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Baris 116:
n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \\
&= n\times(n-1)!\\
\end{align}</math>Contohnya,<math display="block">5! = 5 \times 4 \times 3 \times 2 \times 1 = 5\times 24 = 120. </math>
Faktorial ditemukan dalam beberapa budaya kuno, khususnya di [[matematika India]] dalam tulisan karya [[sastra Jain]], dan
Fungsi faktorial dalam matematika dikembangkan pada akhir abad ke-18 dan awal abad ke-19. [[Aproksimasi Stirling]] menyediakan sebuah hampiran yang akurat mengenai faktorial dari bilangan yang besar, yang memperlihatkan bahwa pertumbuhan nilainya lebih cepat daripada [[pertumbuhan eksponensial]]. Adapula [[rumus Legendre]] yang menjelaskan eksponen bilangan prima dalam [[faktorisasi bilangan prima]] melalui faktorial, dan rumus tersebut dapat dipakai untuk menghitung jejak nol melalui faktorial. [[Daniel Bernoulli]] dan [[Leonhard Euler]] [[menginterpolasi]] fungsi faktorial menjadi sebuah fungsi kontinu pada [[bilangan kompleks]], kecuali pada bilangan bulat negatif. Fungsi tersebut ialah [[fungsi gamma]] (ofset).
== Sejarah ==
Konsep faktorial muncul secara terpisah dalam banyak budaya. In [[Indian mathematics]], one of the earliest known descriptions of factorials comes from the Anuyogadvāra-sūtra,<ref name="datta-singh">{{cite book|last1=Datta|first1=Bibhutibhusan|last2=Singh|first2=Awadhesh Narayan|year=2019|title=Studies in Indian Mathematics and Astronomy: Selected Articles of Kripa Shankar Shukla|publisher=Springer Singapore|editor1-last=Kolachana|editor1-first=Aditya|series=Sources and Studies in the History of Mathematics and Physical Sciences|pages=356–376|contribution=Use of permutations and combinations in India|doi=10.1007/978-981-13-7326-8_18|author1-link=Bibhutibhushan Datta|editor2-last=Mahesh|editor2-first=K.|editor3-last=Ramasubramanian|editor3-first=K.|s2cid=191141516}}. Revised by K. S. Shukla from a paper in ''[[Indian Journal of History of Science]]'' 27 (3): 231–249, 1992, {{MR|1189487}}. See p. 363.</ref> one of the canonical works of [[Jain literature]], which has been assigned dates varying from 300 BCE to 400 CE.<ref>{{cite journal|last=Jadhav|first=Dipak|date=August 2021|title=Jaina Thoughts on Unity Not Being a Number|journal=History of Science in South Asia|publisher=University of Alberta Libraries|volume=9|pages=209–231|doi=10.18732/hssa67|s2cid=238656716}}. See discussion of dating on p. 211.</ref> It separates out the sorted and reversed order of a set of items from the other ("mixed") orders, evaluating the number of mixed orders by subtracting two from the usual product formula for the factorial. The product rule for permutations was also described by 6th-century CE Jain monk [[Jinabhadra]].<ref name="datta-singh" /> Hindu scholars have been using factorial formulas since at least 1150, when [[Bhāskara II]] mentioned factorials in his work [[Līlāvatī]], in connection with a problem of how many ways [[Vishnu]] could hold his four characteristic objects (a [[Shankha|conch shell]], [[Sudarshana Chakra|discus]], [[Kaumodaki|mace]], and [[Sacred lotus in religious art|lotus flower]]) in his four hands, and a similar problem for a ten-handed god.<ref>{{Cite journal|last=Biggs|first=Norman L.|author-link=Norman L. Biggs|date=May 1979|title=The roots of combinatorics|journal=[[Historia Mathematica]]|volume=6|issue=2|pages=109–136|doi=10.1016/0315-0860(79)90074-0|mr=0530622|doi-access=free}}</ref>
In the mathematics of the Middle East, the Hebrew mystic book of creation ''[[Sefer Yetzirah]]'', from the [[Talmud|Talmudic period]] (200 to 500 CE), lists factorials up to 7! as part of an investigation into the number of words that can be formed from the [[Hebrew alphabet]].<ref name="katz">{{cite journal|last=Katz|first=Victor J.|author-link=Victor J. Katz|date=June 1994|title=Ethnomathematics in the classroom|journal=[[For the Learning of Mathematics]]|volume=14|issue=2|pages=26–30|jstor=40248112}}</ref><ref>[[wikisource:Sefer_Yetzirah#CHAPTER_IV|Sefer Yetzirah at Wikisource]], Chapter IV, Section 4</ref> Factorials were also studied for similar reasons by 8th-century Arab grammarian [[Al-Khalil ibn Ahmad al-Farahidi]].<ref name="katz" /> Arab mathematician [[Ibn al-Haytham]] (also known as Alhazen, c. 965 – c. 1040) was the first to formulate [[Wilson's theorem]] connecting the factorials with the [[Prime number|prime numbers]].<ref>{{cite journal|last=Rashed|first=Roshdi|author-link=Roshdi Rashed|year=1980|title=Ibn al-Haytham et le théorème de Wilson|journal=[[Archive for History of Exact Sciences]]|language=fr|volume=22|issue=4|pages=305–321|doi=10.1007/BF00717654|mr=595903|s2cid=120885025}}</ref>
In Europe, although [[Greek mathematics]] included some combinatorics, and [[Plato]] famously used 5040 (a factorial) as the population of an ideal community, in part because of its divisibility properties,<ref>{{cite journal|last=Acerbi|first=F.|year=2003|title=On the shoulders of Hipparchus: a reappraisal of ancient Greek combinatorics|journal=[[Archive for History of Exact Sciences]]|volume=57|issue=6|pages=465–502|doi=10.1007/s00407-003-0067-0|jstor=41134173|mr=2004966|s2cid=122758966}}</ref> there is no direct evidence of ancient Greek study of factorials. Instead, the first work on factorials in Europe was by Jewish scholars such as [[Shabbethai Donnolo]], explicating the Sefer Yetzirah passage.<ref>{{cite book|last=Katz|first=Victor J.|date=2013|title=Combinatorics: Ancient & Modern|publisher=[[Oxford University Press]]|isbn=978-0-19-965659-2|editor1-last=Wilson|editor1-first=Robin|pages=109–121|contribution=Chapter 4: Jewish combinatorics|author-link=Victor J. Katz|editor2-last=Watkins|editor2-first=John J.}} See p. 111.</ref> In 1677, British author [[Fabian Stedman]] described the application of factorials to [[change ringing]], a musical art involving the ringing of several tuned bells.<ref>{{cite journal|last=Hunt|first=Katherine|date=May 2018|title=The Art of Changes: Bell-Ringing, Anagrams, and the Culture of Combination in Seventeenth-Century England|journal=Journal of Medieval and Early Modern Studies|volume=48|issue=2|pages=387–412|doi=10.1215/10829636-4403136}}</ref><ref>{{cite book|last=Stedman|first=Fabian|year=1677|title=Campanalogia|place=London|pages=6–9|author-link=Fabian Stedman}} The publisher is given as "W.S." who may have been William Smith, possibly acting as agent for the [[Ancient Society of College Youths|Society of College Youths]], to which society the "Dedicatory" is addressed.</ref>
From the late 15th century onward, factorials became the subject of study by western mathematicians. In a 1494 treatise, Italian mathematician [[Luca Pacioli]] calculated factorials up to 11!, in connection with a problem of dining table arrangements.<ref>{{cite book|last=Knobloch|first=Eberhard|date=2013|title=Combinatorics: Ancient & Modern|publisher=[[Oxford University Press]]|isbn=978-0-19-965659-2|editor1-last=Wilson|editor1-first=Robin|pages=123–145|contribution=Chapter 5: Renaissance combinatorics|author-link=Eberhard Knobloch|editor2-last=Watkins|editor2-first=John J.}} See p. 126.</ref> [[Christopher Clavius]] discussed factorials in a 1603 commentary on the work of [[Johannes de Sacrobosco]], and in the 1640s, French polymath [[Marin Mersenne]] published large (but not entirely correct) tables of factorials, up to 64!, based on the work of Clavius.{{sfn|Knobloch|2013|pages=130–133}} The [[power series]] for the [[exponential function]], with the reciprocals of factorials for its coefficients, was first formulated in 1676 by [[Isaac Newton]] in a letter to [[Gottfried Wilhelm Leibniz]].<ref name="exponential-series">{{cite book|last1=Ebbinghaus|first1=H.-D.|last2=Hermes|first2=H.|last3=Hirzebruch|first3=F.|last4=Koecher|first4=M.|last5=Mainzer|first5=K.|last6=Neukirch|first6=J.|last7=Prestel|first7=A.|last8=Remmert|first8=R.|year=1990|url=https://books.google.com/books?id=Z53SBwAAQBAJ&pg=PA131|title=Numbers|location=New York|publisher=Springer-Verlag|isbn=0-387-97202-1|series=Graduate Texts in Mathematics|volume=123|page=131|doi=10.1007/978-1-4612-1005-4|mr=1066206|author1-link=Heinz-Dieter Ebbinghaus|author2-link=Hans Hermes|author3-link=Friedrich Hirzebruch|author4-link=Max Koecher|author5-link=Klaus Mainzer|author6-link=Jürgen Neukirch|author8-link=Reinhold Remmert}}</ref> Other important works of early European mathematics on factorials include extensive coverage in a 1685 treatise by [[John Wallis]], a study of their approximate values for large values of <math>n</math> by [[Abraham de Moivre]] in 1721, a 1729 letter from [[James Stirling (mathematician)|James Stirling]] to de Moivre stating what became known as [[Stirling's approximation]], and work at the same time by [[Daniel Bernoulli]] and [[Leonhard Euler]] formulating the continuous extension of the factorial function to the [[gamma function]].<ref>{{cite journal|last=Dutka|first=Jacques|year=1991|title=The early history of the factorial function|journal=[[Archive for History of Exact Sciences]]|volume=43|issue=3|pages=225–249|doi=10.1007/BF00389433|jstor=41133918|mr=1171521|s2cid=122237769}}</ref> [[Adrien-Marie Legendre]] included [[Legendre's formula]], describing the exponents in the [[Integer factorization|factorization]] of factorials into [[Prime power|prime powers]], in an 1808 text on [[number theory]].<ref>{{cite book|last=Dickson|first=Leonard E.|year=1919|title=History of the Theory of Numbers|title-link=History of the Theory of Numbers|publisher=Carnegie Institution of Washington|volume=1|pages=263–278|contribution=Chapter IX: Divisibility of factorials and multinomial coefficients|author-link=Leonard Eugene Dickson|contribution-url=https://archive.org/details/historyoftheoryo01dick/page/262}} See in particular p. 263.</ref>
The notation <math>n!</math> for factorials was introduced by the French mathematician [[Christian Kramp]] in 1808.<ref name="cajori">{{cite book|last=Cajori|first=Florian|year=1929|title=A History of Mathematical Notations, Volume II: Notations Mainly in Higher Mathematics|title-link=A History of Mathematical Notations|publisher=The Open Court Publishing Company|pages=71–77|contribution=448–449. Factorial "{{mvar|n}}"|author-link=Florian Cajori|contribution-url=https://archive.org/details/AHistoryOfMathematicalNotationVolII/page/n93}}</ref> Many other notations have also been used. Another later notation, in which the argument of the factorial was half-enclosed by the left and bottom sides of a box, was popular for some time in Britain and America but fell out of use, perhaps because it is difficult to typeset.<ref name="cajori" /> The word "factorial" (originally French: ''factorielle'') was first used in 1800 by [[Louis François Antoine Arbogast]],<ref>{{cite web|last=Miller|first=Jeff|title=Earliest Known Uses of Some of the Words of Mathematics (F)|url=https://mathshistory.st-andrews.ac.uk/Miller/mathword/f/|work=[[MacTutor History of Mathematics archive]]|publisher=University of St Andrews}}</ref> in the first work on [[Faà di Bruno's formula]],<ref name="craik">{{cite journal|last=Craik|first=Alex D. D.|year=2005|title=Prehistory of Faà di Bruno's formula|journal=[[The American Mathematical Monthly]]|volume=112|issue=2|pages=119–130|doi=10.1080/00029890.2005.11920176|jstor=30037410|mr=2121322|s2cid=45380805}}</ref> but referring to a more general concept of products of [[Arithmetic progression|arithmetic progressions]]. The "factors" that this name refers to are the terms of the product formula for the factorial.<ref>{{cite book|last=Arbogast|first=Louis François Antoine|year=1800|url=https://archive.org/details/ducalculdesdri00arbouoft/page/364|title=Du calcul des dérivations|location=Strasbourg|publisher=L'imprimerie de Levrault, frères|pages=364–365|language=fr|author-link=Louis François Antoine Arbogast}}</ref>
== Referensi ==
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