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== Perumuman ==
Dalam cabang matematika lebih lanjut, saat jarak Euklides dipandang sebagai [[ruang vektor]], jaraknya diiringi dengan [[Norma (matematika)|norma]] yang disebut sebagai [[Norma (matematika)#Norma Euklides|norma Euklides]], didefinisikan sebagai jarak dari masing-masing vektor yang berawal dari [[Titik asal (matematika)|titik asal]]. One of the important properties of this norm, relative to other norms, is that it remains unchanged under arbitrary rotations of space around the origin.<ref>{{citation|title=Relativistic Celestial Mechanics of the Solar System|first1=Sergei|last1=Kopeikin|first2=Michael|last2=Efroimsky|first3=George|last3=Kaplan|publisher=John Wiley & Sons|year=2011|isbn=978-3-527-63457-6|page=106|url=https://books.google.com/books?id=uN5_DQWSR14C&pg=PA106}}</ref> ByMenurut [[teorema Dvoretzky's theorem]], every finite-dimensionalsetiap [[normedruang vectorvektor spacebernorma]] hasdimensi aterhingga high-dimensionalmempunyai subspacesubruang ondimensi whichtinggi thedengan <u>norm is approximately Euclidean</u>; the Euclidean norm is the only norm with this property.<ref>{{citation|last=Matoušek|first=Jiří|author-link=Jiří Matoušek (mathematician)|isbn=978-0-387-95373-1|page=349|publisher=Springer|series=[[Graduate Texts in Mathematics]]|title=Lectures on Discrete Geometry|url=https://books.google.com/books?id=K0fhBwAAQBAJ&pg=PA349|year=2002}}</ref> It can be extended to infinite-dimensional vector spaces as the [[Lp space|L<sup>2</sup> norm]] or L<sup>2</sup> distance.<ref>{{citation|title=Linear and Nonlinear Functional Analysis with Applications|first=Philippe G.|last=Ciarlet|publisher=Society for Industrial and Applied Mathematics|year=2013|isbn=978-1-61197-258-0|page=173|url=https://books.google.com/books?id=AUlWAQAAQBAJ&pg=PA173}}</ref> The Euclidean distance gives Euclidean space the structure of a [[topological space]], the [[Euclidean topology]], with the [[Open ball|open balls]] (subsets of points at less than a given distance from a given point) as its [[Neighbourhood (mathematics)|neighborhoods]].<ref>{{citation|title=General Topology: An Introduction|publisher=De Gruyter|first=Tom|last=Richmond|year=2020|isbn=978-3-11-068657-9|page=32|url=https://books.google.com/books?id=jPgdEAAAQBAJ&pg=PA32}}</ref>
OtherJarak commonumum distanceslainnya ondalam Euclideanruang spacesEuklides andbeserta low-dimensionalruang vectorvektor spacesdimensi includerendah melibatkan:<ref>{{citation|last=Klamroth|first=Kathrin|author-link=Kathrin Klamroth|contribution=Section 1.1: Norms and Metrics|doi=10.1007/0-387-22707-5_1|pages=4–6|publisher=Springer|series=Springer Series in Operations Research|title=Single-Facility Location Problems with Barriers|year=2002}}</ref>
* [[Chebyshev distance]], which measures distance assuming only the most significant dimension is relevant.
== Sejarah ==
EuclideanJarak distanceEuklides isadalah thejarak distance indalam [[Euclideanruang spaceEuklides]]; bothyang conceptsdinamai aredari namedseorang aftermatematikawan ancientYunani Greekkuno mathematicianbernama [[EuclidEuklides]],. whoseKonsep tersebut dijelaskan dalam bukunya, [[Euclid's Elements|''Elements'']], becameyang amenjadi standardbuku textbookcetak instandar geometryselama for many centuriesbertahun.<ref>{{citation|title=Visualization for Information Retrieval|first=Jin|last=Zhang|publisher=Springer|year=2007|isbn=978-3-540-75148-9}}</ref> ConceptsKonsep oftentang [[lengthpanjang]] anddan [[distancejarak]] <u>are widespread across cultures</u>, can be dated to the earliest surviving "protoliterate" bureaucratic documents from [[Sumer]] in the fourth millennium BC (far before Euclid),<ref>{{citation|last=Høyrup|first=Jens|author-link=Jens Høyrup|editor1-last=Jones|editor1-first=Alexander|editor2-last=Taub|editor2-first=Liba|editor2-link=Liba Taub|contribution=Mesopotamian mathematics|contribution-url=https://akira.ruc.dk/~jensh/Publications/2018%7Bj%7D_Mesopotamian%20Mathematics_S.pdf|pages=58–72|publisher=Cambridge University Press|title=The Cambridge History of Science, Volume 1: Ancient Science|year=2018}}</ref> and have been hypothesized to develop in children earlier than the related concepts of speed and time.<ref>{{citation|last1=Acredolo|first1=Curt|last2=Schmid|first2=Jeannine|doi=10.1037/0012-1649.17.4.490|issue=4|journal=[[Developmental Psychology (journal)|Developmental Psychology]]|pages=490–493|title=The understanding of relative speeds, distances, and durations of movement|volume=17|year=1981}}</ref> But the notion of a distance, as a number defined from two points, does not actually appear in Euclid's ''Elements''. Instead, Euclid approaches this concept implicitly, through the [[Congruence (geometry)|congruence]] of line segments, through the comparison of lengths of line segments, and through the concept of [[Proportionality (mathematics)|proportionality]].<ref>{{citation|last=Henderson|first=David W.|author-link=David W. Henderson|journal=[[Bulletin of the American Mathematical Society]]|pages=563–571|title=Review of ''Geometry: Euclid and Beyond'' by Robin Hartshorne|url=https://www.ams.org/journals/bull/2002-39-04/S0273-0979-02-00949-7|volume=39|year=2002|doi=10.1090/S0273-0979-02-00949-7|doi-access=free}}</ref>
The [[Pythagorean theorem]] is also ancient, but it could only take its central role in the measurement of distances after the invention of [[Cartesian coordinates]] by [[René Descartes]] in 1637. The distance formula itself was first published in 1731 by [[Alexis Clairaut]].<ref>{{citation|last=Maor|first=Eli|author-link=Eli Maor|isbn=978-0-691-19688-6|pages=133–134|publisher=Princeton University Press|title=The Pythagorean Theorem: A 4,000-Year History|url=https://books.google.com/books?id=XuWZDwAAQBAJ&pg=PA133|year=2019}}</ref> Because of this formula, Euclidean distance is also sometimes called Pythagorean distance.<ref>{{citation|last1=Rankin|first1=William C.|last2=Markley|first2=Robert P.|last3=Evans|first3=Selby H.|date=March 1970|doi=10.3758/bf03210143|issue=2|journal=[[Perception & Psychophysics]]|pages=103–107|title=Pythagorean distance and the judged similarity of schematic stimuli|volume=7|s2cid=144797925|doi-access=free}}</ref> Although accurate measurements of long distances on the earth's surface, which are not Euclidean, had again been studied in many cultures since ancient times (see [[history of geodesy]]), the idea that Euclidean distance might not be the only way of measuring distances between points in mathematical spaces came even later, with the 19th-century formulation of [[non-Euclidean geometry]].<ref>{{citation|last=Milnor|first=John|author-link=John Milnor|doi=10.1090/S0273-0979-1982-14958-8|issue=1|journal=[[Bulletin of the American Mathematical Society]]|mr=634431|pages=9–24|title=Hyperbolic geometry: the first 150 years|volume=6|year=1982|doi-access=free}}</ref> The definition of the Euclidean norm and Euclidean distance for geometries of more than three dimensions also first appeared in the 19th century, in the work of [[Augustin-Louis Cauchy]].<ref>{{citation|title=Foundations of Hyperbolic Manifolds|volume=149|series=[[Graduate Texts in Mathematics]]|first=John G.|last=Ratcliffe|edition=3rd|publisher=Springer|year=2019|isbn=978-3-030-31597-9|page=32|url=https://books.google.com/books?id=yMO4DwAAQBAJ&pg=PA32}}</ref>
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