The Compendious Book on Calculation by Completion and Balancing

Al - Kitaab al - Mukhtasar fī Hisab al - jabr wa - ʾ l - muqabala (Arabic الكتاب المختصر في حساب الجبر والمقابلة, The concise book about the calculation method by supplementing and balancing ') or Hisab al - jabr wa - ʾ l - muqabala or Kitāb al - jabr wa - ʾ l - muqabala is a math book that was written about 825 by the mathematician Muhammad ibn Musa al - Khwarizmi. The word algebra goes back to the Latin translation of the title ( Ludus algebrae almucgrabalaeque ).

Historical Significance

This book is an important basic text of the classical algebra, the science of solving equations. It determined the character of the algebra as a practical science without axiomatic foundation for centuries.

The work is a good example of the importance of Islamic culture at the height of the Islamic expansion, as it was used in the House of Wisdom in Baghdad and exerted a lasting influence on the history of mathematics. Many Islamic scholars gathered the mathematical knowledge of the ancient Greeks and created a synthesis of Indian mathematics, in particular of Aryabhata and Brahmagupta. This book formed a bridge between the ancient, western and Indian culture. It had greater influence than the work of Diophantus, although it offered content and less than pure word algebra renounced formulas. Al- Khwarizmi here was mainly based on Brahmagupta, but probably knew the corresponding Greek works.

The culmination of the medieval, written in Arabic, algebra was only reached with Omar Chajjams algebra over the evidence of the problems of al - jabr and al - muqabalah ( " resolution of cubic equations by means of conic sections "), but the standard work for the treatment linear and quadratic equations was among the Persians, Arabs, and also later in Medieval Europe, the book Al- Chwarizmis.

Content

According to the testimony of the author, the book includes everything "from the arithmetic is very useful, what people need in inheritance matters as division problems, litigation, commercial, and in general for all mutual relations; or even in the land surveying, digging of canals, geometrical computations and for various other things. "

The book is divided into 3 parts:

• Systematic treatment and resolution of the DC busbars from the first and second degrees (main part of the book ) with final exercises
• Practical surveying tasks
• Solution of Erbteilungsaufgaben

In the equations, no symbols are used, but they are always expressed verbally. All equations are 6 standard types returned (a, b, c are non-negative coefficients; only positive solutions considered ):

• Ax2 = bx
• Ax2 = c
• Bx = c
• Ax2 bx = c
• Ax2 c = bx
• Ax2 = bx c

Each type of equation is solved by a rule which is geometrically demonstrated.

The author also indicates the operations that will help you bring problems to one of its six standard forms:

• Al - Gabr ( " Complete ", " Restore", "Quite Make " ) - eliminate the negative expressions
• Al - muqabalah ( " compensation " ) - Summary of the expressions of the same potency per page

And the four basic operations , -, *, /.

Although the method presented was cumbersome, so that all occurring in practice quadratic equations could be solved.

Tradition

The work is preserved in an Arabic copy and in several Latin translations.

The six types of equations were for centuries the heart of the algebra. Only Michael Stifel was to 1544 negative coefficient and was able to reduce the number of types of equations. And also only at this time (about mid-16th century ) could be cubic in Europe solve equations (see Gerolamo Cardano's Ars magna sive de regulis algebraicis, Nicolo Tartaglia, Scipione del Ferro).