Talk:The Compendious Book on Calculation by Completion and Balancing
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[edit] al-Jabr
The explanation of the al-Jabr operation cannot be correct:
The al-Jabr operation is subtracting a quantity from one side of the equation and adding it to another.
Applying this to:
a = b + c
gives:
a + b = c
Which is clearly inconsistent. Someone with more expertise on this book needs to substitute the correct definition of what the al-Jabr operation was. Furby100 22:07, 2 February 2006 (UTC)
- Fixed some time ago. Mukadderat 23:53, 26 March 2006 (UTC)
[edit] Unsolvable cases
Does anyone know whether al-Khwarismi had any consideration of "unsolvable" cases like x2 + 4 = 0, not covered by the 6 solvable cases? Mukadderat 23:53, 26 March 2006 (UTC)
al-Khwarizmi does not cover them, but such equations occasionally appear in later algebraists, such as Ibn al-Ha'im and al-`Amili. ---Jeff Oaks
[edit] Title, al-jabr, al-muqabala
Before I give some brief comments on this article, I want to say that the errors are not those of the author, but of the secondary sources he/she used.
This article has many errors. The first is that "al-mukhtasar" in the title of al-Khwarizmi's book does not mean "compendious", but is better translated as "condensed", "brief", or "abridged". Also, "al-jabr wa'l-muqabala" was the medieval name for the art of algebra, so the title is better rendered "The condensed book on algebra".
The book does not provide algebraic and geometric solutions to the 6 types of equation. It gives a numerical recipe for the solution, and then offers geometric proofs that the method is valid.
The three types of algebraic number were mal (literally "sum of money", "treasure", etc., for our x^2), jidhr (lit. "root" for our x), and `adad mufrad ("simple number"). In the worked out problems shay' ("thing") was usually used in place of jidhr, and simple numbers were counted in dirhams, a unit of currency. Later algebraists substituted `adad ("number") or ahad ("unit") for dirham.
Regarding the operations al-jabr (restoration) and al-muqabala (confrontation), see my article "Simplifying equations in Arabic algebra" (with Haitham Alkhateeb) in an upcoming issue of Historia Mathematica. The article is already available online:
http://authors.elsevier.com/JournalDetail.html?PubID=622841&Precis=DESC (click on "tables of contents and abstracts")
Medieval mathematicians had no concept of negative numbers. In an expression like "ten less a thing" (10 - x) both the 10 and the x are positive. Do not confuse the simple idea of subtracting positive numbers with the existence of negative numbers. Further, for them "ten less thing" (10 - x) was thought of as a deficient ten, which has had x removed from it. The operation has already been carried out. In an equation like 10 - x = x^2 (ten less a thing equals a mal) one is instructed to "restore" (ajbir, conjugated from al-jabr) the deficient 10 to make it a full 10. Then an x needs to be added to the other side of the equation. This is a two-step process.
Al-muqabala means "confrontation". In an equation like 2x + 10 = 5x the 2x and the 5x are "confronted", which entails bringing them face-to-face and taking their difference.
For a good overview of Arabic algebra, see Ahmed Djebbar's 2005 book _L'algèbre arabe : Genèse d'un art_.
---Jeff Oaks
