The Compendious Book on Calculation by Completion and Balancing
Encyclopedia
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 AD 820 by the Persian
(more exactly, Central Asian) mathematician Muhammad ibn Mūsā al-Khwārizmī
in Baghdad
, the capital of the Abbasid Caliphate at the time.
The book was translated into Latin in the mid 12th century under the title Liber Algebrae et Almucabola. Today's term "algebra
" is derived from the term al-ğabr in the title of this book. The al-ğabr provided an exhaustive account of solving for the positive roots of polynomial equations up to the second degree,.
Several authors have also published texts under the name of Kitāb al-ğabr wa-l-muqābala, including .
J. J. O'Connor and E. F. Robertson wrote in the MacTutor History of Mathematics archive
:
s and for some other problems, and considered to be the foundation of modern algebra
, establishing it as an independent discipline. 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 introduced to the Western world by the Latin translation of Robert of Chester
entitled Liber algebrae et almucabola, hence "algebra".
Since the book does not give any citations to previous authors, 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 mathematics
, as he had written a book entitled (The Book of Addition and Subtraction According to the Hindu Calculation) discussing the Hindu-Arabic numeral system
.
The book classifies 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!" 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:
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.
The al-ğabr (in Arabic script 'الجبر') ("completion" or "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 by al-ğabr into "5x2 = 40x". Repeated application of this rule eliminates negative quantities from calculations.
Al-Muqabala (in Arabic script 'المقابله') ("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, when restricted to positive coefficients and solutions.
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 area
s and volume
s. The last part deals with computations involved in convoluted Islamic rules of inheritance
. None of these parts require the knowledge about solving quadratic equations.
Persian people
The Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient Indo-Iranians and themselves part of...
(more exactly, Central Asian) mathematician Muhammad ibn Mūsā al-Khwārizmī
Muhammad ibn Musa al-Khwarizmi
'There is some confusion in the literature on whether al-Khwārizmī's full name is ' or '. Ibn Khaldun notes in his encyclopedic work: "The first who wrote upon this branch was Abu ʿAbdallah al-Khowarizmi, after whom came Abu Kamil Shojaʿ ibn Aslam." . 'There is some confusion in the literature on...
in Baghdad
Baghdad
Baghdad is the capital of Iraq, as well as the coterminous Baghdad Governorate. The population of Baghdad in 2011 is approximately 7,216,040...
, the capital of the Abbasid Caliphate at the time.
The book was translated into Latin in the mid 12th century under the title Liber Algebrae et Almucabola. Today's term "algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
" is derived from the term al-ğabr in the title of this book. The al-ğabr provided an exhaustive account of solving for the positive roots of polynomial equations up to the second degree,.
Several authors have also published texts under the name of Kitāb al-ğabr wa-l-muqābala, including .
Legacy
R. Rashed and Angela Armstrong write:J. J. O'Connor and E. F. Robertson wrote in the MacTutor History of Mathematics archive
MacTutor History of Mathematics archive
The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland...
:
The book
The book was a compilation and extension of known rules for solving quadratic equationQuadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
s and for some other problems, and considered to be the foundation of modern algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, establishing it as an independent discipline. The word algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
is derived from the name of one of the basic operations with equations (al-ğabr) described in this book. The book was introduced to the Western world by the Latin translation of Robert of Chester
Robert of Chester
Robert of Chester was an English arabist of the 12th century. He translated several historically important books from Arabic to Latin, by authors such as Abu Musa Jabir Ibn Hayyan and Al-Khwarizmi including:...
entitled Liber algebrae et almucabola, hence "algebra".
Since the book does not give any citations to previous authors, 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 mathematics
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
, as he had written a book entitled (The Book of Addition and Subtraction According to the Hindu Calculation) discussing the Hindu-Arabic numeral system
Hindu-Arabic numeral system
The Hindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arab mathematicians , and spread to the western world...
.
The book classifies 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
History of algebra
Algebra is a branch of mathematics concerning the study of structure, relation, and quantity. Elementary algebra is the branch that deals with solving for the operands of arithmetic equations. Modern or abstract algebra has its origins as an abstraction of elementary algebra...
) found in the Greek Arithmetica
Arithmetica
Arithmetica is an ancient Greek text on mathematics written by the mathematician Diophantus in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate equations and indeterminate equations.Equations in the book are called Diophantine equations...
or in Brahmagupta
Brahmagupta
Brahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...
's work. Even the numbers were written out in words rather than symbols!" 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)
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.
The al-ğabr (in Arabic script 'الجبر') ("completion" or "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 by al-ğabr into "5x2 = 40x". Repeated application of this rule eliminates negative quantities from calculations.
Al-Muqabala (in Arabic script 'المقابله') ("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, when restricted to positive coefficients and solutions.
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 area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
s and volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
s. The last part deals with computations involved in convoluted Islamic rules of inheritance
Islamic inheritance jurisprudence
Islamic Inheritance jurisprudence is a field of Islamic Jurisprudence that deals with inheritance, a topic that is prominently dealt with in the Qur'an. It is often called Mīrāth, and its branch of Islamic law is technically known as ʿulm al-farāʾiḍ...
. None of these parts require the knowledge about solving quadratic equations.
Further reading
- Barnabas B. Hughes, ed., Robert of ChesterRobert of ChesterRobert of Chester was an English arabist of the 12th century. He translated several historically important books from Arabic to Latin, by authors such as Abu Musa Jabir Ibn Hayyan and Al-Khwarizmi including:...
'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 - R. Rashed, The development of Arabic mathematics: between arithmetic and algebra, London, 1994.
External links
- 19th Century English Translation
- Al-Khwarizmi
- Annotated excerpt from a translation of the Compendious Book. University of Duisburg-EssenUniversity of Duisburg-EssenThe University Duisburg-Essen is a public university in Duisburg and Essen, North Rhine-Westphalia, Germany and a member of the new founded University Alliance Metropolis Ruhr....
. - The Compendious Book on Calculation by Completion and Balancing In the Arabic original with an English translation (PDF)