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Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala (Arabic for "The Compendious Book on Calculation by Completion and Balancing"), also known under a shorter name spelled as Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala and other transliterations) is a mathematical book written in Arabic, in approximately 820 AD by the Arabic mathematician Al-Khwarizmi (Arabic spelling:الخوارزمي).

Several authors have also published texts under the name of Kitāb al-ğabr wa-l-muqābala, including Abū Ḥanīfa al-Dīnawarī, Abū Kāmil,^[1] Abū Muḥammad al-ʿAdlī, Abū Yūsuf al-Miṣṣīṣī, Ibn Turk, Sind ibn ʿAlī, Sahl ibn Bišr,^[2] and Šarafaddīn al-Ṭūsī.

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 the modern algebra. The word algebra is derived from the name of one of the basic operations with equations (al-ğabr) described in this book. The book was translated in Latin as Liber algebrae et almucabala, hence "algebra".

Since the book does not give any credits, it is not clearly known what earlier works were used by al-Khwarizmi, and modern mathematical historians put forth opinions based on the textual analysis of the book and the overall body of knowledge of the contemporary Muslim world. Most certain are connections with Indian texts.

The book reduces arbitrary quadratic equations to one of the six basic types and provides algebraic and geometric methods to solve the basic ones. Lacking modern abstract notations, "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!"^[3] Thus the equations are verbally described in terms of "squares" (what would today be "x²"), "roots" (what would today be "x") and "numbers" (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)

The al-ğabr ("completion") operation is moving a negative quantity from one side of the equation to the other side and changing its sign. In an al-Khwarizmi's example (in modern notation), "x² = 40x - 4x²" is transformed by al-ğabr into "5x² = 40x". Repeated application of this rule eliminates negative quantities from calculations.

Al-Muqabala ("balancing") 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.

The next part of the book discusses practical examples of the application of the described rules. The following part deals with applied problems of measuring areas and volumes. The last part deals with computations involved in convoluted Islamic rules of inheritance. None of these parts require the knowledge about solving quadratic equations.

• R. Rashed, The development of Arabic mathematics: between arithmetic and algebra, London, 1994.
• Barnabas B. Hughes, ed., Robert of Chester's Latin Translation of Al-Khwarizmi's Al-Jabr: A New Critical Edition, (in Latin language) Wiesbaden: F. Steiner Verlag, 1989. ISBN 3-515-04589-9

• Algebra
• History of algebra

• Al-Khwarizmi