Algorithm - Revision history

David Eppstein: /* By design paradigm */ improve sourcing for divide & conquer / prune & search

2025-05-19T01:34:56Z

By design paradigm: improve sourcing for divide & conquer / prune & search

← Previous revision		Revision as of 01:34, 19 May 2025
Line 124:		Line 124:
	: Brute force is a problem-solving method of systematically trying every possible option until the optimal solution is found. This approach can be very time-consuming, testing every possible combination of variables. It is often used when other methods are unavailable or too complex. Brute force can solve a variety of problems, including finding the shortest path between two points and cracking passwords.		: Brute force is a problem-solving method of systematically trying every possible option until the optimal solution is found. This approach can be very time-consuming, testing every possible combination of variables. It is often used when other methods are unavailable or too complex. Brute force can solve a variety of problems, including finding the shortest path between two points and cracking passwords.
	; Divide and conquer		; Divide and conquer
	: A [[divide-and-conquer algorithm]] repeatedly reduces a problem to one or more smaller instances of itself (usually [[recursion\|recursively]]) until the instances are small enough to solve easily. [[mergesort\|Merge sorting]] is an example of divide and conquer, where an unordered list ~~can~~ be ~~divided~~ into ~~segments~~ ~~containing~~ ~~one~~ ~~item~~ and ~~sorting~~ of ~~the~~ ~~entire~~ ~~list~~ ~~can~~ be ~~obtained~~ by ~~merging~~ ~~the~~ ~~segments~~. A simpler variant of divide and conquer is called a ''decrease-and-conquer algorithm'', which solves one smaller instance of itself, and ~~uses~~ ~~the~~ ~~solution~~ to ~~solve~~ ~~the bigger problem. Divide and conquer divides the problem into multiple subproblems and so the conquer stage is more complex than decrease and conquer algorithms~~.{{~~Citation needed~~\|~~date~~=~~October~~ ~~2024~~}} An example of a ~~decrease~~ and ~~conquer~~ algorithm is the [[binary search algorithm]].		: A [[divide-and-conquer algorithm]] repeatedly reduces a problem to one or more smaller instances of itself (usually [[recursion\|recursively]]) until the instances are small enough to solve easily. [[mergesort\|Merge sorting]] is an example of divide and conquer, where an unordered list is repeatedly split into smaller lists, which are sorted in the same way and then merged.<ref>{{cite book\|title=Algorithm Design: Foundations, Analysis, and Internet Examples\|first1=Michael T.\|last1=Goodrich\|first2=Roberto\|last2=Tamassia\|publisher=John Wiley & Sons\|year=2001\|isbn=9780471383659\|contribution=5.2 Divide and Conquer\|page=263}}</ref> In a simpler variant of divide and conquer called [[prune and search]] or ''decrease-and-conquer algorithm'', which solves one smaller instance of itself, and does not require a merge step.{{sfnp\|Goodrich\|Tamassia\|2001\|loc=4.7.1 Prune-and-search\|p=245}} An example of a prune and search algorithm is the [[binary search algorithm]].
	; Search and enumeration		; Search and enumeration
	: Many problems (such as playing [[Chess\|ches]]s) can be modelled as problems on [[graph theory\|graph]]s. A [[graph exploration algorithm]] specifies rules for moving around a graph and is useful for such problems. This category also includes [[search algorithm]]s, [[branch and bound]] enumeration, and [[backtracking]].		: Many problems (such as playing [[Chess\|ches]]s) can be modelled as problems on [[graph theory\|graph]]s. A [[graph exploration algorithm]] specifies rules for moving around a graph and is useful for such problems. This category also includes [[search algorithm]]s, [[branch and bound]] enumeration, and [[backtracking]].

David Eppstein: /* Ancient algorithms */ the citation is right there in the next sentence

2025-05-19T01:25:12Z

Ancient algorithms: the citation is right there in the next sentence

← Previous revision		Revision as of 01:25, 19 May 2025
Line 42:		Line 42:
	Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in [[Babylonian mathematics]] (around 2500 BC),<ref name="Springer Science & Business Media">{{cite book \|last1=Chabert \|first1=Jean-Luc \|title=A History of Algorithms: From the Pebble to the Microchip \|date=2012 \|publisher=Springer Science & Business Media \|isbn=9783642181924 \|pages=7–8}}</ref> [[Egyptian mathematics]] (around 1550 BC),<ref name="Springer Science & Business Media" /> [[Indian mathematics]] (around 800 BC and later),<ref name=":6">{{cite book \|last1=Sriram \|first1=M. S. \|editor1-last=Emch \|editor1-first=Gerard G. \|editor2-last=Sridharan \|editor2-first=R. \|editor3-last=Srinivas \|editor3-first=M. D. \|title=Contributions to the History of Indian Mathematics \|date=2005 \|publisher=Springer \|isbn=978-93-86279-25-5 \|page=153 \|chapter-url=https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153 \|language=en \|chapter=Algorithms in Indian Mathematics}}</ref><ref>Hayashi, T. (2023, January 1). [https://www.britannica.com/biography/Brahmagupta Brahmagupta]. Encyclopedia Britannica.</ref> the Ifa Oracle (around 500 BC),<ref>{{Cite journal \|last=Zaslavsky \|first=Claudia \|date=1970 \|title=Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria \|url=https://www.jstor.org/stable/3027363 \|journal=The Two-Year College Mathematics Journal \|volume=1 \|issue=2 \|pages=76–99 \|doi=10.2307/3027363 \|jstor=3027363 \|issn=0049-4925}}</ref> [[Greek mathematics]] (around 240 BC),<ref name="Cooke2005">{{cite book\|last=Cooke\|first=Roger L.\|title=The History of Mathematics: A Brief Course\|date=2005\|publisher=John Wiley & Sons\|isbn=978-1-118-46029-0}}</ref> [[Chinese mathematics\|Chinese mathematics (around 200 BC and later)]],<ref>{{Cite journal \|date=1999 \|editor-last=Chabert \|editor-first=Jean-Luc \|title=A History of Algorithms \|url=https://link.springer.com/book/10.1007/978-3-642-18192-4 \|journal=SpringerLink \|language=en \|doi=10.1007/978-3-642-18192-4\|isbn=978-3-540-63369-3 }}</ref> and [[Arabic mathematics]] (around 800 AD).<ref name="Dooley">{{cite book \|last1=Dooley \|first1=John F. \|title=A Brief History of Cryptology and Cryptographic Algorithms \|date=2013 \|publisher=Springer Science & Business Media \|isbn=9783319016283 \|pages=12–3}}</ref>		Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in [[Babylonian mathematics]] (around 2500 BC),<ref name="Springer Science & Business Media">{{cite book \|last1=Chabert \|first1=Jean-Luc \|title=A History of Algorithms: From the Pebble to the Microchip \|date=2012 \|publisher=Springer Science & Business Media \|isbn=9783642181924 \|pages=7–8}}</ref> [[Egyptian mathematics]] (around 1550 BC),<ref name="Springer Science & Business Media" /> [[Indian mathematics]] (around 800 BC and later),<ref name=":6">{{cite book \|last1=Sriram \|first1=M. S. \|editor1-last=Emch \|editor1-first=Gerard G. \|editor2-last=Sridharan \|editor2-first=R. \|editor3-last=Srinivas \|editor3-first=M. D. \|title=Contributions to the History of Indian Mathematics \|date=2005 \|publisher=Springer \|isbn=978-93-86279-25-5 \|page=153 \|chapter-url=https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153 \|language=en \|chapter=Algorithms in Indian Mathematics}}</ref><ref>Hayashi, T. (2023, January 1). [https://www.britannica.com/biography/Brahmagupta Brahmagupta]. Encyclopedia Britannica.</ref> the Ifa Oracle (around 500 BC),<ref>{{Cite journal \|last=Zaslavsky \|first=Claudia \|date=1970 \|title=Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria \|url=https://www.jstor.org/stable/3027363 \|journal=The Two-Year College Mathematics Journal \|volume=1 \|issue=2 \|pages=76–99 \|doi=10.2307/3027363 \|jstor=3027363 \|issn=0049-4925}}</ref> [[Greek mathematics]] (around 240 BC),<ref name="Cooke2005">{{cite book\|last=Cooke\|first=Roger L.\|title=The History of Mathematics: A Brief Course\|date=2005\|publisher=John Wiley & Sons\|isbn=978-1-118-46029-0}}</ref> [[Chinese mathematics\|Chinese mathematics (around 200 BC and later)]],<ref>{{Cite journal \|date=1999 \|editor-last=Chabert \|editor-first=Jean-Luc \|title=A History of Algorithms \|url=https://link.springer.com/book/10.1007/978-3-642-18192-4 \|journal=SpringerLink \|language=en \|doi=10.1007/978-3-642-18192-4\|isbn=978-3-540-63369-3 }}</ref> and [[Arabic mathematics]] (around 800 AD).<ref name="Dooley">{{cite book \|last1=Dooley \|first1=John F. \|title=A Brief History of Cryptology and Cryptographic Algorithms \|date=2013 \|publisher=Springer Science & Business Media \|isbn=9783319016283 \|pages=12–3}}</ref>

	The earliest evidence of algorithms is found in ancient [[Mesopotamia]]n mathematics. A [[Sumer]]ian clay tablet found in [[Shuruppak]] near [[Baghdad]] and dated to {{Circa\|2500 BC}} describes the earliest [[division algorithm]].<ref name="Springer Science & Business Media" /> During the [[First Babylonian dynasty\|Hammurabi dynasty]] {{Circa\|1800\|1600 BC\|lk=no}}, [[Babylonia]]n clay tablets described algorithms for computing formulas.<ref>{{cite journal \|last1=Knuth \|first1=Donald E. \|date=1972 \|title=Ancient Babylonian Algorithms \|url=http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|url-status=dead \|journal=Commun. ACM \|volume=15 \|issue=7 \|pages=671–677 \|doi=10.1145/361454.361514 \|issn=0001-0782 \|s2cid=7829945 \|archive-url=https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|archive-date=2012-12-24}}</ref> Algorithms were also used in [[Babylonian astronomy]].~~{{Citation needed\|date=March 2025}}~~ Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<ref>{{cite book \|last=Aaboe \|first=Asger \|author-link=Asger Aaboe \|title=Episodes from the Early History of Astronomy \|date=2001 \|publisher=Springer \|isbn=978-0-387-95136-2 \|place=New York \|pages=40–62}}</ref>		The earliest evidence of algorithms is found in ancient [[Mesopotamia]]n mathematics. A [[Sumer]]ian clay tablet found in [[Shuruppak]] near [[Baghdad]] and dated to {{Circa\|2500 BC}} describes the earliest [[division algorithm]].<ref name="Springer Science & Business Media" /> During the [[First Babylonian dynasty\|Hammurabi dynasty]] {{Circa\|1800\|1600 BC\|lk=no}}, [[Babylonia]]n clay tablets described algorithms for computing formulas.<ref>{{cite journal \|last1=Knuth \|first1=Donald E. \|date=1972 \|title=Ancient Babylonian Algorithms \|url=http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|url-status=dead \|journal=Commun. ACM \|volume=15 \|issue=7 \|pages=671–677 \|doi=10.1145/361454.361514 \|issn=0001-0782 \|s2cid=7829945 \|archive-url=https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|archive-date=2012-12-24}}</ref> Algorithms were also used in [[Babylonian astronomy]]. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<ref>{{cite book \|last=Aaboe \|first=Asger \|author-link=Asger Aaboe \|title=Episodes from the Early History of Astronomy \|date=2001 \|publisher=Springer \|isbn=978-0-387-95136-2 \|place=New York \|pages=40–62}}</ref>

	Algorithms for arithmetic are also found in ancient [[Egyptian mathematics]], dating back to the [[Rhind Mathematical Papyrus]] {{Circa\|1550 BC\|lk=no}}.<ref name="Springer Science & Business Media" /> Algorithms were later used in ancient [[Hellenistic mathematics]]. Two examples are the [[Sieve of Eratosthenes]], which was described in the ''[[Introduction to Arithmetic]]'' by [[Nicomachus]],<ref>{{cite web \|last=Ast \|first=Courtney \|title=Eratosthenes \|url=http://www.math.wichita.edu/history/men/eratosthenes.html \|url-status=live \|archive-url=https://web.archive.org/web/20150227150653/http://www.math.wichita.edu/history/men/eratosthenes.html \|archive-date=February 27, 2015 \|access-date=February 27, 2015 \|publisher=Wichita State University: Department of Mathematics and Statistics}}</ref><ref name="Cooke2005" />{{rp\|Ch 9.2}} and the [[Euclidean algorithm]], which was first described in ''[[Euclid's Elements]]'' ({{circa\|300 BC\|lk=no}}).<ref name="Cooke2005" />{{rp\|Ch 9.1}}Examples of ancient Indian mathematics included the [[Shulba Sutras]], the [[Kerala school of astronomy and mathematics\|Kerala School]], and the [[Brāhmasphuṭasiddhānta]].<ref name=":6" />		Algorithms for arithmetic are also found in ancient [[Egyptian mathematics]], dating back to the [[Rhind Mathematical Papyrus]] {{Circa\|1550 BC\|lk=no}}.<ref name="Springer Science & Business Media" /> Algorithms were later used in ancient [[Hellenistic mathematics]]. Two examples are the [[Sieve of Eratosthenes]], which was described in the ''[[Introduction to Arithmetic]]'' by [[Nicomachus]],<ref>{{cite web \|last=Ast \|first=Courtney \|title=Eratosthenes \|url=http://www.math.wichita.edu/history/men/eratosthenes.html \|url-status=live \|archive-url=https://web.archive.org/web/20150227150653/http://www.math.wichita.edu/history/men/eratosthenes.html \|archive-date=February 27, 2015 \|access-date=February 27, 2015 \|publisher=Wichita State University: Department of Mathematics and Statistics}}</ref><ref name="Cooke2005" />{{rp\|Ch 9.2}} and the [[Euclidean algorithm]], which was first described in ''[[Euclid's Elements]]'' ({{circa\|300 BC\|lk=no}}).<ref name="Cooke2005" />{{rp\|Ch 9.1}}Examples of ancient Indian mathematics included the [[Shulba Sutras]], the [[Kerala school of astronomy and mathematics\|Kerala School]], and the [[Brāhmasphuṭasiddhānta]].<ref name=":6" />

David Eppstein: more precisely locate the quotation from stone; page 8 (end of section), not page 4 (earlier version of similar def)

2025-05-19T01:21:01Z

more precisely locate the quotation from stone; page 8 (end of section), not page 4 (earlier version of similar def)

← Previous revision		Revision as of 01:21, 19 May 2025
Line 27:		Line 27:
	{{For\|a detailed presentation of the various points of view on the definition of "algorithm"\|Algorithm characterizations}}		{{For\|a detailed presentation of the various points of view on the definition of "algorithm"\|Algorithm characterizations}}

	One informal definition is "a set of rules that precisely defines a sequence of operations",~~<ref>Stone 1973:4</ref>~~{{~~request quotation~~ \| ~~reason =~~ Stone ~~(1972) suggests on page 4: "...any sequence of instructions that a robot can obey, is called an algorithm"~~\|~~date~~=~~July 2020~~}} which would include all [[computer program]]s (including programs that do not perform numeric calculations), and any prescribed [[bureaucratic]] procedure<ref>		One informal definition is "a set of rules that precisely defines a sequence of operations",{{sfnp\|Stone\|1971\|p=8}} which would include all [[computer program]]s (including programs that do not perform numeric calculations), and any prescribed [[bureaucratic]] procedure<ref>
	{{cite book \|last1=Simanowski \|first1=Roberto \|author-link1=Roberto Simanowski \|url=https://books.google.com/books?id=RJV5DwAAQBAJ \|title=The Death Algorithm and Other Digital Dilemmas \|date=2018 \|publisher=MIT Press \|isbn=9780262536370 \|series=Untimely Meditations \|volume=14 \|location=Cambridge, Massachusetts \|page=147 \|translator1-last=Chase \|translator1-first=Jefferson \|quote=[...] the next level of abstraction of central bureaucracy: globally operating algorithms. \|access-date=27 May 2019 \|archive-url=https://web.archive.org/web/20191222120705/https://books.google.com/books?id=RJV5DwAAQBAJ \|archive-date=December 22, 2019 \|url-status=live}}		{{cite book \|last1=Simanowski \|first1=Roberto \|author-link1=Roberto Simanowski \|url=https://books.google.com/books?id=RJV5DwAAQBAJ \|title=The Death Algorithm and Other Digital Dilemmas \|date=2018 \|publisher=MIT Press \|isbn=9780262536370 \|series=Untimely Meditations \|volume=14 \|location=Cambridge, Massachusetts \|page=147 \|translator1-last=Chase \|translator1-first=Jefferson \|quote=[...] the next level of abstraction of central bureaucracy: globally operating algorithms. \|access-date=27 May 2019 \|archive-url=https://web.archive.org/web/20191222120705/https://books.google.com/books?id=RJV5DwAAQBAJ \|archive-date=December 22, 2019 \|url-status=live}}
	</ref>		</ref>
Line 253:		Line 253:
	* {{cite book\| last = Sipser\| first = Michael\| title = Introduction to the Theory of Computation\| year = 2006\| publisher = PWS Publishing Company\| isbn = 978-0-534-94728-6\| url = https://archive.org/details/introductiontoth00sips}}		* {{cite book\| last = Sipser\| first = Michael\| title = Introduction to the Theory of Computation\| year = 2006\| publisher = PWS Publishing Company\| isbn = 978-0-534-94728-6\| url = https://archive.org/details/introductiontoth00sips}}
	* {{cite book \|last1=Sober \|first1=Elliott \|last2=Wilson \|first2=David Sloan \|year=1998 \|title=Unto Others: The Evolution and Psychology of Unselfish Behavior \|url=https://archive.org/details/untoothersevolut00sobe \|url-access=registration \|location=Cambridge \|publisher=Harvard University Press\|isbn=9780674930469 }}		* {{cite book \|last1=Sober \|first1=Elliott \|last2=Wilson \|first2=David Sloan \|year=1998 \|title=Unto Others: The Evolution and Psychology of Unselfish Behavior \|url=https://archive.org/details/untoothersevolut00sobe \|url-access=registration \|location=Cambridge \|publisher=Harvard University Press\|isbn=9780674930469 }}
	* {{Cite book\|last=Stone\|first=Harold S.\|title=Introduction to Computer Organization and Data Structures~~\|edition=1972~~\|publisher=McGraw-Hill, New York\|isbn=~~978-0-07-061726-1~~\|year=~~1972~~}} Cf. in particular the first chapter titled: ''Algorithms, Turing Machines, and Programs''. His succinct informal definition: "...any sequence of instructions that can be obeyed by a robot, is called an ''algorithm''" (p. 4).		* {{Cite book\|last=Stone\|first=Harold S.\|title=Introduction to Computer Organization and Data Structures\|publisher=McGraw-Hill, New York\|isbn=9780070617261\|year=1971}} Cf. in particular the first chapter titled: ''Algorithms, Turing Machines, and Programs''. His succinct informal definition: "...any sequence of instructions that can be obeyed by a robot, is called an ''algorithm''" (p. 4).
	* {{cite book\| last = Tausworthe\| first = Robert C\| title = Standardized Development of Computer Software Part 1 Methods\| year = 1977\| publisher = Prentice–Hall, Inc.\| location = Englewood Cliffs NJ\| isbn = 978-0-13-842195-3 }}		* {{cite book\| last = Tausworthe\| first = Robert C\| title = Standardized Development of Computer Software Part 1 Methods\| year = 1977\| publisher = Prentice–Hall, Inc.\| location = Englewood Cliffs NJ\| isbn = 978-0-13-842195-3 }}
	* {{Cite journal\|last=Turing\|first=Alan M.\|author-link=A. M. Turing\|title=On Computable Numbers, With An Application to the Entscheidungsproblem\|journal=[[Proceedings of the London Mathematical Society]]\|series=Series 2\|volume=42\|pages= 230–265 \|year=1936–37\|doi=10.1112/plms/s2-42.1.230 \|s2cid=73712 }}. Corrections, ibid, vol. 43(1937) pp. 544–546. Reprinted in ''The Undecidable'', p. 116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.		* {{Cite journal\|last=Turing\|first=Alan M.\|author-link=A. M. Turing\|title=On Computable Numbers, With An Application to the Entscheidungsproblem\|journal=[[Proceedings of the London Mathematical Society]]\|series=Series 2\|volume=42\|pages= 230–265 \|year=1936–37\|doi=10.1112/plms/s2-42.1.230 \|s2cid=73712 }}. Corrections, ibid, vol. 43(1937) pp. 544–546. Reprinted in ''The Undecidable'', p. 116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.

David Eppstein: /* Etymology */ clean up attestations and sourcing

2025-05-19T01:12:23Z

Etymology: clean up attestations and sourcing

← Previous revision		Revision as of 01:12, 19 May 2025
Line 11:		Line 11:

	== Etymology ==		== Etymology ==
	Around 825 AD, Persian scientist and polymath [[Al-Khwarizmi\|Muḥammad ibn Mūsā al-Khwārizmī]] wrote ''kitāb al-ḥisāb al-hindī'' ("Book of Indian computation") and ''kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī'' ("Addition and subtraction in Indian arithmetic").~~<ref name=":0" />~~ In the early 12th century, Latin translations of ~~said al-Khwarizmi~~ texts involving the [[Hindu–Arabic numeral system]] and [[arithmetic]] appeared, for example ''Liber Alghoarismi de practica arismetrice'', attributed to [[John of Seville]], and ''Liber Algorismi de numero Indorum'', attributed to [[Adelard of Bath]].<ref name=":1">Blair, Ann, Duguid, Paul, Goeing, Anja-Silvia and Grafton, Anthony. Information: A Historical Companion, Princeton: Princeton University Press, 2021. p. 247</ref> ~~Hereby~~, ''alghoarismi'' or ''algorismi'' is the [[Latinisation of names\|Latinization]] of Al-Khwarizmi's name; the text starts with the phrase ''Dixit Algorismi'', or "Thus spoke Al-Khwarizmi".<ref name=":2" /> Around 1230, the English word ''[[algorism]]'' is attested and then by [[Geoffrey Chaucer\|Chaucer]] in 1391, English adopted the French term.<ref name=":3" /><ref name=":4" />{{Clarification needed\|date=April 2024}} In the 15th century, under the influence of the Greek word ἀριθμός (''arithmos'', "number"; ''cf.'' "arithmetic"), the Latin word was altered to ''algorithmus''.{{Citation needed\|date=April 2024}}		Around 825 AD, Persian scientist and polymath [[Al-Khwarizmi\|Muḥammad ibn Mūsā al-Khwārizmī]] wrote ''kitāb al-ḥisāb al-hindī'' ("Book of Indian computation") and ''kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī'' ("Addition and subtraction in Indian arithmetic"). In the early 12th century, Latin translations of these texts involving the [[Hindu–Arabic numeral system]] and [[arithmetic]] appeared, for example ''Liber Alghoarismi de practica arismetrice'', attributed to [[John of Seville]], and ''Liber Algorismi de numero Indorum'', attributed to [[Adelard of Bath]].<ref name=":1">Blair, Ann, Duguid, Paul, Goeing, Anja-Silvia and Grafton, Anthony. Information: A Historical Companion, Princeton: Princeton University Press, 2021. p. 247</ref> Here, ''alghoarismi'' or ''algorismi'' is the [[Latinisation of names\|Latinization]] of Al-Khwarizmi's name;<ref name=":0" /> the text starts with the phrase ''Dixit Algorismi'', or "Thus spoke Al-Khwarizmi".<ref name=":2" />

			The word ''[[algorism]]'' in English came to mean the use of place-value notation in calculations; it occurs in the ''[[Ancrene Wisse]]'' from circa 1225.<ref>{{cite web\|url=https://www.oed.com/dictionary/algorism_n?tl=true\|title=algorism\|work=Oxford English Dictionary\|access-date=2025-05-18}}</ref> By the time [[Geoffrey Chaucer]] wrote ''[[The Canterbury Tales]]'' in the late 14th century, he used a variant of the same word in describing ''augrym stones'', stones used for place-value calculation.<ref>{{cite web\|url=https://chaucer.fas.harvard.edu/pages/millers-prologue-and-tale\|title=The Miller's Tale\|at=Line 3210\|first=Geoffrey\|last=Chaucer}}</ref><ref>{{cite book\|title=A Glossary of Tudor and Stuart Words: Especially from the Dramatists\|editor-first=Anthony Lawson\|editor-last=Mayhew\|first=Walter William\|last=Skeat\|publisher=Clarendon Press\|year=1914\|contribution=agrim, agrum\|pages=5–6\|contribution-url=https://books.google.com/books?id=z58YAAAAIAAJ&pg=PA5}}</ref> In the 15th century, under the influence of the Greek word ἀριθμός (''arithmos'', "number"; ''cf.'' "arithmetic"), the Latin word was altered to ''algorithmus''.<ref>{{cite book
			\| last = Grabiner \| first = Judith V. \| author-link = Judith Grabiner
			\| editor-last = Matthews \| editor-first = Michael R.
			\| contribution = The role of mathematics in liberal arts education
			\| date = December 2013
			\| doi = 10.1007/978-94-007-7654-8_25
			\| isbn = 9789400776548
			\| pages = 793–836
			\| publisher = Springer
			\| title = International Handbook of Research in History, Philosophy and Science Teaching}}</ref> By 1596, this form of the word was used in English, as ''algorithm'', by [[Thomas Hood (mathematician)\|Thomas Hood]].<ref>{{cite web\|url=https://www.oed.com/dictionary/algorithm_n\|title=algorithm\|work=Oxford English Dictionary\|access-date=2025-05-18}}</ref>

	== Definition ==		== Definition ==

David Eppstein: I don't think this essay-like cleanup banner is valid

2025-05-19T00:39:36Z

I don't think this essay-like cleanup banner is valid

← Previous revision		Revision as of 00:39, 19 May 2025
Line 3:		Line 3:
	{{redirect\|Algorythm\|the album\|Beyond Creation}}		{{redirect\|Algorythm\|the album\|Beyond Creation}}
	{{Use mdy dates\|date=September 2017}}		{{Use mdy dates\|date=September 2017}}

	{{Essay\|date=April 2024}}

	[[File:GCD through successive subtractions.svg\|thumb\|Flowchart of using successive subtractions to find the [[greatest common divisor]] of number ''r'' and ''s''\|alt=In a loop, subtract the larger number against the smaller number. Halt the loop when the subtraction will make a number negative. Assess two numbers, whether one of them is equal to zero or not. If yes, take the other number as the greatest common divisor. If no, put the two numbers in the subtraction loop again.]]		[[File:GCD through successive subtractions.svg\|thumb\|Flowchart of using successive subtractions to find the [[greatest common divisor]] of number ''r'' and ''s''\|alt=In a loop, subtract the larger number against the smaller number. Halt the loop when the subtraction will make a number negative. Assess two numbers, whether one of them is equal to zero or not. If yes, take the other number as the greatest common divisor. If no, put the two numbers in the subtraction loop again.]]
	In [[mathematics]] and [[computer science]], an '''algorithm''' ({{IPAc-en\|audio=en-us-algorithm.ogg\|ˈ\|æ\|l\|ɡ\|ə\|r\|ɪ\|ð\|əm}}) is a finite sequence of [[Rigour#Mathematics\|mathematically rigorous]] instructions, typically used to solve a class of specific [[Computational problem\|problem]]s or to perform a [[computation]].<ref name=":0">{{Cite web\|url=https://www.merriam-webster.com/dictionary/algorithm\|title=Definition of ALGORITHM\|work=Merriam-Webster Online Dictionary \|language=en \|access-date=2019-11-14 \|archive-url=https://web.archive.org/web/20200214074446/https://www.merriam-webster.com/dictionary/algorithm \|archive-date=February 14, 2020\|url-status=live}}</ref> Algorithms are used as specifications for performing [[calculation]]s and [[data processing]]. More advanced algorithms can use [[Conditional (computer programming)\|conditional]]s to divert the code execution through various routes (referred to as [[automated decision-making]]) and deduce valid [[inference]]s (referred to as [[automated reasoning]]).		In [[mathematics]] and [[computer science]], an '''algorithm''' ({{IPAc-en\|audio=en-us-algorithm.ogg\|ˈ\|æ\|l\|ɡ\|ə\|r\|ɪ\|ð\|əm}}) is a finite sequence of [[Rigour#Mathematics\|mathematically rigorous]] instructions, typically used to solve a class of specific [[Computational problem\|problem]]s or to perform a [[computation]].<ref name=":0">{{Cite web\|url=https://www.merriam-webster.com/dictionary/algorithm\|title=Definition of ALGORITHM\|work=Merriam-Webster Online Dictionary \|language=en \|access-date=2019-11-14 \|archive-url=https://web.archive.org/web/20200214074446/https://www.merriam-webster.com/dictionary/algorithm \|archive-date=February 14, 2020\|url-status=live}}</ref> Algorithms are used as specifications for performing [[calculation]]s and [[data processing]]. More advanced algorithms can use [[Conditional (computer programming)\|conditional]]s to divert the code execution through various routes (referred to as [[automated decision-making]]) and deduce valid [[inference]]s (referred to as [[automated reasoning]]).

MrOllie: Restored revision 1284606175 by Remsense (talk): Broken ref

2025-04-29T16:32:38Z

Restored revision 1284606175 by Remsense (talk): Broken ref

← Previous revision		Revision as of 16:32, 29 April 2025
Line 34:		Line 34:
	Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in [[Babylonian mathematics]] (around 2500 BC),<ref name="Springer Science & Business Media">{{cite book \|last1=Chabert \|first1=Jean-Luc \|title=A History of Algorithms: From the Pebble to the Microchip \|date=2012 \|publisher=Springer Science & Business Media \|isbn=9783642181924 \|pages=7–8}}</ref> [[Egyptian mathematics]] (around 1550 BC),<ref name="Springer Science & Business Media" /> [[Indian mathematics]] (around 800 BC and later),<ref name=":6">{{cite book \|last1=Sriram \|first1=M. S. \|editor1-last=Emch \|editor1-first=Gerard G. \|editor2-last=Sridharan \|editor2-first=R. \|editor3-last=Srinivas \|editor3-first=M. D. \|title=Contributions to the History of Indian Mathematics \|date=2005 \|publisher=Springer \|isbn=978-93-86279-25-5 \|page=153 \|chapter-url=https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153 \|language=en \|chapter=Algorithms in Indian Mathematics}}</ref><ref>Hayashi, T. (2023, January 1). [https://www.britannica.com/biography/Brahmagupta Brahmagupta]. Encyclopedia Britannica.</ref> the Ifa Oracle (around 500 BC),<ref>{{Cite journal \|last=Zaslavsky \|first=Claudia \|date=1970 \|title=Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria \|url=https://www.jstor.org/stable/3027363 \|journal=The Two-Year College Mathematics Journal \|volume=1 \|issue=2 \|pages=76–99 \|doi=10.2307/3027363 \|jstor=3027363 \|issn=0049-4925}}</ref> [[Greek mathematics]] (around 240 BC),<ref name="Cooke2005">{{cite book\|last=Cooke\|first=Roger L.\|title=The History of Mathematics: A Brief Course\|date=2005\|publisher=John Wiley & Sons\|isbn=978-1-118-46029-0}}</ref> [[Chinese mathematics\|Chinese mathematics (around 200 BC and later)]],<ref>{{Cite journal \|date=1999 \|editor-last=Chabert \|editor-first=Jean-Luc \|title=A History of Algorithms \|url=https://link.springer.com/book/10.1007/978-3-642-18192-4 \|journal=SpringerLink \|language=en \|doi=10.1007/978-3-642-18192-4\|isbn=978-3-540-63369-3 }}</ref> and [[Arabic mathematics]] (around 800 AD).<ref name="Dooley">{{cite book \|last1=Dooley \|first1=John F. \|title=A Brief History of Cryptology and Cryptographic Algorithms \|date=2013 \|publisher=Springer Science & Business Media \|isbn=9783319016283 \|pages=12–3}}</ref>		Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in [[Babylonian mathematics]] (around 2500 BC),<ref name="Springer Science & Business Media">{{cite book \|last1=Chabert \|first1=Jean-Luc \|title=A History of Algorithms: From the Pebble to the Microchip \|date=2012 \|publisher=Springer Science & Business Media \|isbn=9783642181924 \|pages=7–8}}</ref> [[Egyptian mathematics]] (around 1550 BC),<ref name="Springer Science & Business Media" /> [[Indian mathematics]] (around 800 BC and later),<ref name=":6">{{cite book \|last1=Sriram \|first1=M. S. \|editor1-last=Emch \|editor1-first=Gerard G. \|editor2-last=Sridharan \|editor2-first=R. \|editor3-last=Srinivas \|editor3-first=M. D. \|title=Contributions to the History of Indian Mathematics \|date=2005 \|publisher=Springer \|isbn=978-93-86279-25-5 \|page=153 \|chapter-url=https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153 \|language=en \|chapter=Algorithms in Indian Mathematics}}</ref><ref>Hayashi, T. (2023, January 1). [https://www.britannica.com/biography/Brahmagupta Brahmagupta]. Encyclopedia Britannica.</ref> the Ifa Oracle (around 500 BC),<ref>{{Cite journal \|last=Zaslavsky \|first=Claudia \|date=1970 \|title=Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria \|url=https://www.jstor.org/stable/3027363 \|journal=The Two-Year College Mathematics Journal \|volume=1 \|issue=2 \|pages=76–99 \|doi=10.2307/3027363 \|jstor=3027363 \|issn=0049-4925}}</ref> [[Greek mathematics]] (around 240 BC),<ref name="Cooke2005">{{cite book\|last=Cooke\|first=Roger L.\|title=The History of Mathematics: A Brief Course\|date=2005\|publisher=John Wiley & Sons\|isbn=978-1-118-46029-0}}</ref> [[Chinese mathematics\|Chinese mathematics (around 200 BC and later)]],<ref>{{Cite journal \|date=1999 \|editor-last=Chabert \|editor-first=Jean-Luc \|title=A History of Algorithms \|url=https://link.springer.com/book/10.1007/978-3-642-18192-4 \|journal=SpringerLink \|language=en \|doi=10.1007/978-3-642-18192-4\|isbn=978-3-540-63369-3 }}</ref> and [[Arabic mathematics]] (around 800 AD).<ref name="Dooley">{{cite book \|last1=Dooley \|first1=John F. \|title=A Brief History of Cryptology and Cryptographic Algorithms \|date=2013 \|publisher=Springer Science & Business Media \|isbn=9783319016283 \|pages=12–3}}</ref>

	The earliest evidence of algorithms is found in ancient [[Mesopotamia]]n mathematics. A [[Sumer]]ian clay tablet found in [[Shuruppak]] near [[Baghdad]] and dated to {{Circa\|2500 BC}} describes the earliest [[division algorithm]].<ref name="Springer Science & Business Media" /> During the [[First Babylonian dynasty\|Hammurabi dynasty]] {{Circa\|1800\|1600 BC\|lk=no}}, [[Babylonia]]n clay tablets described algorithms for computing formulas.<ref>{{cite journal \|last1=Knuth \|first1=Donald E. \|date=1972 \|title=Ancient Babylonian Algorithms \|url=http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|url-status=dead \|journal=Commun. ACM \|volume=15 \|issue=7 \|pages=671–677 \|doi=10.1145/361454.361514 \|issn=0001-0782 \|s2cid=7829945 \|archive-url=https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|archive-date=2012-12-24}}</ref> Algorithms were also used in [[Babylonian astronomy]].~~<ref>~~{{~~Cite book~~ \|~~last~~= ~~\|first= \|title= \|publisher= \|year= \|isbn= \|edition= \|language=~~}}~~</ref>~~ Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<ref>{{cite book \|last=Aaboe \|first=Asger \|author-link=Asger Aaboe \|title=Episodes from the Early History of Astronomy \|date=2001 \|publisher=Springer \|isbn=978-0-387-95136-2 \|place=New York \|pages=40–62}}</ref>		The earliest evidence of algorithms is found in ancient [[Mesopotamia]]n mathematics. A [[Sumer]]ian clay tablet found in [[Shuruppak]] near [[Baghdad]] and dated to {{Circa\|2500 BC}} describes the earliest [[division algorithm]].<ref name="Springer Science & Business Media" /> During the [[First Babylonian dynasty\|Hammurabi dynasty]] {{Circa\|1800\|1600 BC\|lk=no}}, [[Babylonia]]n clay tablets described algorithms for computing formulas.<ref>{{cite journal \|last1=Knuth \|first1=Donald E. \|date=1972 \|title=Ancient Babylonian Algorithms \|url=http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|url-status=dead \|journal=Commun. ACM \|volume=15 \|issue=7 \|pages=671–677 \|doi=10.1145/361454.361514 \|issn=0001-0782 \|s2cid=7829945 \|archive-url=https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|archive-date=2012-12-24}}</ref> Algorithms were also used in [[Babylonian astronomy]].{{Citation needed\|date=March 2025}} Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<ref>{{cite book \|last=Aaboe \|first=Asger \|author-link=Asger Aaboe \|title=Episodes from the Early History of Astronomy \|date=2001 \|publisher=Springer \|isbn=978-0-387-95136-2 \|place=New York \|pages=40–62}}</ref>

	Algorithms for arithmetic are also found in ancient [[Egyptian mathematics]], dating back to the [[Rhind Mathematical Papyrus]] {{Circa\|1550 BC\|lk=no}}.<ref name="Springer Science & Business Media" /> Algorithms were later used in ancient [[Hellenistic mathematics]]. Two examples are the [[Sieve of Eratosthenes]], which was described in the ''[[Introduction to Arithmetic]]'' by [[Nicomachus]],<ref>{{cite web \|last=Ast \|first=Courtney \|title=Eratosthenes \|url=http://www.math.wichita.edu/history/men/eratosthenes.html \|url-status=live \|archive-url=https://web.archive.org/web/20150227150653/http://www.math.wichita.edu/history/men/eratosthenes.html \|archive-date=February 27, 2015 \|access-date=February 27, 2015 \|publisher=Wichita State University: Department of Mathematics and Statistics}}</ref><ref name="Cooke2005" />{{rp\|Ch 9.2}} and the [[Euclidean algorithm]], which was first described in ''[[Euclid's Elements]]'' ({{circa\|300 BC\|lk=no}}).<ref name="Cooke2005" />{{rp\|Ch 9.1}}Examples of ancient Indian mathematics included the [[Shulba Sutras]], the [[Kerala school of astronomy and mathematics\|Kerala School]], and the [[Brāhmasphuṭasiddhānta]].<ref name=":6" />		Algorithms for arithmetic are also found in ancient [[Egyptian mathematics]], dating back to the [[Rhind Mathematical Papyrus]] {{Circa\|1550 BC\|lk=no}}.<ref name="Springer Science & Business Media" /> Algorithms were later used in ancient [[Hellenistic mathematics]]. Two examples are the [[Sieve of Eratosthenes]], which was described in the ''[[Introduction to Arithmetic]]'' by [[Nicomachus]],<ref>{{cite web \|last=Ast \|first=Courtney \|title=Eratosthenes \|url=http://www.math.wichita.edu/history/men/eratosthenes.html \|url-status=live \|archive-url=https://web.archive.org/web/20150227150653/http://www.math.wichita.edu/history/men/eratosthenes.html \|archive-date=February 27, 2015 \|access-date=February 27, 2015 \|publisher=Wichita State University: Department of Mathematics and Statistics}}</ref><ref name="Cooke2005" />{{rp\|Ch 9.2}} and the [[Euclidean algorithm]], which was first described in ''[[Euclid's Elements]]'' ({{circa\|300 BC\|lk=no}}).<ref name="Cooke2005" />{{rp\|Ch 9.1}}Examples of ancient Indian mathematics included the [[Shulba Sutras]], the [[Kerala school of astronomy and mathematics\|Kerala School]], and the [[Brāhmasphuṭasiddhānta]].<ref name=":6" />

Buggybops: added citation about Babylonian Astronomy

2025-04-29T16:29:51Z

added citation about Babylonian Astronomy

← Previous revision		Revision as of 16:29, 29 April 2025
Line 34:		Line 34:
	Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in [[Babylonian mathematics]] (around 2500 BC),<ref name="Springer Science & Business Media">{{cite book \|last1=Chabert \|first1=Jean-Luc \|title=A History of Algorithms: From the Pebble to the Microchip \|date=2012 \|publisher=Springer Science & Business Media \|isbn=9783642181924 \|pages=7–8}}</ref> [[Egyptian mathematics]] (around 1550 BC),<ref name="Springer Science & Business Media" /> [[Indian mathematics]] (around 800 BC and later),<ref name=":6">{{cite book \|last1=Sriram \|first1=M. S. \|editor1-last=Emch \|editor1-first=Gerard G. \|editor2-last=Sridharan \|editor2-first=R. \|editor3-last=Srinivas \|editor3-first=M. D. \|title=Contributions to the History of Indian Mathematics \|date=2005 \|publisher=Springer \|isbn=978-93-86279-25-5 \|page=153 \|chapter-url=https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153 \|language=en \|chapter=Algorithms in Indian Mathematics}}</ref><ref>Hayashi, T. (2023, January 1). [https://www.britannica.com/biography/Brahmagupta Brahmagupta]. Encyclopedia Britannica.</ref> the Ifa Oracle (around 500 BC),<ref>{{Cite journal \|last=Zaslavsky \|first=Claudia \|date=1970 \|title=Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria \|url=https://www.jstor.org/stable/3027363 \|journal=The Two-Year College Mathematics Journal \|volume=1 \|issue=2 \|pages=76–99 \|doi=10.2307/3027363 \|jstor=3027363 \|issn=0049-4925}}</ref> [[Greek mathematics]] (around 240 BC),<ref name="Cooke2005">{{cite book\|last=Cooke\|first=Roger L.\|title=The History of Mathematics: A Brief Course\|date=2005\|publisher=John Wiley & Sons\|isbn=978-1-118-46029-0}}</ref> [[Chinese mathematics\|Chinese mathematics (around 200 BC and later)]],<ref>{{Cite journal \|date=1999 \|editor-last=Chabert \|editor-first=Jean-Luc \|title=A History of Algorithms \|url=https://link.springer.com/book/10.1007/978-3-642-18192-4 \|journal=SpringerLink \|language=en \|doi=10.1007/978-3-642-18192-4\|isbn=978-3-540-63369-3 }}</ref> and [[Arabic mathematics]] (around 800 AD).<ref name="Dooley">{{cite book \|last1=Dooley \|first1=John F. \|title=A Brief History of Cryptology and Cryptographic Algorithms \|date=2013 \|publisher=Springer Science & Business Media \|isbn=9783319016283 \|pages=12–3}}</ref>		Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in [[Babylonian mathematics]] (around 2500 BC),<ref name="Springer Science & Business Media">{{cite book \|last1=Chabert \|first1=Jean-Luc \|title=A History of Algorithms: From the Pebble to the Microchip \|date=2012 \|publisher=Springer Science & Business Media \|isbn=9783642181924 \|pages=7–8}}</ref> [[Egyptian mathematics]] (around 1550 BC),<ref name="Springer Science & Business Media" /> [[Indian mathematics]] (around 800 BC and later),<ref name=":6">{{cite book \|last1=Sriram \|first1=M. S. \|editor1-last=Emch \|editor1-first=Gerard G. \|editor2-last=Sridharan \|editor2-first=R. \|editor3-last=Srinivas \|editor3-first=M. D. \|title=Contributions to the History of Indian Mathematics \|date=2005 \|publisher=Springer \|isbn=978-93-86279-25-5 \|page=153 \|chapter-url=https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153 \|language=en \|chapter=Algorithms in Indian Mathematics}}</ref><ref>Hayashi, T. (2023, January 1). [https://www.britannica.com/biography/Brahmagupta Brahmagupta]. Encyclopedia Britannica.</ref> the Ifa Oracle (around 500 BC),<ref>{{Cite journal \|last=Zaslavsky \|first=Claudia \|date=1970 \|title=Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria \|url=https://www.jstor.org/stable/3027363 \|journal=The Two-Year College Mathematics Journal \|volume=1 \|issue=2 \|pages=76–99 \|doi=10.2307/3027363 \|jstor=3027363 \|issn=0049-4925}}</ref> [[Greek mathematics]] (around 240 BC),<ref name="Cooke2005">{{cite book\|last=Cooke\|first=Roger L.\|title=The History of Mathematics: A Brief Course\|date=2005\|publisher=John Wiley & Sons\|isbn=978-1-118-46029-0}}</ref> [[Chinese mathematics\|Chinese mathematics (around 200 BC and later)]],<ref>{{Cite journal \|date=1999 \|editor-last=Chabert \|editor-first=Jean-Luc \|title=A History of Algorithms \|url=https://link.springer.com/book/10.1007/978-3-642-18192-4 \|journal=SpringerLink \|language=en \|doi=10.1007/978-3-642-18192-4\|isbn=978-3-540-63369-3 }}</ref> and [[Arabic mathematics]] (around 800 AD).<ref name="Dooley">{{cite book \|last1=Dooley \|first1=John F. \|title=A Brief History of Cryptology and Cryptographic Algorithms \|date=2013 \|publisher=Springer Science & Business Media \|isbn=9783319016283 \|pages=12–3}}</ref>

	The earliest evidence of algorithms is found in ancient [[Mesopotamia]]n mathematics. A [[Sumer]]ian clay tablet found in [[Shuruppak]] near [[Baghdad]] and dated to {{Circa\|2500 BC}} describes the earliest [[division algorithm]].<ref name="Springer Science & Business Media" /> During the [[First Babylonian dynasty\|Hammurabi dynasty]] {{Circa\|1800\|1600 BC\|lk=no}}, [[Babylonia]]n clay tablets described algorithms for computing formulas.<ref>{{cite journal \|last1=Knuth \|first1=Donald E. \|date=1972 \|title=Ancient Babylonian Algorithms \|url=http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|url-status=dead \|journal=Commun. ACM \|volume=15 \|issue=7 \|pages=671–677 \|doi=10.1145/361454.361514 \|issn=0001-0782 \|s2cid=7829945 \|archive-url=https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|archive-date=2012-12-24}}</ref> Algorithms were also used in [[Babylonian astronomy]].{{~~Citation~~ ~~needed~~\|~~date~~=~~March~~ ~~2025~~}} Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<ref>{{cite book \|last=Aaboe \|first=Asger \|author-link=Asger Aaboe \|title=Episodes from the Early History of Astronomy \|date=2001 \|publisher=Springer \|isbn=978-0-387-95136-2 \|place=New York \|pages=40–62}}</ref>		The earliest evidence of algorithms is found in ancient [[Mesopotamia]]n mathematics. A [[Sumer]]ian clay tablet found in [[Shuruppak]] near [[Baghdad]] and dated to {{Circa\|2500 BC}} describes the earliest [[division algorithm]].<ref name="Springer Science & Business Media" /> During the [[First Babylonian dynasty\|Hammurabi dynasty]] {{Circa\|1800\|1600 BC\|lk=no}}, [[Babylonia]]n clay tablets described algorithms for computing formulas.<ref>{{cite journal \|last1=Knuth \|first1=Donald E. \|date=1972 \|title=Ancient Babylonian Algorithms \|url=http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|url-status=dead \|journal=Commun. ACM \|volume=15 \|issue=7 \|pages=671–677 \|doi=10.1145/361454.361514 \|issn=0001-0782 \|s2cid=7829945 \|archive-url=https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|archive-date=2012-12-24}}</ref> Algorithms were also used in [[Babylonian astronomy]].<ref>{{Cite book \|last= \|first= \|title= \|publisher= \|year= \|isbn= \|edition= \|language=}}</ref> Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<ref>{{cite book \|last=Aaboe \|first=Asger \|author-link=Asger Aaboe \|title=Episodes from the Early History of Astronomy \|date=2001 \|publisher=Springer \|isbn=978-0-387-95136-2 \|place=New York \|pages=40–62}}</ref>

	Algorithms for arithmetic are also found in ancient [[Egyptian mathematics]], dating back to the [[Rhind Mathematical Papyrus]] {{Circa\|1550 BC\|lk=no}}.<ref name="Springer Science & Business Media" /> Algorithms were later used in ancient [[Hellenistic mathematics]]. Two examples are the [[Sieve of Eratosthenes]], which was described in the ''[[Introduction to Arithmetic]]'' by [[Nicomachus]],<ref>{{cite web \|last=Ast \|first=Courtney \|title=Eratosthenes \|url=http://www.math.wichita.edu/history/men/eratosthenes.html \|url-status=live \|archive-url=https://web.archive.org/web/20150227150653/http://www.math.wichita.edu/history/men/eratosthenes.html \|archive-date=February 27, 2015 \|access-date=February 27, 2015 \|publisher=Wichita State University: Department of Mathematics and Statistics}}</ref><ref name="Cooke2005" />{{rp\|Ch 9.2}} and the [[Euclidean algorithm]], which was first described in ''[[Euclid's Elements]]'' ({{circa\|300 BC\|lk=no}}).<ref name="Cooke2005" />{{rp\|Ch 9.1}}Examples of ancient Indian mathematics included the [[Shulba Sutras]], the [[Kerala school of astronomy and mathematics\|Kerala School]], and the [[Brāhmasphuṭasiddhānta]].<ref name=":6" />		Algorithms for arithmetic are also found in ancient [[Egyptian mathematics]], dating back to the [[Rhind Mathematical Papyrus]] {{Circa\|1550 BC\|lk=no}}.<ref name="Springer Science & Business Media" /> Algorithms were later used in ancient [[Hellenistic mathematics]]. Two examples are the [[Sieve of Eratosthenes]], which was described in the ''[[Introduction to Arithmetic]]'' by [[Nicomachus]],<ref>{{cite web \|last=Ast \|first=Courtney \|title=Eratosthenes \|url=http://www.math.wichita.edu/history/men/eratosthenes.html \|url-status=live \|archive-url=https://web.archive.org/web/20150227150653/http://www.math.wichita.edu/history/men/eratosthenes.html \|archive-date=February 27, 2015 \|access-date=February 27, 2015 \|publisher=Wichita State University: Department of Mathematics and Statistics}}</ref><ref name="Cooke2005" />{{rp\|Ch 9.2}} and the [[Euclidean algorithm]], which was first described in ''[[Euclid's Elements]]'' ({{circa\|300 BC\|lk=no}}).<ref name="Cooke2005" />{{rp\|Ch 9.1}}Examples of ancient Indian mathematics included the [[Shulba Sutras]], the [[Kerala school of astronomy and mathematics\|Kerala School]], and the [[Brāhmasphuṭasiddhānta]].<ref name=":6" />

Remsense: Reverted 1 edit by Kz11522 (talk) to last revision by Janan2025

2025-04-08T17:24:16Z

Reverted 1 edit by Kz11522 (talk) to last revision by Janan2025

← Previous revision		Revision as of 17:24, 8 April 2025
Line 1:		Line 1:
	{{Short description\|Sequence of operations for a task}}		{{Short description\|Sequence of operations for a task}}
	{{Redirect\|Algorithms\|the ~~sub field~~ of computer science\|Analysis of algorithms\|other uses\|Algorithm (disambiguation)}}		{{Redirect\|Algorithms\|the subfield of computer science\|Analysis of algorithms\|other uses\|Algorithm (disambiguation)}}
	{{redirect\|Algorythm\|the album\|Beyond Creation}}		{{redirect\|Algorythm\|the album\|Beyond Creation}}
	{{Use mdy dates\|date=September 2017}}		{{Use mdy dates\|date=September 2017}}
Line 53:		Line 53:
	[[File:Diagram for the computation of Bernoulli numbers.jpg\|thumb\|[[Ada Lovelace]]'s diagram from "[[Note G]]", the first published computer algorithm]]		[[File:Diagram for the computation of Bernoulli numbers.jpg\|thumb\|[[Ada Lovelace]]'s diagram from "[[Note G]]", the first published computer algorithm]]

	In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve the ''[[Entscheidungsproblem]] ''(decision problem) posed by [[David Hilbert]]. Later formalizations were framed as attempts to define "[[effective calculability]]"<ref>Kleene 1943 in Davis 1965:274</ref> or "effective method".<ref>Rosser 1939 in Davis 1965:225</ref> Those formalizations included the [[Kurt Gödel\|Gödel]]-[[Jacques Herbrand\|Herbrand]]-[[Stephen Cole Kleene\|Kleene]] recursive functions of 1930, 1934 and 1935, [[Alonzo Church]]'s [[lambda calculus]] of 1936, [[Emil Post]]'s [[Formulation 1]] of 1936, and [[Alan Turing]]'s [[Turing machines]] of 1936–37 and 1939.		In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve the ''[[Entscheidungsproblem]] ''(decision problem) posed by [[David Hilbert]]. Later formalizations were framed as attempts to define "[[effective calculability]]"<ref>Kleene 1943 in Davis 1965:274</ref> or "effective method".<ref>Rosser 1939 in Davis 1965:225</ref> Those formalizations included the [[Kurt Gödel\|Gödel]]–[[Jacques Herbrand\|Herbrand]]–[[Stephen Cole Kleene\|Kleene]] recursive functions of 1930, 1934 and 1935, [[Alonzo Church]]'s [[lambda calculus]] of 1936, [[Emil Post]]'s [[Formulation 1]] of 1936, and [[Alan Turing]]'s [[Turing machines]] of 1936–37 and 1939.

	==Representations==		==Representations==

Kz11522: changed hyphens

2025-04-08T16:49:37Z

changed hyphens

← Previous revision		Revision as of 16:49, 8 April 2025
Line 1:		Line 1:
	{{Short description\|Sequence of operations for a task}}		{{Short description\|Sequence of operations for a task}}
	{{Redirect\|Algorithms\|the ~~subfield~~ of computer science\|Analysis of algorithms\|other uses\|Algorithm (disambiguation)}}		{{Redirect\|Algorithms\|the sub field of computer science\|Analysis of algorithms\|other uses\|Algorithm (disambiguation)}}
	{{redirect\|Algorythm\|the album\|Beyond Creation}}		{{redirect\|Algorythm\|the album\|Beyond Creation}}
	{{Use mdy dates\|date=September 2017}}		{{Use mdy dates\|date=September 2017}}
Line 53:		Line 53:
	[[File:Diagram for the computation of Bernoulli numbers.jpg\|thumb\|[[Ada Lovelace]]'s diagram from "[[Note G]]", the first published computer algorithm]]		[[File:Diagram for the computation of Bernoulli numbers.jpg\|thumb\|[[Ada Lovelace]]'s diagram from "[[Note G]]", the first published computer algorithm]]

	In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve the ''[[Entscheidungsproblem]] ''(decision problem) posed by [[David Hilbert]]. Later formalizations were framed as attempts to define "[[effective calculability]]"<ref>Kleene 1943 in Davis 1965:274</ref> or "effective method".<ref>Rosser 1939 in Davis 1965:225</ref> Those formalizations included the [[Kurt Gödel\|Gödel]]–[[Jacques Herbrand\|Herbrand]]–[[Stephen Cole Kleene\|Kleene]] recursive functions of 1930, 1934 and 1935, [[Alonzo Church]]'s [[lambda calculus]] of 1936, [[Emil Post]]'s [[Formulation 1]] of 1936, and [[Alan Turing]]'s [[Turing machines]] of 1936–37 and 1939.		In 1928, a partial formalization of the modern concept of algorithms began with attempts to solve the ''[[Entscheidungsproblem]] ''(decision problem) posed by [[David Hilbert]]. Later formalizations were framed as attempts to define "[[effective calculability]]"<ref>Kleene 1943 in Davis 1965:274</ref> or "effective method".<ref>Rosser 1939 in Davis 1965:225</ref> Those formalizations included the [[Kurt Gödel\|Gödel]]-[[Jacques Herbrand\|Herbrand]]-[[Stephen Cole Kleene\|Kleene]] recursive functions of 1930, 1934 and 1935, [[Alonzo Church]]'s [[lambda calculus]] of 1936, [[Emil Post]]'s [[Formulation 1]] of 1936, and [[Alan Turing]]'s [[Turing machines]] of 1936–37 and 1939.

	==Representations==		==Representations==

Janan2025: This edit appears unconstructive.

2025-03-30T14:48:28Z

This edit appears unconstructive.

← Previous revision		Revision as of 14:48, 30 March 2025
Line 7:		Line 7:

	[[File:GCD through successive subtractions.svg\|thumb\|Flowchart of using successive subtractions to find the [[greatest common divisor]] of number ''r'' and ''s''\|alt=In a loop, subtract the larger number against the smaller number. Halt the loop when the subtraction will make a number negative. Assess two numbers, whether one of them is equal to zero or not. If yes, take the other number as the greatest common divisor. If no, put the two numbers in the subtraction loop again.]]		[[File:GCD through successive subtractions.svg\|thumb\|Flowchart of using successive subtractions to find the [[greatest common divisor]] of number ''r'' and ''s''\|alt=In a loop, subtract the larger number against the smaller number. Halt the loop when the subtraction will make a number negative. Assess two numbers, whether one of them is equal to zero or not. If yes, take the other number as the greatest common divisor. If no, put the two numbers in the subtraction loop again.]]
	In [[mathematics]] and [[computer science]], an '''algorithm''' ({{IPAc-en\|ˈ}}) is a finite sequence of [[Rigour#Mathematics\|mathematically rigorous]] instructions, typically used to solve a class of specific [[Computational problem\|problem]]s or to perform a [[computation]].<ref name=":0">{{Cite web\|url=https://www.merriam-webster.com/dictionary/algorithm\|title=Definition of ALGORITHM\|work=Merriam-Webster Online Dictionary \|language=en \|access-date=2019-11-14 \|archive-url=https://web.archive.org/web/20200214074446/https://www.merriam-webster.com/dictionary/algorithm \|archive-date=February 14, 2020\|url-status=live}}</ref> Algorithms are used as specifications for performing [[calculation]]s and [[data processing]]. More advanced algorithms can use [[Conditional (computer programming)\|conditional]]s to divert the code execution through various routes (referred to as [[automated decision-making]]) and deduce valid [[inference]]s (referred to as [[automated reasoning]]).		In [[mathematics]] and [[computer science]], an '''algorithm''' ({{IPAc-en\|audio=en-us-algorithm.ogg\|ˈ\|æ\|l\|ɡ\|ə\|r\|ɪ\|ð\|əm}}) is a finite sequence of [[Rigour#Mathematics\|mathematically rigorous]] instructions, typically used to solve a class of specific [[Computational problem\|problem]]s or to perform a [[computation]].<ref name=":0">{{Cite web\|url=https://www.merriam-webster.com/dictionary/algorithm\|title=Definition of ALGORITHM\|work=Merriam-Webster Online Dictionary \|language=en \|access-date=2019-11-14 \|archive-url=https://web.archive.org/web/20200214074446/https://www.merriam-webster.com/dictionary/algorithm \|archive-date=February 14, 2020\|url-status=live}}</ref> Algorithms are used as specifications for performing [[calculation]]s and [[data processing]]. More advanced algorithms can use [[Conditional (computer programming)\|conditional]]s to divert the code execution through various routes (referred to as [[automated decision-making]]) and deduce valid [[inference]]s (referred to as [[automated reasoning]]).

	In contrast, a [[Heuristic (computer science)\|heuristic]] is an approach to solving problems without well-defined correct or optimal results.<ref name=":2">David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, {{isbn\|1402030045}}</ref> For example, although social media [[recommender system]]s are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation.		In contrast, a [[Heuristic (computer science)\|heuristic]] is an approach to solving problems without well-defined correct or optimal results.<ref name=":2">David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, {{isbn\|1402030045}}</ref> For example, although social media [[recommender system]]s are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation.
Line 284:		Line 284:
	{{Algorithmic paradigms}}		{{Algorithmic paradigms}}
	{{Authority control}}		{{Authority control}}

	__NOEDITSECTION__
			[[Category:Algorithms\| ]]
			[[Category:Articles with example pseudocode]]
			[[Category:Mathematical logic]]
			[[Category:Theoretical computer science]]

← Previous revision		Revision as of 01:25, 19 May 2025
Line 42:		Line 42:
	Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in [[Babylonian mathematics]] (around 2500 BC),<ref name="Springer Science & Business Media">{{cite book \|last1=Chabert \|first1=Jean-Luc \|title=A History of Algorithms: From the Pebble to the Microchip \|date=2012 \|publisher=Springer Science & Business Media \|isbn=9783642181924 \|pages=7–8}}</ref> [[Egyptian mathematics]] (around 1550 BC),<ref name="Springer Science & Business Media" /> [[Indian mathematics]] (around 800 BC and later),<ref name=":6">{{cite book \|last1=Sriram \|first1=M. S. \|editor1-last=Emch \|editor1-first=Gerard G. \|editor2-last=Sridharan \|editor2-first=R. \|editor3-last=Srinivas \|editor3-first=M. D. \|title=Contributions to the History of Indian Mathematics \|date=2005 \|publisher=Springer \|isbn=978-93-86279-25-5 \|page=153 \|chapter-url=https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153 \|language=en \|chapter=Algorithms in Indian Mathematics}}</ref><ref>Hayashi, T. (2023, January 1). [https://www.britannica.com/biography/Brahmagupta Brahmagupta]. Encyclopedia Britannica.</ref> the Ifa Oracle (around 500 BC),<ref>{{Cite journal \|last=Zaslavsky \|first=Claudia \|date=1970 \|title=Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria \|url=https://www.jstor.org/stable/3027363 \|journal=The Two-Year College Mathematics Journal \|volume=1 \|issue=2 \|pages=76–99 \|doi=10.2307/3027363 \|jstor=3027363 \|issn=0049-4925}}</ref> [[Greek mathematics]] (around 240 BC),<ref name="Cooke2005">{{cite book\|last=Cooke\|first=Roger L.\|title=The History of Mathematics: A Brief Course\|date=2005\|publisher=John Wiley & Sons\|isbn=978-1-118-46029-0}}</ref> [[Chinese mathematics\|Chinese mathematics (around 200 BC and later)]],<ref>{{Cite journal \|date=1999 \|editor-last=Chabert \|editor-first=Jean-Luc \|title=A History of Algorithms \|url=https://link.springer.com/book/10.1007/978-3-642-18192-4 \|journal=SpringerLink \|language=en \|doi=10.1007/978-3-642-18192-4\|isbn=978-3-540-63369-3 }}</ref> and [[Arabic mathematics]] (around 800 AD).<ref name="Dooley">{{cite book \|last1=Dooley \|first1=John F. \|title=A Brief History of Cryptology and Cryptographic Algorithms \|date=2013 \|publisher=Springer Science & Business Media \|isbn=9783319016283 \|pages=12–3}}</ref>		Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in [[Babylonian mathematics]] (around 2500 BC),<ref name="Springer Science & Business Media">{{cite book \|last1=Chabert \|first1=Jean-Luc \|title=A History of Algorithms: From the Pebble to the Microchip \|date=2012 \|publisher=Springer Science & Business Media \|isbn=9783642181924 \|pages=7–8}}</ref> [[Egyptian mathematics]] (around 1550 BC),<ref name="Springer Science & Business Media" /> [[Indian mathematics]] (around 800 BC and later),<ref name=":6">{{cite book \|last1=Sriram \|first1=M. S. \|editor1-last=Emch \|editor1-first=Gerard G. \|editor2-last=Sridharan \|editor2-first=R. \|editor3-last=Srinivas \|editor3-first=M. D. \|title=Contributions to the History of Indian Mathematics \|date=2005 \|publisher=Springer \|isbn=978-93-86279-25-5 \|page=153 \|chapter-url=https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153 \|language=en \|chapter=Algorithms in Indian Mathematics}}</ref><ref>Hayashi, T. (2023, January 1). [https://www.britannica.com/biography/Brahmagupta Brahmagupta]. Encyclopedia Britannica.</ref> the Ifa Oracle (around 500 BC),<ref>{{Cite journal \|last=Zaslavsky \|first=Claudia \|date=1970 \|title=Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria \|url=https://www.jstor.org/stable/3027363 \|journal=The Two-Year College Mathematics Journal \|volume=1 \|issue=2 \|pages=76–99 \|doi=10.2307/3027363 \|jstor=3027363 \|issn=0049-4925}}</ref> [[Greek mathematics]] (around 240 BC),<ref name="Cooke2005">{{cite book\|last=Cooke\|first=Roger L.\|title=The History of Mathematics: A Brief Course\|date=2005\|publisher=John Wiley & Sons\|isbn=978-1-118-46029-0}}</ref> [[Chinese mathematics\|Chinese mathematics (around 200 BC and later)]],<ref>{{Cite journal \|date=1999 \|editor-last=Chabert \|editor-first=Jean-Luc \|title=A History of Algorithms \|url=https://link.springer.com/book/10.1007/978-3-642-18192-4 \|journal=SpringerLink \|language=en \|doi=10.1007/978-3-642-18192-4\|isbn=978-3-540-63369-3 }}</ref> and [[Arabic mathematics]] (around 800 AD).<ref name="Dooley">{{cite book \|last1=Dooley \|first1=John F. \|title=A Brief History of Cryptology and Cryptographic Algorithms \|date=2013 \|publisher=Springer Science & Business Media \|isbn=9783319016283 \|pages=12–3}}</ref>

	The earliest evidence of algorithms is found in ancient [[Mesopotamia]]n mathematics. A [[Sumer]]ian clay tablet found in [[Shuruppak]] near [[Baghdad]] and dated to {{Circa\|2500 BC}} describes the earliest [[division algorithm]].<ref name="Springer Science & Business Media" /> During the [[First Babylonian dynasty\|Hammurabi dynasty]] {{Circa\|1800\|1600 BC\|lk=no}}, [[Babylonia]]n clay tablets described algorithms for computing formulas.<ref>{{cite journal \|last1=Knuth \|first1=Donald E. \|date=1972 \|title=Ancient Babylonian Algorithms \|url=http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|url-status=dead \|journal=Commun. ACM \|volume=15 \|issue=7 \|pages=671–677 \|doi=10.1145/361454.361514 \|issn=0001-0782 \|s2cid=7829945 \|archive-url=https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|archive-date=2012-12-24}}</ref> Algorithms were also used in [[Babylonian astronomy]].~~{{Citation needed\|date=March 2025}}~~ Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<ref>{{cite book \|last=Aaboe \|first=Asger \|author-link=Asger Aaboe \|title=Episodes from the Early History of Astronomy \|date=2001 \|publisher=Springer \|isbn=978-0-387-95136-2 \|place=New York \|pages=40–62}}</ref>		The earliest evidence of algorithms is found in ancient [[Mesopotamia]]n mathematics. A [[Sumer]]ian clay tablet found in [[Shuruppak]] near [[Baghdad]] and dated to {{Circa\|2500 BC}} describes the earliest [[division algorithm]].<ref name="Springer Science & Business Media" /> During the [[First Babylonian dynasty\|Hammurabi dynasty]] {{Circa\|1800\|1600 BC\|lk=no}}, [[Babylonia]]n clay tablets described algorithms for computing formulas.<ref>{{cite journal \|last1=Knuth \|first1=Donald E. \|date=1972 \|title=Ancient Babylonian Algorithms \|url=http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|url-status=dead \|journal=Commun. ACM \|volume=15 \|issue=7 \|pages=671–677 \|doi=10.1145/361454.361514 \|issn=0001-0782 \|s2cid=7829945 \|archive-url=https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|archive-date=2012-12-24}}</ref> Algorithms were also used in [[Babylonian astronomy]]. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<ref>{{cite book \|last=Aaboe \|first=Asger \|author-link=Asger Aaboe \|title=Episodes from the Early History of Astronomy \|date=2001 \|publisher=Springer \|isbn=978-0-387-95136-2 \|place=New York \|pages=40–62}}</ref>

	Algorithms for arithmetic are also found in ancient [[Egyptian mathematics]], dating back to the [[Rhind Mathematical Papyrus]] {{Circa\|1550 BC\|lk=no}}.<ref name="Springer Science & Business Media" /> Algorithms were later used in ancient [[Hellenistic mathematics]]. Two examples are the [[Sieve of Eratosthenes]], which was described in the ''[[Introduction to Arithmetic]]'' by [[Nicomachus]],<ref>{{cite web \|last=Ast \|first=Courtney \|title=Eratosthenes \|url=http://www.math.wichita.edu/history/men/eratosthenes.html \|url-status=live \|archive-url=https://web.archive.org/web/20150227150653/http://www.math.wichita.edu/history/men/eratosthenes.html \|archive-date=February 27, 2015 \|access-date=February 27, 2015 \|publisher=Wichita State University: Department of Mathematics and Statistics}}</ref><ref name="Cooke2005" />{{rp\|Ch 9.2}} and the [[Euclidean algorithm]], which was first described in ''[[Euclid's Elements]]'' ({{circa\|300 BC\|lk=no}}).<ref name="Cooke2005" />{{rp\|Ch 9.1}}Examples of ancient Indian mathematics included the [[Shulba Sutras]], the [[Kerala school of astronomy and mathematics\|Kerala School]], and the [[Brāhmasphuṭasiddhānta]].<ref name=":6" />		Algorithms for arithmetic are also found in ancient [[Egyptian mathematics]], dating back to the [[Rhind Mathematical Papyrus]] {{Circa\|1550 BC\|lk=no}}.<ref name="Springer Science & Business Media" /> Algorithms were later used in ancient [[Hellenistic mathematics]]. Two examples are the [[Sieve of Eratosthenes]], which was described in the ''[[Introduction to Arithmetic]]'' by [[Nicomachus]],<ref>{{cite web \|last=Ast \|first=Courtney \|title=Eratosthenes \|url=http://www.math.wichita.edu/history/men/eratosthenes.html \|url-status=live \|archive-url=https://web.archive.org/web/20150227150653/http://www.math.wichita.edu/history/men/eratosthenes.html \|archive-date=February 27, 2015 \|access-date=February 27, 2015 \|publisher=Wichita State University: Department of Mathematics and Statistics}}</ref><ref name="Cooke2005" />{{rp\|Ch 9.2}} and the [[Euclidean algorithm]], which was first described in ''[[Euclid's Elements]]'' ({{circa\|300 BC\|lk=no}}).<ref name="Cooke2005" />{{rp\|Ch 9.1}}Examples of ancient Indian mathematics included the [[Shulba Sutras]], the [[Kerala school of astronomy and mathematics\|Kerala School]], and the [[Brāhmasphuṭasiddhānta]].<ref name=":6" />

← Previous revision		Revision as of 01:21, 19 May 2025
Line 27:		Line 27:
	{{For\|a detailed presentation of the various points of view on the definition of "algorithm"\|Algorithm characterizations}}		{{For\|a detailed presentation of the various points of view on the definition of "algorithm"\|Algorithm characterizations}}

	One informal definition is "a set of rules that precisely defines a sequence of operations",~~<ref>Stone 1973:4</ref>~~{{~~request quotation~~ \| ~~reason =~~ Stone ~~(1972) suggests on page 4: "...any sequence of instructions that a robot can obey, is called an algorithm"~~\|~~date~~=~~July 2020~~}} which would include all [[computer program]]s (including programs that do not perform numeric calculations), and any prescribed [[bureaucratic]] procedure<ref>		One informal definition is "a set of rules that precisely defines a sequence of operations",{{sfnp\|Stone\|1971\|p=8}} which would include all [[computer program]]s (including programs that do not perform numeric calculations), and any prescribed [[bureaucratic]] procedure<ref>
	{{cite book \|last1=Simanowski \|first1=Roberto \|author-link1=Roberto Simanowski \|url=https://books.google.com/books?id=RJV5DwAAQBAJ \|title=The Death Algorithm and Other Digital Dilemmas \|date=2018 \|publisher=MIT Press \|isbn=9780262536370 \|series=Untimely Meditations \|volume=14 \|location=Cambridge, Massachusetts \|page=147 \|translator1-last=Chase \|translator1-first=Jefferson \|quote=[...] the next level of abstraction of central bureaucracy: globally operating algorithms. \|access-date=27 May 2019 \|archive-url=https://web.archive.org/web/20191222120705/https://books.google.com/books?id=RJV5DwAAQBAJ \|archive-date=December 22, 2019 \|url-status=live}}		{{cite book \|last1=Simanowski \|first1=Roberto \|author-link1=Roberto Simanowski \|url=https://books.google.com/books?id=RJV5DwAAQBAJ \|title=The Death Algorithm and Other Digital Dilemmas \|date=2018 \|publisher=MIT Press \|isbn=9780262536370 \|series=Untimely Meditations \|volume=14 \|location=Cambridge, Massachusetts \|page=147 \|translator1-last=Chase \|translator1-first=Jefferson \|quote=[...] the next level of abstraction of central bureaucracy: globally operating algorithms. \|access-date=27 May 2019 \|archive-url=https://web.archive.org/web/20191222120705/https://books.google.com/books?id=RJV5DwAAQBAJ \|archive-date=December 22, 2019 \|url-status=live}}
	</ref>		</ref>
Line 253:		Line 253:
	* {{cite book\| last = Sipser\| first = Michael\| title = Introduction to the Theory of Computation\| year = 2006\| publisher = PWS Publishing Company\| isbn = 978-0-534-94728-6\| url = https://archive.org/details/introductiontoth00sips}}		* {{cite book\| last = Sipser\| first = Michael\| title = Introduction to the Theory of Computation\| year = 2006\| publisher = PWS Publishing Company\| isbn = 978-0-534-94728-6\| url = https://archive.org/details/introductiontoth00sips}}
	* {{cite book \|last1=Sober \|first1=Elliott \|last2=Wilson \|first2=David Sloan \|year=1998 \|title=Unto Others: The Evolution and Psychology of Unselfish Behavior \|url=https://archive.org/details/untoothersevolut00sobe \|url-access=registration \|location=Cambridge \|publisher=Harvard University Press\|isbn=9780674930469 }}		* {{cite book \|last1=Sober \|first1=Elliott \|last2=Wilson \|first2=David Sloan \|year=1998 \|title=Unto Others: The Evolution and Psychology of Unselfish Behavior \|url=https://archive.org/details/untoothersevolut00sobe \|url-access=registration \|location=Cambridge \|publisher=Harvard University Press\|isbn=9780674930469 }}
	* {{Cite book\|last=Stone\|first=Harold S.\|title=Introduction to Computer Organization and Data Structures~~\|edition=1972~~\|publisher=McGraw-Hill, New York\|isbn=~~978-0-07-061726-1~~\|year=~~1972~~}} Cf. in particular the first chapter titled: ''Algorithms, Turing Machines, and Programs''. His succinct informal definition: "...any sequence of instructions that can be obeyed by a robot, is called an ''algorithm''" (p. 4).		* {{Cite book\|last=Stone\|first=Harold S.\|title=Introduction to Computer Organization and Data Structures\|publisher=McGraw-Hill, New York\|isbn=9780070617261\|year=1971}} Cf. in particular the first chapter titled: ''Algorithms, Turing Machines, and Programs''. His succinct informal definition: "...any sequence of instructions that can be obeyed by a robot, is called an ''algorithm''" (p. 4).
	* {{cite book\| last = Tausworthe\| first = Robert C\| title = Standardized Development of Computer Software Part 1 Methods\| year = 1977\| publisher = Prentice–Hall, Inc.\| location = Englewood Cliffs NJ\| isbn = 978-0-13-842195-3 }}		* {{cite book\| last = Tausworthe\| first = Robert C\| title = Standardized Development of Computer Software Part 1 Methods\| year = 1977\| publisher = Prentice–Hall, Inc.\| location = Englewood Cliffs NJ\| isbn = 978-0-13-842195-3 }}
	* {{Cite journal\|last=Turing\|first=Alan M.\|author-link=A. M. Turing\|title=On Computable Numbers, With An Application to the Entscheidungsproblem\|journal=[[Proceedings of the London Mathematical Society]]\|series=Series 2\|volume=42\|pages= 230–265 \|year=1936–37\|doi=10.1112/plms/s2-42.1.230 \|s2cid=73712 }}. Corrections, ibid, vol. 43(1937) pp. 544–546. Reprinted in ''The Undecidable'', p. 116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.		* {{Cite journal\|last=Turing\|first=Alan M.\|author-link=A. M. Turing\|title=On Computable Numbers, With An Application to the Entscheidungsproblem\|journal=[[Proceedings of the London Mathematical Society]]\|series=Series 2\|volume=42\|pages= 230–265 \|year=1936–37\|doi=10.1112/plms/s2-42.1.230 \|s2cid=73712 }}. Corrections, ibid, vol. 43(1937) pp. 544–546. Reprinted in ''The Undecidable'', p. 116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.

← Previous revision		Revision as of 00:39, 19 May 2025
Line 3:		Line 3:
	{{redirect\|Algorythm\|the album\|Beyond Creation}}		{{redirect\|Algorythm\|the album\|Beyond Creation}}
	{{Use mdy dates\|date=September 2017}}		{{Use mdy dates\|date=September 2017}}

	{{Essay\|date=April 2024}}

	[[File:GCD through successive subtractions.svg\|thumb\|Flowchart of using successive subtractions to find the [[greatest common divisor]] of number ''r'' and ''s''\|alt=In a loop, subtract the larger number against the smaller number. Halt the loop when the subtraction will make a number negative. Assess two numbers, whether one of them is equal to zero or not. If yes, take the other number as the greatest common divisor. If no, put the two numbers in the subtraction loop again.]]		[[File:GCD through successive subtractions.svg\|thumb\|Flowchart of using successive subtractions to find the [[greatest common divisor]] of number ''r'' and ''s''\|alt=In a loop, subtract the larger number against the smaller number. Halt the loop when the subtraction will make a number negative. Assess two numbers, whether one of them is equal to zero or not. If yes, take the other number as the greatest common divisor. If no, put the two numbers in the subtraction loop again.]]
	In [[mathematics]] and [[computer science]], an '''algorithm''' ({{IPAc-en\|audio=en-us-algorithm.ogg\|ˈ\|æ\|l\|ɡ\|ə\|r\|ɪ\|ð\|əm}}) is a finite sequence of [[Rigour#Mathematics\|mathematically rigorous]] instructions, typically used to solve a class of specific [[Computational problem\|problem]]s or to perform a [[computation]].<ref name=":0">{{Cite web\|url=https://www.merriam-webster.com/dictionary/algorithm\|title=Definition of ALGORITHM\|work=Merriam-Webster Online Dictionary \|language=en \|access-date=2019-11-14 \|archive-url=https://web.archive.org/web/20200214074446/https://www.merriam-webster.com/dictionary/algorithm \|archive-date=February 14, 2020\|url-status=live}}</ref> Algorithms are used as specifications for performing [[calculation]]s and [[data processing]]. More advanced algorithms can use [[Conditional (computer programming)\|conditional]]s to divert the code execution through various routes (referred to as [[automated decision-making]]) and deduce valid [[inference]]s (referred to as [[automated reasoning]]).		In [[mathematics]] and [[computer science]], an '''algorithm''' ({{IPAc-en\|audio=en-us-algorithm.ogg\|ˈ\|æ\|l\|ɡ\|ə\|r\|ɪ\|ð\|əm}}) is a finite sequence of [[Rigour#Mathematics\|mathematically rigorous]] instructions, typically used to solve a class of specific [[Computational problem\|problem]]s or to perform a [[computation]].<ref name=":0">{{Cite web\|url=https://www.merriam-webster.com/dictionary/algorithm\|title=Definition of ALGORITHM\|work=Merriam-Webster Online Dictionary \|language=en \|access-date=2019-11-14 \|archive-url=https://web.archive.org/web/20200214074446/https://www.merriam-webster.com/dictionary/algorithm \|archive-date=February 14, 2020\|url-status=live}}</ref> Algorithms are used as specifications for performing [[calculation]]s and [[data processing]]. More advanced algorithms can use [[Conditional (computer programming)\|conditional]]s to divert the code execution through various routes (referred to as [[automated decision-making]]) and deduce valid [[inference]]s (referred to as [[automated reasoning]]).

← Previous revision		Revision as of 16:32, 29 April 2025
Line 34:		Line 34:
	Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in [[Babylonian mathematics]] (around 2500 BC),<ref name="Springer Science & Business Media">{{cite book \|last1=Chabert \|first1=Jean-Luc \|title=A History of Algorithms: From the Pebble to the Microchip \|date=2012 \|publisher=Springer Science & Business Media \|isbn=9783642181924 \|pages=7–8}}</ref> [[Egyptian mathematics]] (around 1550 BC),<ref name="Springer Science & Business Media" /> [[Indian mathematics]] (around 800 BC and later),<ref name=":6">{{cite book \|last1=Sriram \|first1=M. S. \|editor1-last=Emch \|editor1-first=Gerard G. \|editor2-last=Sridharan \|editor2-first=R. \|editor3-last=Srinivas \|editor3-first=M. D. \|title=Contributions to the History of Indian Mathematics \|date=2005 \|publisher=Springer \|isbn=978-93-86279-25-5 \|page=153 \|chapter-url=https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153 \|language=en \|chapter=Algorithms in Indian Mathematics}}</ref><ref>Hayashi, T. (2023, January 1). [https://www.britannica.com/biography/Brahmagupta Brahmagupta]. Encyclopedia Britannica.</ref> the Ifa Oracle (around 500 BC),<ref>{{Cite journal \|last=Zaslavsky \|first=Claudia \|date=1970 \|title=Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria \|url=https://www.jstor.org/stable/3027363 \|journal=The Two-Year College Mathematics Journal \|volume=1 \|issue=2 \|pages=76–99 \|doi=10.2307/3027363 \|jstor=3027363 \|issn=0049-4925}}</ref> [[Greek mathematics]] (around 240 BC),<ref name="Cooke2005">{{cite book\|last=Cooke\|first=Roger L.\|title=The History of Mathematics: A Brief Course\|date=2005\|publisher=John Wiley & Sons\|isbn=978-1-118-46029-0}}</ref> [[Chinese mathematics\|Chinese mathematics (around 200 BC and later)]],<ref>{{Cite journal \|date=1999 \|editor-last=Chabert \|editor-first=Jean-Luc \|title=A History of Algorithms \|url=https://link.springer.com/book/10.1007/978-3-642-18192-4 \|journal=SpringerLink \|language=en \|doi=10.1007/978-3-642-18192-4\|isbn=978-3-540-63369-3 }}</ref> and [[Arabic mathematics]] (around 800 AD).<ref name="Dooley">{{cite book \|last1=Dooley \|first1=John F. \|title=A Brief History of Cryptology and Cryptographic Algorithms \|date=2013 \|publisher=Springer Science & Business Media \|isbn=9783319016283 \|pages=12–3}}</ref>		Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in [[Babylonian mathematics]] (around 2500 BC),<ref name="Springer Science & Business Media">{{cite book \|last1=Chabert \|first1=Jean-Luc \|title=A History of Algorithms: From the Pebble to the Microchip \|date=2012 \|publisher=Springer Science & Business Media \|isbn=9783642181924 \|pages=7–8}}</ref> [[Egyptian mathematics]] (around 1550 BC),<ref name="Springer Science & Business Media" /> [[Indian mathematics]] (around 800 BC and later),<ref name=":6">{{cite book \|last1=Sriram \|first1=M. S. \|editor1-last=Emch \|editor1-first=Gerard G. \|editor2-last=Sridharan \|editor2-first=R. \|editor3-last=Srinivas \|editor3-first=M. D. \|title=Contributions to the History of Indian Mathematics \|date=2005 \|publisher=Springer \|isbn=978-93-86279-25-5 \|page=153 \|chapter-url=https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153 \|language=en \|chapter=Algorithms in Indian Mathematics}}</ref><ref>Hayashi, T. (2023, January 1). [https://www.britannica.com/biography/Brahmagupta Brahmagupta]. Encyclopedia Britannica.</ref> the Ifa Oracle (around 500 BC),<ref>{{Cite journal \|last=Zaslavsky \|first=Claudia \|date=1970 \|title=Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria \|url=https://www.jstor.org/stable/3027363 \|journal=The Two-Year College Mathematics Journal \|volume=1 \|issue=2 \|pages=76–99 \|doi=10.2307/3027363 \|jstor=3027363 \|issn=0049-4925}}</ref> [[Greek mathematics]] (around 240 BC),<ref name="Cooke2005">{{cite book\|last=Cooke\|first=Roger L.\|title=The History of Mathematics: A Brief Course\|date=2005\|publisher=John Wiley & Sons\|isbn=978-1-118-46029-0}}</ref> [[Chinese mathematics\|Chinese mathematics (around 200 BC and later)]],<ref>{{Cite journal \|date=1999 \|editor-last=Chabert \|editor-first=Jean-Luc \|title=A History of Algorithms \|url=https://link.springer.com/book/10.1007/978-3-642-18192-4 \|journal=SpringerLink \|language=en \|doi=10.1007/978-3-642-18192-4\|isbn=978-3-540-63369-3 }}</ref> and [[Arabic mathematics]] (around 800 AD).<ref name="Dooley">{{cite book \|last1=Dooley \|first1=John F. \|title=A Brief History of Cryptology and Cryptographic Algorithms \|date=2013 \|publisher=Springer Science & Business Media \|isbn=9783319016283 \|pages=12–3}}</ref>

	The earliest evidence of algorithms is found in ancient [[Mesopotamia]]n mathematics. A [[Sumer]]ian clay tablet found in [[Shuruppak]] near [[Baghdad]] and dated to {{Circa\|2500 BC}} describes the earliest [[division algorithm]].<ref name="Springer Science & Business Media" /> During the [[First Babylonian dynasty\|Hammurabi dynasty]] {{Circa\|1800\|1600 BC\|lk=no}}, [[Babylonia]]n clay tablets described algorithms for computing formulas.<ref>{{cite journal \|last1=Knuth \|first1=Donald E. \|date=1972 \|title=Ancient Babylonian Algorithms \|url=http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|url-status=dead \|journal=Commun. ACM \|volume=15 \|issue=7 \|pages=671–677 \|doi=10.1145/361454.361514 \|issn=0001-0782 \|s2cid=7829945 \|archive-url=https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|archive-date=2012-12-24}}</ref> Algorithms were also used in [[Babylonian astronomy]].~~<ref>~~{{~~Cite book~~ \|~~last~~= ~~\|first= \|title= \|publisher= \|year= \|isbn= \|edition= \|language=~~}}~~</ref>~~ Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<ref>{{cite book \|last=Aaboe \|first=Asger \|author-link=Asger Aaboe \|title=Episodes from the Early History of Astronomy \|date=2001 \|publisher=Springer \|isbn=978-0-387-95136-2 \|place=New York \|pages=40–62}}</ref>		The earliest evidence of algorithms is found in ancient [[Mesopotamia]]n mathematics. A [[Sumer]]ian clay tablet found in [[Shuruppak]] near [[Baghdad]] and dated to {{Circa\|2500 BC}} describes the earliest [[division algorithm]].<ref name="Springer Science & Business Media" /> During the [[First Babylonian dynasty\|Hammurabi dynasty]] {{Circa\|1800\|1600 BC\|lk=no}}, [[Babylonia]]n clay tablets described algorithms for computing formulas.<ref>{{cite journal \|last1=Knuth \|first1=Donald E. \|date=1972 \|title=Ancient Babylonian Algorithms \|url=http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|url-status=dead \|journal=Commun. ACM \|volume=15 \|issue=7 \|pages=671–677 \|doi=10.1145/361454.361514 \|issn=0001-0782 \|s2cid=7829945 \|archive-url=https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|archive-date=2012-12-24}}</ref> Algorithms were also used in [[Babylonian astronomy]].{{Citation needed\|date=March 2025}} Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<ref>{{cite book \|last=Aaboe \|first=Asger \|author-link=Asger Aaboe \|title=Episodes from the Early History of Astronomy \|date=2001 \|publisher=Springer \|isbn=978-0-387-95136-2 \|place=New York \|pages=40–62}}</ref>

	Algorithms for arithmetic are also found in ancient [[Egyptian mathematics]], dating back to the [[Rhind Mathematical Papyrus]] {{Circa\|1550 BC\|lk=no}}.<ref name="Springer Science & Business Media" /> Algorithms were later used in ancient [[Hellenistic mathematics]]. Two examples are the [[Sieve of Eratosthenes]], which was described in the ''[[Introduction to Arithmetic]]'' by [[Nicomachus]],<ref>{{cite web \|last=Ast \|first=Courtney \|title=Eratosthenes \|url=http://www.math.wichita.edu/history/men/eratosthenes.html \|url-status=live \|archive-url=https://web.archive.org/web/20150227150653/http://www.math.wichita.edu/history/men/eratosthenes.html \|archive-date=February 27, 2015 \|access-date=February 27, 2015 \|publisher=Wichita State University: Department of Mathematics and Statistics}}</ref><ref name="Cooke2005" />{{rp\|Ch 9.2}} and the [[Euclidean algorithm]], which was first described in ''[[Euclid's Elements]]'' ({{circa\|300 BC\|lk=no}}).<ref name="Cooke2005" />{{rp\|Ch 9.1}}Examples of ancient Indian mathematics included the [[Shulba Sutras]], the [[Kerala school of astronomy and mathematics\|Kerala School]], and the [[Brāhmasphuṭasiddhānta]].<ref name=":6" />		Algorithms for arithmetic are also found in ancient [[Egyptian mathematics]], dating back to the [[Rhind Mathematical Papyrus]] {{Circa\|1550 BC\|lk=no}}.<ref name="Springer Science & Business Media" /> Algorithms were later used in ancient [[Hellenistic mathematics]]. Two examples are the [[Sieve of Eratosthenes]], which was described in the ''[[Introduction to Arithmetic]]'' by [[Nicomachus]],<ref>{{cite web \|last=Ast \|first=Courtney \|title=Eratosthenes \|url=http://www.math.wichita.edu/history/men/eratosthenes.html \|url-status=live \|archive-url=https://web.archive.org/web/20150227150653/http://www.math.wichita.edu/history/men/eratosthenes.html \|archive-date=February 27, 2015 \|access-date=February 27, 2015 \|publisher=Wichita State University: Department of Mathematics and Statistics}}</ref><ref name="Cooke2005" />{{rp\|Ch 9.2}} and the [[Euclidean algorithm]], which was first described in ''[[Euclid's Elements]]'' ({{circa\|300 BC\|lk=no}}).<ref name="Cooke2005" />{{rp\|Ch 9.1}}Examples of ancient Indian mathematics included the [[Shulba Sutras]], the [[Kerala school of astronomy and mathematics\|Kerala School]], and the [[Brāhmasphuṭasiddhānta]].<ref name=":6" />

← Previous revision		Revision as of 16:29, 29 April 2025
Line 34:		Line 34:
	Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in [[Babylonian mathematics]] (around 2500 BC),<ref name="Springer Science & Business Media">{{cite book \|last1=Chabert \|first1=Jean-Luc \|title=A History of Algorithms: From the Pebble to the Microchip \|date=2012 \|publisher=Springer Science & Business Media \|isbn=9783642181924 \|pages=7–8}}</ref> [[Egyptian mathematics]] (around 1550 BC),<ref name="Springer Science & Business Media" /> [[Indian mathematics]] (around 800 BC and later),<ref name=":6">{{cite book \|last1=Sriram \|first1=M. S. \|editor1-last=Emch \|editor1-first=Gerard G. \|editor2-last=Sridharan \|editor2-first=R. \|editor3-last=Srinivas \|editor3-first=M. D. \|title=Contributions to the History of Indian Mathematics \|date=2005 \|publisher=Springer \|isbn=978-93-86279-25-5 \|page=153 \|chapter-url=https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153 \|language=en \|chapter=Algorithms in Indian Mathematics}}</ref><ref>Hayashi, T. (2023, January 1). [https://www.britannica.com/biography/Brahmagupta Brahmagupta]. Encyclopedia Britannica.</ref> the Ifa Oracle (around 500 BC),<ref>{{Cite journal \|last=Zaslavsky \|first=Claudia \|date=1970 \|title=Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria \|url=https://www.jstor.org/stable/3027363 \|journal=The Two-Year College Mathematics Journal \|volume=1 \|issue=2 \|pages=76–99 \|doi=10.2307/3027363 \|jstor=3027363 \|issn=0049-4925}}</ref> [[Greek mathematics]] (around 240 BC),<ref name="Cooke2005">{{cite book\|last=Cooke\|first=Roger L.\|title=The History of Mathematics: A Brief Course\|date=2005\|publisher=John Wiley & Sons\|isbn=978-1-118-46029-0}}</ref> [[Chinese mathematics\|Chinese mathematics (around 200 BC and later)]],<ref>{{Cite journal \|date=1999 \|editor-last=Chabert \|editor-first=Jean-Luc \|title=A History of Algorithms \|url=https://link.springer.com/book/10.1007/978-3-642-18192-4 \|journal=SpringerLink \|language=en \|doi=10.1007/978-3-642-18192-4\|isbn=978-3-540-63369-3 }}</ref> and [[Arabic mathematics]] (around 800 AD).<ref name="Dooley">{{cite book \|last1=Dooley \|first1=John F. \|title=A Brief History of Cryptology and Cryptographic Algorithms \|date=2013 \|publisher=Springer Science & Business Media \|isbn=9783319016283 \|pages=12–3}}</ref>		Step-by-step procedures for solving mathematical problems have been recorded since antiquity. This includes in [[Babylonian mathematics]] (around 2500 BC),<ref name="Springer Science & Business Media">{{cite book \|last1=Chabert \|first1=Jean-Luc \|title=A History of Algorithms: From the Pebble to the Microchip \|date=2012 \|publisher=Springer Science & Business Media \|isbn=9783642181924 \|pages=7–8}}</ref> [[Egyptian mathematics]] (around 1550 BC),<ref name="Springer Science & Business Media" /> [[Indian mathematics]] (around 800 BC and later),<ref name=":6">{{cite book \|last1=Sriram \|first1=M. S. \|editor1-last=Emch \|editor1-first=Gerard G. \|editor2-last=Sridharan \|editor2-first=R. \|editor3-last=Srinivas \|editor3-first=M. D. \|title=Contributions to the History of Indian Mathematics \|date=2005 \|publisher=Springer \|isbn=978-93-86279-25-5 \|page=153 \|chapter-url=https://books.google.com/books?id=qfJdDwAAQBAJ&pg=PA153 \|language=en \|chapter=Algorithms in Indian Mathematics}}</ref><ref>Hayashi, T. (2023, January 1). [https://www.britannica.com/biography/Brahmagupta Brahmagupta]. Encyclopedia Britannica.</ref> the Ifa Oracle (around 500 BC),<ref>{{Cite journal \|last=Zaslavsky \|first=Claudia \|date=1970 \|title=Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria \|url=https://www.jstor.org/stable/3027363 \|journal=The Two-Year College Mathematics Journal \|volume=1 \|issue=2 \|pages=76–99 \|doi=10.2307/3027363 \|jstor=3027363 \|issn=0049-4925}}</ref> [[Greek mathematics]] (around 240 BC),<ref name="Cooke2005">{{cite book\|last=Cooke\|first=Roger L.\|title=The History of Mathematics: A Brief Course\|date=2005\|publisher=John Wiley & Sons\|isbn=978-1-118-46029-0}}</ref> [[Chinese mathematics\|Chinese mathematics (around 200 BC and later)]],<ref>{{Cite journal \|date=1999 \|editor-last=Chabert \|editor-first=Jean-Luc \|title=A History of Algorithms \|url=https://link.springer.com/book/10.1007/978-3-642-18192-4 \|journal=SpringerLink \|language=en \|doi=10.1007/978-3-642-18192-4\|isbn=978-3-540-63369-3 }}</ref> and [[Arabic mathematics]] (around 800 AD).<ref name="Dooley">{{cite book \|last1=Dooley \|first1=John F. \|title=A Brief History of Cryptology and Cryptographic Algorithms \|date=2013 \|publisher=Springer Science & Business Media \|isbn=9783319016283 \|pages=12–3}}</ref>

	The earliest evidence of algorithms is found in ancient [[Mesopotamia]]n mathematics. A [[Sumer]]ian clay tablet found in [[Shuruppak]] near [[Baghdad]] and dated to {{Circa\|2500 BC}} describes the earliest [[division algorithm]].<ref name="Springer Science & Business Media" /> During the [[First Babylonian dynasty\|Hammurabi dynasty]] {{Circa\|1800\|1600 BC\|lk=no}}, [[Babylonia]]n clay tablets described algorithms for computing formulas.<ref>{{cite journal \|last1=Knuth \|first1=Donald E. \|date=1972 \|title=Ancient Babylonian Algorithms \|url=http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|url-status=dead \|journal=Commun. ACM \|volume=15 \|issue=7 \|pages=671–677 \|doi=10.1145/361454.361514 \|issn=0001-0782 \|s2cid=7829945 \|archive-url=https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|archive-date=2012-12-24}}</ref> Algorithms were also used in [[Babylonian astronomy]].{{~~Citation~~ ~~needed~~\|~~date~~=~~March~~ ~~2025~~}} Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<ref>{{cite book \|last=Aaboe \|first=Asger \|author-link=Asger Aaboe \|title=Episodes from the Early History of Astronomy \|date=2001 \|publisher=Springer \|isbn=978-0-387-95136-2 \|place=New York \|pages=40–62}}</ref>		The earliest evidence of algorithms is found in ancient [[Mesopotamia]]n mathematics. A [[Sumer]]ian clay tablet found in [[Shuruppak]] near [[Baghdad]] and dated to {{Circa\|2500 BC}} describes the earliest [[division algorithm]].<ref name="Springer Science & Business Media" /> During the [[First Babylonian dynasty\|Hammurabi dynasty]] {{Circa\|1800\|1600 BC\|lk=no}}, [[Babylonia]]n clay tablets described algorithms for computing formulas.<ref>{{cite journal \|last1=Knuth \|first1=Donald E. \|date=1972 \|title=Ancient Babylonian Algorithms \|url=http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|url-status=dead \|journal=Commun. ACM \|volume=15 \|issue=7 \|pages=671–677 \|doi=10.1145/361454.361514 \|issn=0001-0782 \|s2cid=7829945 \|archive-url=https://web.archive.org/web/20121224100137/http://steiner.math.nthu.edu.tw/disk5/js/computer/1.pdf \|archive-date=2012-12-24}}</ref> Algorithms were also used in [[Babylonian astronomy]].<ref>{{Cite book \|last= \|first= \|title= \|publisher= \|year= \|isbn= \|edition= \|language=}}</ref> Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.<ref>{{cite book \|last=Aaboe \|first=Asger \|author-link=Asger Aaboe \|title=Episodes from the Early History of Astronomy \|date=2001 \|publisher=Springer \|isbn=978-0-387-95136-2 \|place=New York \|pages=40–62}}</ref>

	Algorithms for arithmetic are also found in ancient [[Egyptian mathematics]], dating back to the [[Rhind Mathematical Papyrus]] {{Circa\|1550 BC\|lk=no}}.<ref name="Springer Science & Business Media" /> Algorithms were later used in ancient [[Hellenistic mathematics]]. Two examples are the [[Sieve of Eratosthenes]], which was described in the ''[[Introduction to Arithmetic]]'' by [[Nicomachus]],<ref>{{cite web \|last=Ast \|first=Courtney \|title=Eratosthenes \|url=http://www.math.wichita.edu/history/men/eratosthenes.html \|url-status=live \|archive-url=https://web.archive.org/web/20150227150653/http://www.math.wichita.edu/history/men/eratosthenes.html \|archive-date=February 27, 2015 \|access-date=February 27, 2015 \|publisher=Wichita State University: Department of Mathematics and Statistics}}</ref><ref name="Cooke2005" />{{rp\|Ch 9.2}} and the [[Euclidean algorithm]], which was first described in ''[[Euclid's Elements]]'' ({{circa\|300 BC\|lk=no}}).<ref name="Cooke2005" />{{rp\|Ch 9.1}}Examples of ancient Indian mathematics included the [[Shulba Sutras]], the [[Kerala school of astronomy and mathematics\|Kerala School]], and the [[Brāhmasphuṭasiddhānta]].<ref name=":6" />		Algorithms for arithmetic are also found in ancient [[Egyptian mathematics]], dating back to the [[Rhind Mathematical Papyrus]] {{Circa\|1550 BC\|lk=no}}.<ref name="Springer Science & Business Media" /> Algorithms were later used in ancient [[Hellenistic mathematics]]. Two examples are the [[Sieve of Eratosthenes]], which was described in the ''[[Introduction to Arithmetic]]'' by [[Nicomachus]],<ref>{{cite web \|last=Ast \|first=Courtney \|title=Eratosthenes \|url=http://www.math.wichita.edu/history/men/eratosthenes.html \|url-status=live \|archive-url=https://web.archive.org/web/20150227150653/http://www.math.wichita.edu/history/men/eratosthenes.html \|archive-date=February 27, 2015 \|access-date=February 27, 2015 \|publisher=Wichita State University: Department of Mathematics and Statistics}}</ref><ref name="Cooke2005" />{{rp\|Ch 9.2}} and the [[Euclidean algorithm]], which was first described in ''[[Euclid's Elements]]'' ({{circa\|300 BC\|lk=no}}).<ref name="Cooke2005" />{{rp\|Ch 9.1}}Examples of ancient Indian mathematics included the [[Shulba Sutras]], the [[Kerala school of astronomy and mathematics\|Kerala School]], and the [[Brāhmasphuṭasiddhānta]].<ref name=":6" />