Editing Integer (section)

== History ==

The word integer comes from the [[Latin]] [[wikt:integer#Latin|''integer'']] meaning "whole" or (literally) "untouched", from ''in'' ("not") plus ''tangere'' ("to touch"). "[[wikt:entire|Entire]]" derives from the same origin via the [[French language|French]] word ''[[wikt:entier|entier]]'', which means both ''entire'' and ''integer''.<ref>{{cite book |first=Nick |last=Evans |contribution=A-Quantifiers and Scope |editor-first=Emmon W. |editor-last=Bach |title=Quantification in Natural Languages |isbn=978-0-7923-3352-4 |year=1995 |pages=262 |url=https://books.google.com/books?id=NlQL97qBSZkC |location=Dordrecht, The Netherlands; Boston, MA |publisher=Kluwer Academic Publishers}}</ref> Historically the term was used for a [[number]] that was a multiple of 1,<ref>{{cite book |last1=Smedley |first1=Edward |last2=Rose |first2=Hugh James |last3=Rose |first3=Henry John |title=Encyclopædia Metropolitana |date=1845 |publisher=B. Fellowes |page=537 |url=https://books.google.com/books?id=ZVI_AQAAMAAJ&pg=PA537 |language=en|quote=An integer is a multiple of unity}}</ref><ref>{{harvnb|Encyclopaedia Britannica|1771|p=[https://books.google.com/books?id=d50qAQAAMAAJ&pg=PA367 367]}}</ref> or to the whole part of a [[mixed number]].<ref>{{cite book| title = Incipit liber Abbaci compositus to Lionardo filio Bonaccii Pisano in year Mccij | type=Manuscript | trans-title=The Book of Calculation | last1 = Pisano | first1 = Leonardo  | author1-link=Fibonacci | publisher = Museo Galileo | date = 1202 | url = https://bibdig.museogalileo.it/tecanew/opera?bid=1072400&seq=30 |lang=la|last2=Boncompagni|first2=Baldassarre (transliteration)|translator-last = Sigler |translator-first = Laurence E.|page=30|quote=Nam rupti uel fracti semper ponendi sunt post integra, quamuis prius integra quam rupti pronuntiari debeant.|trans-quote=And the fractions are always put after the whole, thus first the integer is written, and then the fraction}}</ref><ref>{{harvnb|Encyclopaedia Britannica|1771|p=[https://books.google.com/books?id=d50qAQAAMAAJ&pg=PA83 83]}}</ref> Only positive integers were considered, making the term synonymous with the [[natural number]]s. The definition of integer expanded over time to include [[negative number]]s as their usefulness was recognized.<ref name="negmath">{{cite book|last=Martinez|first=Alberto|title=Negative Math|pages=80–109|date=2014|publisher=Princeton University Press}}</ref> For example [[Leonhard Euler]] in his 1765 ''[[Elements of Algebra]]'' defined integers to include both positive and negative numbers.<ref>{{cite book |last1=Euler |first1=Leonhard |title=Vollstandige Anleitung Zur Algebra|lang=de|trans-title=Complete Introduction to Algebra|volume=1|date=1771 |url=https://archive.org/details/1770LEULERVollstandigeAnleitungZurAlgebraVol1/page/n31/mode/2up|page=10|quote=Alle diese Zahlen, so wohl positive als negative, führen den bekannten Nahmen der gantzen Zahlen, welche also entweder größer oder kleiner sind als nichts. Man nennt dieselbe gantze Zahlen, um sie von den gebrochenen, und noch vielerley andern Zahlen, wovon unten gehandelt werden wird, zu unterscheiden.|trans-quote=All these numbers, both positive and negative, are called whole numbers, which are either greater or lesser than nothing. We call them whole numbers, to distinguish them from fractions, and from several other kinds of numbers of which we shall hereafter speak.}}</ref>

The phrase ''the set of the integers'' was not used before the end of the 19th century, when [[Georg Cantor]] introduced the concept of [[infinite set]]s and [[set theory]]. The use of the letter Z to denote the set of integers comes from the [[German language|German]] word ''[[wikt:Zahlen|Zahlen]]'' ("numbers")<ref name="earliest">{{cite web |url=http://jeff560.tripod.com/nth.html |title=Earliest Uses of Symbols of Number Theory |access-date=2010-09-20 |date=2010-08-29 |first=Jeff |last=Miller |archive-url=https://web.archive.org/web/20100131022510/http://jeff560.tripod.com/nth.html |archive-date=2010-01-31 |url-status=dead }}</ref><ref name="Cameron1998">{{cite book |author=Peter Jephson Cameron |title=Introduction to Algebra |url=https://books.google.com/books?id=syYYl-NVM5IC&pg=PA4 |year=1998 |publisher=Oxford University Press |isbn=978-0-19-850195-4 |page=4 |access-date=2016-02-15 |archive-url=https://web.archive.org/web/20161208142220/https://books.google.com/books?id=syYYl-NVM5IC&pg=PA4 |archive-date=2016-12-08 |url-status=live }}</ref> and has been attributed to [[David Hilbert]].<ref>{{cite book |title=The University of Leeds Review |date=1989 |publisher=University of Leeds. |page=46 |url=https://books.google.com/books?id=Z-7kAAAAMAAJ|language=en|volume=31-32|quote=Incidentally, Z comes from "Zahl": the notation was created by Hilbert.}}</ref> The earliest known use of the notation in a textbook occurs in [[Éléments de mathématique|Algèbre]] written by the collective [[Nicolas Bourbaki]], dating to 1947.<ref name="earliest"/><ref>{{cite book |last1=Bourbaki |first1=Nicolas |title=Algèbre, Chapter 1 |date=1951|edition=2nd |publisher=Hermann|location=Paris |page=27|language=fr|url=https://archive.org/details/algebrebour00bour/page/26/mode/2up|quote=Le symétrisé de '''N''' se note '''Z'''; ses éléments sont appelés entiers rationnels.|trans-quote=The group of differences of '''N''' is denoted by '''Z'''; its elements are called the rational integers.}}</ref> The notation was not adopted immediately. For example, another textbook used the letter J,<ref>{{cite book |last1=Birkhoff |first1=Garrett |title=Lattice Theory |date=1948 |publisher=American Mathematical Society |page=63 |edition=Revised |url=https://archive.org/details/in.ernet.dli.2015.166886/page/n63/mode/2up|quote=the set ''J'' of all integers}}</ref> and a 1960 paper used Z to denote the non-negative integers.<ref>{{cite book |last1=Society |first1=Canadian Mathematical |title=Canadian Journal of Mathematics |date=1960 |publisher=Canadian Mathematical Society |page=374 |url=https://books.google.com/books?id=uMAXOmLTCGsC&dq=integer%20set%20Z&pg=PA374 |language=en|quote=Consider the set ''Z'' of non-negative integers}}</ref> But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers.<ref>{{cite book |last1=Bezuszka |first1=Stanley |title=Contemporary Progress in Mathematics: Teacher Supplement [to] Part 1 and Part 2 |date=1961 |publisher=Boston College |page=69 |url=https://books.google.com/books?id=dhJPAQAAMAAJ&q=integer+set+Z |language=en|quote=Modern Algebra texts generally designate the set of integers by the capital letter Z.}}</ref>

The symbol <math>\mathbb{Z}</math> is often annotated to denote various sets, with varying usage amongst different authors: <math>\mathbb{Z}^+</math>, <math>\mathbb{Z}_+</math>, or <math>\mathbb{Z}^{>}</math> for the positive integers, <math>\mathbb{Z}^{0+}</math> or <math>\mathbb{Z}^{\geq}</math> for non-negative integers, and <math>\mathbb{Z}^{\neq}</math> for non-zero integers. Some authors use <math>\mathbb{Z}^{*}</math> for non-zero integers, while others use it for non-negative integers, or for {−1,1} (the [[group of units]] of <math>\mathbb{Z}</math>). Additionally, <math>\mathbb{Z}_{p}</math> is used to denote either the set of [[integers modulo n|integers modulo {{math|''p''}}]] (i.e., the set of [[congruence relation|congruence classes]] of integers), or the set of [[p-adic integer|{{math|''p''}}-adic integers]].<ref name=edexcelc1>Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008</ref><ref>LK Turner, FJ BUdden, D Knighton, "Advanced Mathematics", Book 2, Longman 1975.</ref>

{{anchor|Whole numbers}}
The ''whole numbers'' were synonymous with the integers up until the early 1950s.<ref>{{cite book |last1=Mathews |first1=George Ballard |title=Theory of Numbers |date=1892 |publisher=Deighton, Bell and Company |page=2 |url=https://books.google.com/books?id=iQ_vAAAAMAAJ&pg=PA2 |language=en}}</ref><ref>{{cite book |last1=Betz |first1=William |title=Junior Mathematics for Today |date=1934 |publisher=Ginn |url=https://books.google.com/books?id=RzNCAAAAIAAJ |language=en |quote=The whole numbers, or integers, when arranged in their natural order, such as 1, 2, 3, are called consecutive integers.}}</ref><ref>{{cite book |last1=Peck |first1=Lyman C. |title=Elements of Algebra |date=1950 |publisher=McGraw-Hill |page=3 |url=https://books.google.com/books?id=tclXAAAAYAAJ&q=integers+whole+numbers |language=en |quote=The numbers which so arise are called positive whole numbers, or positive integers.}}</ref> In the late 1950s, as part of the [[New Math]] movement,<ref>{{cite thesis|type=PhD |url= https://dr.lib.iastate.edu/handle/20.500.12876/80303 |title=A history of the "new math" movement in the United States|date=1981|last=Hayden|first=Robert|publisher=Iowa State University |doi=10.31274/rtd-180813-5631|page=145|quote=A much more influential force in bringing news of the "new math" to high school teachers and administrators was the National Council of Teachers of Mathematics (NCTM).|doi-access=free}}</ref> American elementary school teachers began teaching that ''whole numbers'' referred to the [[natural number]]s, excluding negative numbers, while ''integer'' included the negative numbers.<ref>{{cite book |title=The Growth of Mathematical Ideas, Grades K-12: 24th Yearbook |date=1959 |publisher=National Council of Teachers of Mathematics |page=14 |isbn=9780608166186 |url=https://books.google.com/books?id=OO9RAQAAIAAJ&pg=PA14 |language=en}}</ref><ref>{{cite book |last1=Deans |first1=Edwina |title=Elementary School Mathematics: New Directions |date=1963 |publisher=U.S. Department of Health, Education, and Welfare, Office of Education |page=42 |url=https://books.google.com/books?id=bAUJAQAAMAAJ&pg=PA42 |language=en}}</ref> The ''whole numbers'' remain ambiguous to the present day.<ref>{{cite web |title=entry: whole number |url=https://www.ahdictionary.com/word/search.html?q=whole+number |website=The American Heritage Dictionary |publisher=HarperCollins}}</ref>