View source for Al-Jabr

{{Short description|9th-century Arabic work on algebra}}
{{other uses|Jaber (disambiguation)}}
{{Use dmy dates|date=September 2020}}
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{{Infobox book
| name              = The Concise Book of Calculation by Restoration and Balancing
| image             = Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg
| alt               = title page in Arabic writing and calligraphy; hand-drawn ornamental frame; parchment is gilded and stained from age
| caption           = Title page, 9th century
| author            = [[Al-Khwarizmi|Muhammad ibn Musa al-Khwarizmi]]
| title_orig        = كتاب المختصر في حساب الجبر والمقابلة
| orig_lang_code    = ar
| illustrator       = Muhammad ibn Musa al-Khwarizmi
| country           = [[Abbasid Caliphate]]
| language          = Arabic
| subject           = [[Algebra]]
| pub_date          = 820
| native_wikisource = الكتاب المختصر في حساب الجبر والمقابلة
| wikisource        = The Compendious Book on Calculation by Completion and Balancing
| notes             = 
}}
'''''The Concise Book of Calculation by Restoration and Balancing''''' ({{langx|ar|الكتاب المختصر في حساب الجبر والمقابلة}}, {{transliteration|ar|al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah}};{{efn|The Arabic title is sometimes condensed to ''Ḥisāb al-Jabr wal-Muqābalah'' or ''Kitāb al-Jabr wal-Muqābalah}'' or given under other [[romanization of Arabic|transliterations]].}} or {{langx|la|Liber Algebræ et Almucabola}}), commonly abbreviated '''''Al-Jabr''''' or '''''Algebra''''' ([[Arabic]]: {{lang|ar|الجبر}}), is an [[Arabic mathematics|Arabic mathematical]] treatise on [[algebra]] written in [[Baghdad]] around 820 by the Persian polymath [[Al-Khwarizmi]]. It was a landmark work in the [[history of mathematics]], with its title being the ultimate etymology of the word "algebra" itself, later borrowed into [[Medieval Latin]] as {{lang|la|algebrāica}}.

''Al-Jabr'' provided an exhaustive account of solving for the positive [[root of a function|roots]] of [[polynomial equations]] up to the second degree.<ref name="Boyer 1991">{{cite book|first=Carl B.|last=Boyer|author-link=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|isbn=0-471-54397-7|url-access=registration|url=https://archive.org/details/historyofmathema00boye |pages=228–229}}</ref>{{efn|"The Arabs in general loved a good clear argument from premise to conclusion, as well as systematic organization – respects in which neither Diophantus nor the Hindus excelled."{{r|Boyer 1991}}}} It was the first text to teach [[elementary algebra]], and the first to teach algebra for its own sake.{{efn|"In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".<ref name="Gandz236">{{cite book|last1=Gandz|last2=Saloman|year=1936|title=The sources of al-Khwarizmi's algebra|publisher=Osiris|volume=I|pages=263–277}}</ref>}} It also introduced the fundamental concept of "reduction" and "balancing" (which the term ''al-jabr'' originally referred to), the transposition of subtracted terms to the other side of an equation, i.e. the cancellation of like terms on opposite sides of the equation.{{efn|"It is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the translation above. The word ''al-jabr'' presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation, which is evident in the treatise; the word ''muqabalah'' is said to refer to "reduction" or "balancing"—that is, the cancellation of like terms on opposite sides of the equation."{{r|Boyer 1991}}}} The mathematics historian [[Victor J. Katz]] regards ''Al-Jabr'' as the first true algebra text that is still extant.{{efn|"The first true algebra text which is still extant is the work on al-jabr and al-muqabala by Mohammad ibn Musa al-Khwarizmi, written in Baghdad around 825."<ref>{{cite journal
 | last1 = Katz | first1 = Victor J.
 | last2 = Barton | first2 = Bill
 | date = December 2006
 | doi = 10.1007/s10649-006-9023-7
 | issue = 2
 | journal = Educational Studies in Mathematics
 | pages = 185–201
 | title = Stages in the history of algebra with implications for teaching
 | volume = 66}} See p. 190.</ref>}} Translated into Latin by [[Robert of Chester]] in 1145, it was used until the sixteenth century as the principal mathematical textbook of European universities.<ref>{{cite book|url=https://archive.org/details/isbn_9780333631423|url-access=registration|title=History of the Arabs|publisher=Macmillan International Higher Education|last=Philip Khuri Hitti|author-link=Philip Khuri Hitti|year=2002|pages=[https://archive.org/details/isbn_9780333631423/page/379 379]|isbn=9780333631423 }}</ref>{{efn|"The Compendious Book on Calculation by Completion and Balancing" (Hisab al-Jabr wa H-Muqabala) on the development of the subject cannot be underestimated. Translated into Latin during the twelfth century, it remained the principal mathematics textbook in European universities until the sixteenth century"<ref>{{Cite book|url=https://archive.org/details/isbn_9780781810159|url-access=registration|title=A History of the Islamic World|publisher=Hippocrene Books|last=Fred James Hill, Nicholas Awde|year=2003|pages=[https://archive.org/details/isbn_9780781810159/page/55 55]|isbn=9780781810159 }}</ref>}}<ref>{{Cite web|url=http://www.ms.uky.edu/~carl/ma330/project2/al-khwa21.html|title=Al-Khwarizmi|first1=Shawn|last1= Overbay|first2= Jimmy |last2=Schorer|first3= Heather|last3= Conger|publisher=University of Kentucky|author-link=University of Kentucky}}</ref><ref>{{Cite web|url=http://www.sjsu.edu/people/patricia.backer/history/islam.htm|title=Islam Spain and the history of technology|website=www.sjsu.edu|access-date=2018-01-24}}</ref>

Several authors have also published texts under this name, including [[Abu Hanifa Dinawari]], [[Abu Kamil]], Abū Muḥammad al-ʿAdlī, Abū Yūsuf al-Miṣṣīṣī, [['Abd al-Hamīd ibn Turk]], Sind ibn ʿAlī, Sahl ibn Bišr, and [[Sharaf al-Dīn al-Tūsī|Šarafaddīn al-Ṭūsī]].

==Legacy==
R. Rashed and Angela Armstrong write:
{{blockquote|Al-Khwarizmi's text can be seen to be distinct not only from the [[Babylonian mathematics|Babylonian tablets]], but also from the [[Diophantus]]' ''[[Arithmetica]]''. It no longer concerns a series of [[Mathematical problem|problem]]s to be resolved, but an [[Expository writing|exposition]] which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems.<ref>{{Cite book | last1=Rashed | first1=R. | last2=Armstrong | first2=Angela | year=1994 | title=The Development of Arabic Mathematics | publisher=[[Springer Science+Business Media|Springer]] | isbn=0-7923-2565-6 | oclc=29181926 | pages=11–12 }}</ref>}}

J. J. O'Connor and E. F. Robertson wrote in the ''[[MacTutor History of Mathematics Archive]]'':
{{blockquote|Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed [[rational numbers]], [[irrational number]]s, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before.<ref>{{MacTutor|class=HistTopics|id=Arabic_mathematics|title=Arabic mathematics: forgotten brilliance?|year=1999}}</ref>}}

==The book==
The book was a compilation and extension of known rules for solving [[quadratic equation]]s and for some other problems, and considered to be the foundation of algebra, establishing it as an independent discipline. The word ''algebra'' is derived from the name of one of the basic operations with equations described in this book, following its Latin translation by [[Robert of Chester]].<ref name=Algebra>{{cite book |last = Robert of Chester  |title = Algebra of al-Khowarizmi  |url = http://library.albany.edu/preservation/brittle_bks/khuwarizmi_robertofchester/ |publisher = Macmillan |year = 1915 |archive-url=https://web.archive.org/web/20181118154937/http://library.albany.edu/preservation/brittle_bks/khuwarizmi_robertofchester/ |archive-date=18 November 2018 |url-status=dead}}</ref>

== Quadratic equations ==
[[File:Bodleian MS. Huntington 214 roll332 frame36.jpg|thumbnail|right|Pages from a 14th-century Arabic copy of the book, showing geometric solutions to two quadratic equations.]]
The book classifies quadratic equations to one of the six basic types and provides algebraic and geometric methods to solve the basic ones. Historian Carl Boyer notes the following regarding the lack of modern abstract notations in the book:
{{blockquote|... the algebra of al-Khwarizmi is thoroughly rhetorical, with none of the syncopation (see [[History of algebra]]) found in the Greek ''[[Arithmetica]]'' or in [[Brahmagupta]]'s work. Even the numbers were written out in words rather than symbols! | Carl B. Boyer | A History of Mathematics{{r|Boyer 1991}}}}

Thus the equations are verbally described in terms of "squares" (what would today be "''x''<sup>2</sup>"), "roots" (what would today be "''x''") and "numbers" ("constants": ordinary spelled out numbers, like 'forty-two'). The six types, with modern notations, are:
# squares equal roots (''ax''<sup>2</sup> = ''bx'')
# squares equal number (''ax''<sup>2</sup> = ''c'')
# roots equal number (''bx'' = ''c'')
# squares and roots equal number (''ax''<sup>2</sup> + ''bx'' = ''c'')
# squares and number equal roots (''ax''<sup>2</sup> + ''c'' = ''bx'')
# roots and number equal squares (''bx'' + ''c'' = ''ax''<sup>2</sup>)

Islamic mathematicians, unlike the Hindus, did not deal with negative numbers at all; hence an equation like ''bx'' + ''c'' = 0 does not appear in the classification, because it has no positive solutions if all the coefficients are positive. Similarly equation types 4, 5 and 6, which look equivalent to the modern eye, were distinguished because the coefficients must all be positive.<ref>{{cite book
 | last = Berggren | first = J. L.
 | contribution = Algebra in Islam
 | doi = 10.1007/978-1-4612-4608-4_4
 | isbn = 9781461246084
 | pages = 99–126
 | publisher = Springer
 | title = Episodes in the Mathematics of Medieval Islam
 | year = 1986}} See p. 103</ref>

''Al-Jabr'' ("forcing", "restoring") operation is moving a deficient quantity from one side of the equation to the other side. In an al-Khwarizmi's example (in modern notation), "''x''<sup>2</sup> = 40''x''&nbsp;−&nbsp;4''x''<sup>2</sup>" is transformed by ''al-Jabr'' into "5''x''<sup>2</sup> = 40''x''". Repeated application of this rule eliminates negative quantities from calculations.

''Al-Muqābala'' ({{lang|ar|المقابله}}, "balancing" or "corresponding") means subtraction of the same positive quantity from both sides: "''x''<sup>2</sup> + 5 = 40''x'' + 4''x''<sup>2</sup>" is turned into "5 = 40''x'' + 3''x''<sup>2</sup>". Repeated application of this rule makes quantities of each type ("square"/"root"/"number") appear in the equation at most once, which helps to see that there are only 6 basic solvable types of the problem, when restricted to positive coefficients and solutions. <!-- see talk -->

Subsequent parts of the book do not rely on solving quadratic equations.

== Area and volume ==
The second chapter of the book catalogues methods of finding [[area]] and [[volume]]. These include approximations of [[pi]] (π), given three ways, as 3 1/7, √10, and 62832/20000. This latter approximation, equalling 3.1416, earlier appeared in the Indian ''[[Āryabhaṭīya]]'' (499 CE).<ref name=BLvdW>B.L. van der Waerden, ''A History of Algebra: From al-Khwārizmī to Emmy Noether''; Berlin: Springer-Verlag, 1985. {{ISBN|3-540-13610-X}}</ref>

== Other topics ==
Al-Khwārizmī explicates the [[Jewish calendar]] and the [[Metonic cycle|19-year cycle]] described by the convergence of lunar months and solar years.<ref name=BLvdW />

About half of the book deals with [[Islamic inheritance jurisprudence|Islamic rules of inheritance]], which are complex and require skill in first-order algebraic equations.<ref>{{cite encyclopedia|encyclopedia=Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences|volume=1|editor=I. Grattan-Guinness
|publisher=JHU Press|year=2003|title=Mathematics applied to aspects of religious ritual in Islam|author=David A. King|url=https://books.google.com/books?id=2hDvzITtfdAC&pg=PA83|page=83|isbn=9780801873966}}</ref>

== References ==
=== Notes ===
{{Notelist|30em}}

===Citations===
{{Reflist}}

== Further reading ==
* Hughes, Barnabas B. ed., ''[[Robert of Chester]]'s Latin Translation of Al-Khwarizmi's Al-Jabr: A New Critical Edition'', (in [[Latin]]) Wiesbaden: F. Steiner Verlag, 1989. {{ISBN|3-515-04589-9}}
* Rashed, R. ''The development of Arabic mathematics: between arithmetic and algebra'', London, 1994.{{ISBN?}}

==External links==
{{Wikisource|The Compendious Book on Calculation by Completion and Balancing}}
*[https://archive.org/details/algebraofmohamme00khuwrich 19th Century English Translation] at the [[Internet Archive]]
*[http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Al-Khwarizmi.html Al-Khwarizmi]
*[http://www.uni-due.de/imperia/md/content/didmath/ag_jahnke/musa.pdf Annotated excerpt from a translation of the Compendious Book]. [[University of Duisburg-Essen]].
*[http://www.wilbourhall.org/index.html#algebra The Compendious Book on Calculation by Completion and Balancing] – in the Arabic original with an English translation (PDF)
*{{cite web|url=http://www.muslimheritage.com/topics/default.cfm?ArticleID=637 |title=The Science of Restoring and Balancing – The Science of Algebra |first=Mahbub |last=Ghani |date=5 January 2007 |work=Muslim Heritage}}

{{Islamic mathematics}}
{{Mathematics in Iran}}

{{Authority control}}

[[Category:9th-century Arabic-language books]]
[[Category:History of algebra]]
[[Category:Mathematical works of the medieval Islamic world]]
[[Category:Mathematics textbooks]]
[[Category:Scientific works of the Abbasid Caliphate]]