Editing Absolute value (section)

==Terminology and notation==
In 1806, [[Jean-Robert Argand]] introduced the term ''module'', meaning ''unit of measure'' in French, specifically for the ''complex'' absolute value,<ref name=oed>[[Oxford English Dictionary]], Draft Revision, June 2008</ref><ref>Nahin, [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Argand.html O'Connor and Robertson], and [http://functions.wolfram.com/ComplexComponents/Abs/35/ functions.Wolfram.com.]; for the French sense, see [[Dictionnaire de la langue française (Littré)|Littré]], 1877</ref> and it was borrowed into English in 1866 as the Latin equivalent ''modulus''.<ref name=oed /> The term ''absolute value'' has been used in this sense from at least 1806 in French<ref>[[Lazare Nicolas Marguerite Carnot|Lazare Nicolas M. Carnot]], ''Mémoire sur la relation qui existe entre les distances respectives de cinq point quelconques pris dans l'espace'', p.&nbsp;105 [https://books.google.com/books?id=YyIOAAAAQAAJ&pg=PA105 at Google Books]</ref> and 1857 in English.<ref>James Mill Peirce, ''A Text-book of Analytic Geometry'' [https://archive.org/details/atextbookanalyt00peirgoog/page/n60 <!-- pg=42 --> at Internet Archive].  The oldest citation in the 2nd edition of the Oxford English Dictionary is from 1907. The term ''absolute value'' is also used in contrast to ''relative value''.</ref> The notation {{math|{{abs|{{mvar|x}}}}}}, with a [[vertical bar]] on each side, was introduced by [[Karl Weierstrass]] in 1841.<ref>Nicholas J. Higham, ''Handbook of writing for the mathematical sciences'', SIAM. {{ISBN|0-89871-420-6}}, p.&nbsp;25</ref> Other names for ''absolute value'' include ''numerical value''<ref name=oed /> and ''magnitude''.<ref name=oed /> The absolute value of <math>x</math> has also been denoted <math>\operatorname{abs} x</math> in some mathematical publications,<ref>{{cite journal
 | last = Siegel | first = Carl Ludwig
 | doi = 10.2307/1968953
 | journal = Annals of Mathematics
 | jstor = 1968953
 | mr = 8095
 | pages = 613–616
 | series = Second Series
 | title = Note on automorphic functions of several variables
 | volume = 43
 | year = 1942| issue = 4
 }}</ref> and in [[spreadsheet]]s, programming languages, and computational software packages, the absolute value of <math display="inline">x</math> is generally represented by <code>abs(''x'')</code>, or a similar expression,<ref>{{cite book|title=Excel Formulas and Functions For Dummies|first=Ken|last=Bluttman|publisher=John Wiley & Sons|year=2015|isbn=9781119076780|page=135|contribution=Ignoring signs|contribution-url=https://books.google.com/books?id=3pVxBgAAQBAJ&pg=PA135}}</ref> as it has been since the earliest days of [[high-level programming language]]s.<ref>{{citation
 | last = Knuth | first = D. E. | author-link = Donald Knuth
 | contribution = Invited papers: History of writing compilers
 | doi = 10.1145/800198.806098
 | page = 43, 126
 | publisher = ACM Press
 | title = Proceedings of the 1962 ACM National Conference
 | year = 1962}}</ref>

The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it denotes its [[cardinality]]; when applied to a [[Matrix (math)|matrix]], it denotes its [[determinant]].<ref>{{cite report|url=https://www.unicode.org/notes/tn28/UTN28-PlainTextMath-v3.3.pdf|type=Unicode report 28|title=A Nearly Plain-Text Encoding of Mathematics|first=Murray III|last=Sargent|date=January 22, 2025|access-date=2025-02-23}}</ref> Vertical bars denote the absolute value only for algebraic objects for which the notion of an absolute value is defined, notably an [[Element (mathematics)|element]] of a [[normed division algebra]], for example a real number, a complex number, or a quaternion.  A closely related but distinct notation is the use of vertical bars for either the [[Euclidean norm]]<ref>{{Cite book|title=Calculus on Manifolds|last=Spivak|first=Michael|publisher=Westview|year=1965|isbn=0805390219|location=Boulder, CO|pages=1}}</ref> or [[sup norm]]<ref>{{Cite book|title=Analysis on Manifolds|last=Munkres|first=James|publisher=Westview|year=1991|isbn=0201510359|location=Boulder, CO|pages=4}}</ref> of a vector {{nowrap|in <math>\R^n</math>,}} although double vertical bars with subscripts {{nowrap|(<math>\|\cdot\|_2</math>}} {{nowrap|and <math>\|\cdot\|_\infty</math>,}} respectively) are a more common and less ambiguous notation.