Editing Binomial theorem (section)

==History==
Special cases of the binomial theorem were known since at least the 4th century BC when [[Greek mathematics|Greek mathematician]] [[Euclid]] mentioned the special case of the binomial theorem for exponent <math>n=2</math>.<ref name="Coolidge">{{cite journal|title=The Story of the Binomial Theorem|first=J. L.|last=Coolidge|journal=The American Mathematical Monthly| volume=56| issue=3|date=1949|pages=147–157|doi=10.2307/2305028|jstor = 2305028}}</ref> Greek mathematician [[Diophantus]] cubed various binomials, including <math>x-1</math>.<ref name="Coolidge" />  Indian mathematician [[Aryabhata]]'s method for finding cube roots, from around 510 AD, suggests that he knew the binomial formula for exponent <math>n=3</math>.<ref name="Coolidge" />

Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting {{mvar|k}} objects out of {{mvar|n}} without replacement ([[combinations]]), were of interest to ancient Indian mathematicians. The [[Jainism|Jain]] ''[[Bhagavati Sutra]]'' (c. 300 BC) describes the number of combinations of philosophical categories, senses, or other things, with correct results up through {{tmath|1= n = 4}} (probably obtained by listing all possibilities and counting them)<ref name=biggs>{{cite journal |last=Biggs |first=Norman L. |author-link=Norman L. Biggs |title=The roots of combinatorics |journal=Historia Mathematica |volume=6 |date=1979 |issue=2 |pages=109–136 |doi=10.1016/0315-0860(79)90074-0 |doi-access=free}}</ref> and a suggestion that higher combinations could likewise be found.<ref>{{cite journal |last=Datta |first=Bibhutibhushan |author-link=Bibhutibhushan Datta |url=https://archive.org/details/in.ernet.dli.2015.165748/page/n139/ |title=The Jaina School of Mathematics |journal=Bulletin of the Calcutta Mathematical Society |volume=27 |year=1929 |at=5. 115–145 (esp. 133–134) }} Reprinted as "The Mathematical Achievements of the Jainas" in {{cite book|editor-last=Chattopadhyaya |editor-first=Debiprasad |title=Studies in the History of Science in India |volume=2 |place=New Delhi |publisher=Editorial Enterprises |year=1982 |pages=684–716}}</ref> The ''[[Chandaḥśāstra]]'' by the Indian lyricist [[Piṅgala]] (3rd or 2nd century BC) somewhat cryptically describes a method of arranging two types of syllables to form [[metre (poetry)|metre]]s of various lengths and counting them; as interpreted and elaborated by Piṅgala's 10th-century commentator [[Halāyudha]] his "method of pyramidal expansion" (''meru-prastāra'') for counting metres is equivalent to [[Pascal's triangle]].<ref>{{cite journal |last=Bag |first=Amulya Kumar |title=Binomial theorem in ancient India |journal=Indian Journal of History of Science |volume=1 |number=1 |year=1966 |pages=68–74 |url=http://repository.ias.ac.in/70374/1/10-pub.pdf }} {{pb}} {{cite journal |last=Shah |first=Jayant |year=2013 |journal=Gaṇita Bhāratī |volume=35 |number=1–4 |pages=43–96 |title=A History of Piṅgala's Combinatorics |id={{ResearchGatePub|353496244}} }} ([https://ia800306.us.archive.org/19/items/Pingala/Pingala.pdf Preprint]) {{pb}} Survey sources:  {{pb}} {{cite book |last=Edwards |first=A. W. F. |author-link=A. W. F. Edwards |year=1987 |chapter=The combinatorial numbers in India |title=Pascal's Arithmetical Triangle |place=London |publisher=Charles Griffin |isbn=0-19-520546-4 |chapter-url=https://archive.org/details/pascalsarithmeti0000edwa/page/27 |pages=27–33 |chapter-url-access=limited }} {{pb}} {{cite book |last=Divakaran |first=P. P. |year=2018 |title=The Mathematics of India: Concepts, Methods, Connections |chapter=Combinatorics |at=§5.5 {{pgs|135–140}} |publisher=Springer; Hindustan Book Agency |doi=10.1007/978-981-13-1774-3_5 |isbn=978-981-13-1773-6 }} {{pb}} {{cite book |last=Roy |first=Ranjan |author-link=Ranjan Roy |year=2021 |title=Series and Products in the Development of Mathematics |edition=2 |volume=1 |publisher=Cambridge University Press |chapter=The Binomial Theorem |at=Ch. 4, {{pgs|77–104}} |isbn=978-1-108-70945-3 |doi=10.1017/9781108709453.005 }}</ref> [[Varāhamihira]] (6th century AD) describes another method for computing combination counts by adding numbers in columns.<ref name=gupta>{{cite journal |last=Gupta |first=Radha Charan |author-link=Radha Charan Gupta |title=Varāhamihira's Calculation of {{tmath|{}^nC_r}} and the Discovery of Pascal's Triangle |journal=Gaṇita Bhāratī |volume=14 |number=1–4 |year=1992 |pages=45–49 }} Reprinted in {{cite book |editor-last=Ramasubramanian |editor-first=K. |year=2019 |title=Gaṇitānanda |publisher=Springer |doi=10.1007/978-981-13-1229-8_29 |pages=285–289 }}</ref> By the 9th century at latest Indian mathematicians learned to express this as a product of fractions {{tmath| \tfrac{n}1 \times \tfrac{n - 1}2 \times \cdots \times \tfrac{n - k + 1}{n-k} }}, and clear statements of this rule can be found in [[Śrīdhara]]'s ''Pāṭīgaṇita'' (8th–9th century), [[Mahāvīra (mathematician)|Mahāvīra]]'s ''[[Gaṇita-sāra-saṅgraha]]'' (c. 850), and [[Bhāskara II]]'s ''Līlāvatī'' (12th century).{{r|gupta}}{{r|biggs}}<ref>{{cite book |year=1959|title=The Patiganita of Sridharacarya |editor-last=Shukla |editor-first=Kripa Shankar |editor-link= Kripa Shankar Shukla |publisher=Lucknow University |chapter-url=https://archive.org/details/Patiganita/page/n294/mode/1up |chapter=Combinations of Savours |at=Vyavahāras 1.9, {{pgs|97}} (text), {{pgs|58–59}} (translation) }}</ref>

The Persian mathematician [[al-Karajī]] (953–1029) wrote a now-lost book containing the binomial theorem and a table of binomial coefficients, often credited as their first appearance.<ref name=yadegari>{{cite journal |last=Yadegari |first=Mohammad  |year=1980 |title=The Binomial Theorem: A Widespread Concept in Medieval Islamic Mathematics |journal=Historia Mathematica |volume=7 |issue=4 |pages=401–406 |doi=10.1016/0315-0860(80)90004-X |doi-access=free }}</ref><ref name=rashed>{{cite journal |last=Rashed |first=Roshdi |author-link=Roshdi Rashed |year=1972 |title=L'induction mathématique: al-Karajī, al-Samawʾal |journal=Archive for History of Exact Sciences |volume=9 |issue=1 |pages=1–21 |jstor=41133347 |doi=10.1007/BF00348537 |language=fr }} Translated into English by A. F. W. Armstrong in {{Cite book |last=Rashed |first=Roshdi |year=1994 |title=The Development of Arabic Mathematics: Between Arithmetic and Algebra |chapter=Mathematical Induction: al-Karajī and al-Samawʾal |chapter-url=https://archive.org/details/RoshdiRashedauth.TheDevelopmentOfArabicMathematicsBetweenArithmeticAndAlgebraSpringerNetherlands1994/page/n71/ |at=§1.4, {{pgs|62–81}} |doi=10.1007/978-94-017-3274-1_2 |publisher=Kluwer |isbn=0-7923-2565-6 |quote="The first formulation of the binomial and the table of binomial coefficients, to our knowledge, is to be found in a text by al-Karajī, cited by al-Samawʾal in ''al-Bāhir''." }}</ref><ref>{{Cite encyclopedia |title=Al-Karajī |encyclopedia=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures |last=Sesiano |first=Jacques |editor-last=Selin |editor-first=Helaine |editor-link=Helaine Selin |year=1997 |publisher=Springer |doi=10.1007/978-94-017-1416-7_11 |isbn=978-94-017-1418-1 |pages=475–476 |quote=Another [lost work of Karajī's] contained the first known explanation of the arithmetical (Pascal's) triangle; the passage in question survived through al-Samawʾal's ''Bāhir'' (twelfth century) which heavily drew from the ''Badīʿ''. }}</ref><ref>
{{cite journal |last=Berggren |first=John Lennart |year=1985 |title=History of mathematics in the Islamic world: The present state of the art |journal=Review of Middle East Studies |volume=19 |number=1 |pages=9–33 |doi=10.1017/S0026318400014796 }} Republished in {{Cite book |title=From Alexandria, Through Baghdad |editor1-last=Sidoli |editor1-first=Nathan |editor2-last=Brummelen |editor2-first=Glen Van |editor2-link=Glen Van Brummelen |year=2014 |publisher=Springer |isbn=978-3-642-36735-9 |doi=10.1007/978-3-642-36736-6_4 |pages=51–71 |quote=[...] since the table of binomial coefficients had been previously found in such late works as those of al-Kāshī (fifteenth century) and Naṣīr al-Dīn al-Ṭūsī (thirteenth century), some had suggested that the table was a Chinese import. However, the use of the binomial coefficients by Islamic mathematicians of the eleventh century, in a context which had deep roots in Islamic mathematics, suggests strongly that the table was a local discovery – most probably of al-Karajī.}}</ref>
An explicit statement of the binomial theorem appears in [[al-Samawʾal]]'s ''al-Bāhir'' (12th century), there credited to al-Karajī.{{r|yadegari}}{{r|rashed}} Al-Samawʾal algebraically expanded the square, cube, and fourth power of a binomial, each in terms of the previous power, and noted that similar proofs could be provided for higher powers, an early form of [[mathematical induction]]. He then provided al-Karajī's table of binomial coefficients (Pascal's triangle turned on its side) up to {{tmath|1= n = 12}} and a rule for generating them equivalent to the [[recurrence relation]] {{tmath|1=\textstyle \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k} }}.{{r|rashed}}<ref name=Karaji>{{MacTutor|id=Al-Karaji|title=Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji}}</ref> The Persian poet and mathematician [[Omar Khayyam]] was probably familiar with the formula to higher orders, although many of his mathematical works are lost.<ref name="Coolidge" /> The binomial expansions of small degrees were known in the 13th century mathematical works of [[Yang Hui]]<ref>{{cite web | last = Landau | first = James A. | title = Historia Matematica Mailing List Archive: Re: [HM] Pascal's Triangle | work = Archives of Historia Matematica | format = mailing list email | access-date = 2007-04-13 | date = 1999-05-08 | url = http://archives.math.utk.edu/hypermail/historia/may99/0073.html | archive-date = 2021-02-24 | archive-url = https://web.archive.org/web/20210224081637/http://archives.math.utk.edu/hypermail/historia/may99/0073.html | url-status = dead }}</ref> and also [[Chu Shih-Chieh]].<ref name="Coolidge" />  Yang Hui attributes the method to a much earlier 11th century text of [[Jia Xian]], although those writings are now also lost.<ref>{{cite book |title=A History of Chinese Mathematics |chapter=Jia Xian and Liu Yi |last=Martzloff |first=Jean-Claude |translator-last=Wilson |translator-first=Stephen S. |publisher=Springer |year=1997 |orig-year=French ed. 1987 |isbn=3-540-54749-5 |page=142 |chapter-url=https://archive.org/details/historyofchinese0000mart_g2q8/page/142/mode/2up?&q=%22depends+on+the+binomial+expansion%22 |chapter-url-access=limited }}</ref>

In Europe, descriptions of the construction of Pascal's triangle can be found as early as [[Jordanus de Nemore]]'s ''De arithmetica'' (13th century).<ref>{{cite journal |last=Hughes |first=Barnabas|year=1989 |title=The arithmetical triangle of Jordanus de Nemore |journal=Historia Mathematica |volume=16 |number=3 |pages=213–223 |doi=10.1016/0315-0860(89)90018-9 }}</ref> In 1544, [[Michael Stifel]] introduced the term "binomial coefficient" and showed how to use them to express <math>(1+x)^n</math> in terms of <math>(1+x)^{n-1}</math>, via "Pascal's triangle".<ref name=Kline>{{cite book|title=History of mathematical thought|first=Morris| last=Kline| author-link=Morris Kline|page=273|publisher=Oxford University Press|year=1972}}</ref> Other 16th century mathematicians including [[Niccolò Fontana Tartaglia]] and [[Simon Stevin]] also knew of it.<ref name=Kline /> 17th-century mathematician [[Blaise Pascal]] studied the eponymous triangle comprehensively in his ''Traité du triangle arithmétique''.<ref>{{Cite book |last=Katz |first=Victor |author-link=Victor Katz |title=A History of Mathematics: An Introduction |edition=3rd |publisher=Addison-Wesley |year=2009 |orig-year=1993 |isbn=978-0-321-38700-4 |at=§ 14.3, {{pgs|487–497}} |chapter=Elementary Probability }}</ref>

By the early 17th century, some specific cases of the generalized binomial theorem, such as for <math>n=\tfrac{1}{2}</math>, can be found in the work of [[Henry Briggs (mathematician)|Henry Briggs]]' ''Arithmetica Logarithmica'' (1624).{{r|stillwell}} [[Isaac Newton]] is generally credited with discovering the generalized binomial theorem, valid for any real exponent, in 1665, inspired by the work of [[John Wallis]]'s ''Arithmetic Infinitorum'' and his method of interpolation.<ref name=Kline /><ref>{{cite book |title=Elements of the History of Mathematics |date=1994 |first=N. |last=Bourbaki |author-link=Nicolas Bourbaki  |translator=J. Meldrum |translator-link=John D. P. Meldrum |publisher=Springer |isbn=3-540-19376-6 |url-access=registration |url=https://archive.org/details/elementsofhistor0000bour}}</ref><ref name="Coolidge" /><ref>{{Cite journal |last=Whiteside |first=D. T. |author-link=Tom Whiteside |date=1961 |title=Newton's Discovery of the General Binomial Theorem |url=https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/newtons-discovery-of-the-general-binomial-theorem/19B5921B0248598CFB6441FCE085D113 |journal=The Mathematical Gazette |language=en |volume=45 |issue=353 |pages=175–180 |doi=10.2307/3612767 |jstor=3612767 }}</ref>{{r|stillwell}} A logarithmic version of the theorem for fractional exponents was discovered independently by [[James Gregory (mathematician)|James Gregory]] who wrote down his formula in 1670.<ref name=stillwell>{{cite book |last=Stillwell |first=John |author-link=John Stillwell |title=Mathematics and its history |date=2010 |publisher=Springer |isbn=978-1-4419-6052-8 |page=186 |edition=3rd}}</ref>