eric W. "Natural Number". mathworld.wolfram.com. Retrieved 11 August 2020.
Weisstein, Eric W. "Counting Number". MathWorld.
Weisstein, Eric W. "Multiplication". mathworld.wolfram.com. Retrieved 27 July 2020.

functions.wolfram.com

"Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions". functions.wolfram.com. Retrieved 27 July 2020.

worldcat.org (Global: 5^th place; English: 5^th place)

search.worldcat.org

Mueller, Ian (2006). Philosophy of mathematics and deductive structure in Euclid's Elements. Mineola, New York: Dover Publications. p. 58. ISBN 978-0-486-45300-2. OCLC 69792712.
Kirby, Laurie; Paris, Jeff (1982). "Accessible Independence Results for Peano Arithmetic". Bulletin of the London Mathematical Society. 14 (4). Wiley: 285–293. doi:10.1112/blms/14.4.285. ISSN 0024-6093.

Natural number (English Wikipedia)

ams.org (Global: 451st place; English: 277th place)

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clarku.edu (Global: 6,602nd place; English: 6,109th place)

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digizeitschriften.de (Global: 5,626th place; English: 8,880th place)

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

encyclopediaofmath.org (Global: 3,863rd place; English: 2,637th place)

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iso.org (Global: 629th place; English: 610th place)

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jstor.org (Global: 26th place; English: 20th place)

msu.edu (Global: 1,844th place; English: 1,231st place)

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nih.gov (Global: 4th place; English: 4th place)

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wolfram.com (Global: 513th place; English: 537th place)

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ams.org (Global: 451^st place; English: 277^th place)

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bnf.fr (Global: 124^th place; English: 544^th place)

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

clarku.edu (Global: 6,602^nd place; English: 6,109^th place)

digizeitschriften.de (Global: 5,626^th place; English: 8,880^th place)

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

encyclopediaofmath.org (Global: 3,863^rd place; English: 2,637^th place)

google.com (Global: 163^rd place; English: 185^th place)

iso.org (Global: 629^th place; English: 610^th place)

jstor.org (Global: 26^th place; English: 20^th place)

msu.edu (Global: 1,844^th place; English: 1,231^st place)

msu.ru (Global: 1,129^th place; English: 6,103^rd place)

nih.gov (Global: 4^th place; English: 4^th place)

st-and.ac.uk (Global: 3,503^rd place; English: 3,612^th place)

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