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Al-Jabr

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From Wikipedia, the free encyclopedia

9th-century Arabic work on algebra

For other uses, seeJaber (disambiguation).

The Concise Book of Calculation by Restoration and Balancing
Title page, 9th century
Author	Muhammad ibn 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 ArabicWikisource
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] orLatin:Liber Algebræ et Almucabola), commonly abbreviatedAl-Jabr orAlgebra (Arabic:الجبر), is anArabic-language mathematical treatise onalgebra written inBaghdad around 820 by thePersian polymath Al-Khwarizmi. It was a landmark work in thehistory of mathematics, with its title being the ultimate etymology of the word "algebra" itself, later borrowed intoMedieval Latin asalgebrāica.

Al-Jabr provided an exhaustive account of solving for the positiveroots ofpolynomial equations up to the second degree.^[1]^[b] It was the first text to teachelementary algebra, and the first to teach algebra for its own sake.^[c] It also introduced the fundamental concept of "reduction" and "balancing" (which the termal-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 historianVictor J. Katz regardsAl-Jabr as the first true algebra text that is still extant.^[e] Translated into Latin byRobert 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, includingAbu Hanifa Dinawari,Abu Kamil, Abū Muḥammad al-ʿAdlī, Abū Yūsuf al-Miṣṣīṣī,'Abd al-Hamīd ibn Turk, Sind ibn ʿAlī, Sahl ibn Bišr, andŠarafaddīn al-Ṭūsī.

Legacy

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R. Rashed and Angela Armstrong write:

Al-Khwarizmi's text can be seen to be distinct not only from theBabylonian tablets, but also from theDiophantus'Arithmetica. It no longer concerns a series ofproblems to be resolved, but anexposition 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.^[8]

J. J. O'Connor and E. F. Robertson wrote in theMacTutor 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 allowedrational 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.^[9]

Content

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The book was a compilation and extension of known rules for solvingquadratic equations and for some other problems, and considered to be the foundation of algebra, establishing it as an independent discipline. The wordalgebra is derived from the name of one of the basic operations with equations described in this book, following its Latin translation byRobert of Chester.^[10]

Quadratic equations

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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 (seeHistory of algebra) found in the GreekArithmetica or inBrahmagupta'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 "x²"), "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 (ax² =bx)
squares equal number (ax² =c)
roots equal number (bx =c)
squares and roots equal number (ax² +bx =c)
squares and number equal roots (ax² +c =bx)
roots and number equal squares (bx +c =ax²)

Islamic mathematicians, unlike the Hindus, did not deal with negative numbers at all; hence an equation likebx +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.^[11]

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), "x² = 40x − 4x²" is transformed byal-Jabr into "5x² = 40x". 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: "x² + 5 = 40x + 4x²" is turned into "5 = 40x + 3x²". 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

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The second chapter of the book catalogues methods of findingarea andvolume. These include approximations ofpi (π), given three ways, as 3 1/7, √10, and 62832/20000. This latter approximation, equalling 3.1416, earlier appeared in the IndianĀryabhaṭīya (499 CE).^[12]

Notes

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^The Arabic title is sometimes condensed toḤisāb al-Jabr wal-Muqābalah orKitāb al-Jabr wal-Muqābalah} or given under othertransliterations.
^"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 termsal-jabr andmuqabalah mean, but the usual interpretation is similar to that implied in the translation above. The wordal-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 wordmuqabalah 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

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^^a ^b ^c ^dBoyer, 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. Retrieved24 January 2018.
^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.
^Robert of Chester (1915).Algebra of al-Khowarizmi. Macmillan. Archived fromthe 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
^^a ^bB.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.

External links

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EnglishWikisource has original text related to this article:

The Compendious Book on Calculation by Completion and Balancing

19th Century English Translation at theInternet 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.

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