Pascal's triangle: Difference between revisions

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[[File:TrianguloPascal.jpg|thumb|right|upright=1.25|[[Blaise Pascal|Pascal]]'s version of the triangle]]

The pattern of numbers that forms Pascal's triangle was known well before Pascal's time. Some sources claim that the Persian mathematician [[Al-Karaji]] (953–1029) wrote a now-lost book which contained the first description of Pascal's triangle<ref>{{Cite web |title=The Binomial Theorem |url=https://mathcenter.oxford.emory.edu/site/math108/binomialTheorem/ |access-date=2024-12-02 |website=mathcenter.oxford.emory.edu|quote=That said, he was not the first person to study it. The Persian mathematician and engineer Al-Karaji, who lived from 935 to 1029 is currently credited with its discovery. (''Interesting tidbit: Al-Karaji also introduced the powerful idea of arguing by mathematical induction.'')}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=kt9DIY1g9HYC&q=al+karaji+pascal%27s+triangle&pg=PA132|title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures|last=Selin|first=Helaine|date=2008-03-12|publisher=Springer Science & Business Media|isbn=9781402045592|language=en|page=132|bibcode=2008ehst.book.....S|quote=Other, lost works of al-Karaji's are known to have dealt with inderterminate algebra, arithmetic, inheritance algebra, and the construction of buildings. Another contained the first known explanation of the arithmetical (Pascal's) triangle; the passage in question survived through al-Sama'wal's Bahir (twelfth century) which heavily drew from the Badi.}}</ref><ref>[https://books.google.com/books?id=vSkClSvU_9AC&pg=PA62 The Development of Arabic Mathematics Between Arithmetic and Algebra - R. Rashed|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-Karaji, cited by al-Sama'wal in al-Bahir."] "Page 63"</ref><ref>{{Cite book|url=https://books.google.com/books?id=kAjABAAAQBAJ&q=al+karaji+binomial+theorem&pg=PA54|title=From Alexandria, Through Baghdad: Surveys and Studies in the Ancient Greek and Medieval Islamic Mathematical Sciences in Honor of J.L. Berggren|last1=Sidoli|first1=Nathan|last2=Brummelen|first2=Glen Van|date=2013-10-30|publisher=Springer Science & Business Media|isbn=9783642367366|language=en|page=54|quote=However, the use of binomial coefficients by Islamic mathematicians of the eleventh century, in a context which had deep roots in Islamic mathematics, suggests strongly the table was a local discovery - most probably of al-Karaji."}}</ref> while others say that the ''[[Chandaḥśāstra]]'', by the Indian lyricist [[Piṅgala]], (3rd or 2nd century BC) somewhat crypically 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]] in his "method of pyramidal expansion" (''meru-prastāra'') for counting metres, is equivalent to [[Pascal's triangle]].<ref>{{cite journal |last=Alsdorf |first=Ludwig |year=1991 |orig-year=1933 |title=The Pratyayas: Indian Contribution to Combinatorics |journal=Indian Journal of History of Science |volume=26 |number=1 |pages=17–61 |url=https://insa.nic.in/(S(f0a4mvblfb5vfmialsdau2an))/writereaddata/UpLoadedFiles/IJHS/Vol26_1_3_SRSarma.pdf}} Translated by S. R. Sarma from {{cite journal |last=Alsdorf |first=Ludwig |display-authors=0 |title={{mvar|π}} Die Pratyayas. Ein Beitrag zur indischen Mathematik |journal=Zeitschrift fur Indologie und Iranistik |volume=9 |year=1933 |pages=97–157 }} {{pb}} {{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}} Tertiary sources: {{pb}} {{cite book |title=A Concise History Of Science In India |year=1971 |publisher=Indian National Science Academy |editor-last=Bose |editor-first=D. M. |editor-link=Debendra Mohan Bose |last=Sen |first=Samarendra Nath |chapter=Mathematics |chapter-url=https://archive.org/details/in.ernet.dli.2015.502083/page/n178 |at=Ch. 3, {{pgs|136–212}}, esp. "Permutations, Combinations and Pascal Triangle", {{pgs|156–157}} }} {{pb}} {{cite journal |title=The Binomial Coefficient Function |last=Fowler |first=David H. |author-link= David Fowler (mathematician) |journal=The American Mathematical Monthly |year=1996 |volume=103 |number=1 |pages=1–17, esp. §4 "A Historical Note", {{pgs|10–17}} |jstor=2975209 }} {{pb}} {{cite book |last=Roy |first=Ranjan |author-link=Ranjan Roy |year=2021 |orig-year=1st ed. 2011 |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> It was later repeated by [[Omar Khayyám]] (1048–1131), another Persian mathematician; thus the triangle is also referred to as '''Khayyam's triangle''' ({{langx|fa|مثلث خیام|label=none}}) in Iran.<ref>{{cite book |author=Kennedy, E. |title=Omar Khayyam. The Mathematics Teacher 1958 |jstor=i27957284|year=1966 |publisher=National Council of Teachers of Mathematics |pages=140–142}}</ref> Several theorems related to the triangle were known, including the [[binomial theorem]]. Khayyam used a method of finding [[nth root|''n''th roots]] based on the binomial expansion, and therefore on the binomial coefficients.<ref name=":0">{{citation

| last = Coolidge | first = J. L. | author-link = Julian Coolidge

| journal = [[The American Mathematical Monthly]]