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bn musa al khwarizmi original title كتاب المختصر في حساب الجبر والمقابلة illustrator muhammad ibn musa al khwarizmi language arabic subject algebra publication date 820 publication place abbasid caliphate original text كتاب المختصر في حساب الجبر والمقابلة at arabic wikisource translation the concise book of calculation by restoration and balancing at wikisource the concise book of calculation by restoration and balancing arabic الكتاب المختصر في حساب الجبر والمقابلة al kitāb al mukhtaṣar fī ḥisāb al jabr wal muqābalah a or latin liber algebræ et almucabola commonly abbreviated al jabr or algebra arabic الجبر is an arabic language mathematical treatise on algebra written in baghdad around 820 by 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 algebrāica al jabr provided an exhaustive account of solving for the positive roots of polynomial equations up to the second degree 1 b it was the first text to teach elementary algebra and the first to teach algebra for its own sake c 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 d the mathematics historian victor j katz regards al jabr as the first true algebra text that is still extant e translated into latin by robert of chester in 1145 it was used until the sixteenth century as the principal mathematical textbook of european universities 4 f 6 7 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 šarafaddīn al ṭūsī content edit the book was a compilation and extension of known rules for solving quadratic equations 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 8 quadratic equations edit 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 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 1 thus the equations are verbally described in terms of squares what would today be x 2 roots what would today be x and numbers constants ordinary spelled out numbers like forty two the six types with modern notations are 1 squares equal roots a x 2 b x displaystyle ax 2 bx 2 squares equal number a x 2 c displaystyle ax 2 c 3 roots equal number b x c displaystyle bx c 4 squares and roots equal number a x 2 b x c displaystyle ax 2 bx c 5 squares and number equal roots a x 2 c b x displaystyle ax 2 c bx 6 roots and number equal squares b x c a x 2 displaystyle bx c ax 2 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 9 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 2 40 x 4 x 2 is transformed by al jabr into 5 x 2 40 x repeated application of this rule eliminates negative quantities from calculations al muqābala المقابله balancing or corresponding means subtraction of the same positive quantity from both sides x 2 5 40 x 4 x 2 is turned into 5 40 x 3 x 2 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 subsequent parts of the book do not rely on solving quadratic equations area and volume edit 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 displaystyle 3 tfrac 1 7 10 displaystyle sqrt 10 and 62832 20000 displaystyle 62832 20000 this latter approximation equalling 3 1416 earlier appeared in the indian āryabhaṭīya 499 ce 10 other topics edit al khwārizmī explicates the jewish calendar and the 19 year cycle described by the convergence of lunar months and solar years 10 about half of the book deals with islamic rules of inheritance which are complex and require skill in first order algebraic equations 11 legacy edit al jabr introduced balancing and reduction to mathematical expressions and founded algebra as an independent mathematical discipline which is taught worldwide typically as part of an elementary and high school curriculum 12 r rashed and angela armstrong write al khwarizmi s text can be seen to be distinct not only from the babylonian tablets but also from the diophantus arithmetica it no longer concerns a series of problems to be resolved but an 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 13 j j o connor and e f robertson wrote in the mactutor history of mathematics archive 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 numbers 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 14 notes edit the arabic title is sometimes condensed to ḥisāb al jabr wal muqābalah or kitāb al jabr wal muqābalah or given under other transliterations 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 1 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 2 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 1 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 3 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 5 references edit 1 2 3 4 boyer carl b 1991 the arabic hegemony a history of mathematics second ed john wiley sons inc pp 228 229 isbn 0 471 54397 7 gandz saloman 1936 the sources of al khwarizmi s algebra vol i osiris pp 263 277 katz victor j barton bill december 2006 stages in the history of algebra with implications for teaching educational studies in mathematics 66 2 185 201 doi 10 1007 s10649 006 9023 7 see p 190 philip khuri hitti 2002 history of the arabs macmillan international higher education pp 379 isbn 9780333631423 fred james hill nicholas awde 2003 a history of the islamic world hippocrene books pp 55 isbn 9780781810159 overbay shawn schorer jimmy conger heather al khwarizmi university of kentucky islam spain and the history of technology www sjsu edu retrieved 24 january 2018 robert of chester 1915 algebra of al khowarizmi macmillan archived from the original on 18 november 2018 berggren j l 1986 algebra in islam episodes in the mathematics of medieval islam springer pp 99 126 doi 10 1007 978 1 4612 4608 4_4 isbn 9781461246084 see p 103 1 2 b l van der waerden a history of algebra from al khwārizmī to emmy noether berlin springer verlag 1985 isbn 3 540 13610 x david a king 2003 mathematics applied to aspects of religious ritual in islam in i grattan guinness ed companion encyclopedia of the history and philosophy of the mathematical sciences vol 1 jhu press p 83 isbn 9780801873966 al khwarizmi the father of algeb www irfi org retrieved 10 april 2026 rashed r armstrong angela 1994 the development of arabic mathematics springer pp 11 12 isbn 0 7923 2565 6 oclc 29181926 o connor john j robertson edmund f 1999 arabic mathematics forgotten brilliance mactutor history of mathematics archive university of st andrews further reading edit 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 missing external links edit english wikisource has original text related to this article the compendious book on calculation by completion and balancing 19th century english translation at the internet archive al khwarizmi annotated excerpt from a translation of the compendious book university of duisburg essen the compendious book on calculation by completion and balancing in the arabic original with an english translation pdf ghani mahbub 5 january 2007 the science of restoring and balancing the science of algebra muslim heritage v t e mathematics in the medieval islamic world mathematicians 9th century abd al hamīd ibn turk sanad ibn ali al jawharī al ḥajjāj ibn yūsuf al kindi qusta ibn luqa al mahani al dinawari banū mūsā brothers hunayn ibn ishaq al khwarizmi yusuf al khuri ishaq ibn hunayn na im ibn musa thābit ibn qurra al marwazi abu said gorgani 10th century abu al wafa al khazin al qabisi abu kamil ahmad ibn yusuf aṣ ṣaidanānī sinān ibn al fatḥ al khojandi al nayrizi al saghani brethren of purity ibn sahl ibn yunus al uqlidisi al battani sinan ibn thabit ibrahim ibn sinan al isfahani nazif ibn yumn al qūhī abu al jud al sijzi al karaji al majriti al jabali 11th century abu nasr mansur alhazen kushyar gilani al biruni ibn al samh abu mansur al baghdadi avicenna al jayyānī al nasawī al zarqālī ibn hud al isfizari omar khayyam muhammad al baghdadi 12th century jabir ibn aflah al kharaqī al khazini al samawal al maghribi al hassar sharaf al din al tusi ibn al yasamin 13th century ibn al ha im al ishbili ahmad al buni ibn munim alam al din al hanafi ibn adlan al urdi nasir al din al tusi al abhari muhyi al din al maghribi al hasan al marrakushi qutb al din al shirazi shams al din al samarqandi ibn al banna kamāl al dīn al fārisī 14th century nizam al din al nisapuri ibn al shatir ibn al durayhim al khalili al umawi 15th century ibn al majdi al rūmī al kāshī ulugh beg ali qushji al wafa i al qalaṣādī sibt al maridini ibn ghazi al miknasi 16th century al birjandi muhammad baqir yazdi taqi ad din ibn hamza al maghribi mathematical works the compendious book on calculation by completion and balancing de gradibus principles of hindu reckoning book of optics the book of healing book on the measurement of plane and spherical figures encyclopedia of the brethren of purity toledan tables tabula rogeriana zij concepts alhazen s problem islamic geometric patterns centers al azhar university al mustansiriya university house of knowledge house of wisdom constantinople observatory of taqi ad din madrasa maragheh observatory university of al qarawiyyin influences babylonian mathematics greek mathematics indian mathematics influenced byzantine mathematics european mathematics indian mathematics related hindu arabic numeral system arabic numerals eastern arabic numerals western arabic numerals trigonometric functions history of trigonometry history of algebra v t e mathematics in iran mathematicians before 20th century abu al wafa buzjani jamshid al kashi al kashi s theorem omar khayyam khayyam pascal s triangle khayyam saccheri quadrilateral khayyam s solution of cubic equations al mahani muhammad baqir yazdi nizam al din al nisapuri al nayrizi kushyar gilani ayn al quzat hamadani al isfahani al isfizari al khwarizmi al jabr najm al din al qazwini al katibi nasir al din al tusi al biruni modern maryam mirzakhani caucher birkar sara zahedi farideh firoozbakht firoozbakht s conjecture s l hakimi havel hakimi algorithm siamak yassemi freydoon shahidi langlands shahidi method hamid naderi yeganeh esmail babolian ramin takloo bighash lotfi a zadeh fuzzy mathematics fuzzy set fuzzy logic ebadollah s mahmoodian reza sarhangi the bridges organization siavash shahshahani gholamhossein mosaheb amin shokrollahi mohammad mehdi zahedi mohsen hashtroodi hossein zakeri amir ali ahmadi prize recipients fields medal maryam mirzakhani 2014 caucher birkar 2018 ems prize sara zahedi 2016 satter prize maryam mirzakhani 2013 organizations iranian mathematical society institutions institute for research in fundamental sciences authority control databases gnd retrieved from https en wikipedia org w index php title al jabr oldid 1368647525 categories 9th century arabic language books history of algebra mathematical works of the medieval islamic world mathematics textbooks scientific works of the abbasid caliphate hidden categories articles with 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