The Compendious Book on Calculation by Completion and Balancing

Legacy

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."

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."

The book

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.

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. There are indications of connections with Indian mathematics, as he had written a book entitled The Book of Bringing Together and Separating According to the Hindu Calculation (Kitāb al-Jamʿ wa-l-tafrīq bi-ḥisāb al-Hind), discussing the Hindu-Arabic numeral system.

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!

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:

1. squares equal roots (ax² = bx)
2. squares equal number (ax² = c)
3. roots equal number (bx = c)
4. squares and roots equal number (ax² + bx = c)
5. squares and number equal roots (ax² + c = bx)
6. 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 ("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 by al-ğabr into "5x² = 40x". Repeated application of this rule eliminates negative quantities from calculations.

Al-Muqabala (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

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).

Other topics

Al-Khwārizmī explicates the Jewish calendar and the 19-year cycle described by the convergence of lunar months and solar years.

The last part deals with computations involved in Islamic rules of inheritance.