Abu Ja'far Muhammad ibn Musa Al-Khwarizmi

Abu Ja'far Muhammad ibn Musa Al-Khwarizmi

 

We know few details of Abu Ja'far Muhammad ibn Musa al-Khwarizmi's life. One unfortunate effect of this lack of knowledge seems to be the temptation to make guesses based on very little evidence. In [1] Toomer suggests that the name al-Khwarizmi may indicate that he came from Khwarizm south of the Aral Sea in central Asia. He then writes:-

But the historian al-Tabari gives him the additional epithet "al-Qutrubbulli", indicating that he came from Qutrubbull, a district between the Tigris and Euphrates not far from Baghdad, so perhaps his ancestors, rather than he himself, came from Khwarizm ... Another epithet given to him by al-Tabari, "al-Majusi", would seem to indicate that he was an adherent of the old Zoroastrian religion. ... the pious preface to al-Khwarizmi's "Algebra" shows that he was an orthodox Muslim, so Al-Tabari's epithet could mean no more than that his forebears, and perhaps he in his youth, had been Zoroastrians.

However, Rashed [7], put a rather different interpretation on the same words by Al-Tabari:-

... Al-Tabari's words should read: "Muhammad ibn Musa al-Khwarizmi and al-Majusi al-Qutrubbulli ...", (and that there are two people al-Khwarizmi and al-Majusi al-Qutrubbulli): the letter "wa" was omitted in the early copy. This would not be worth mentioning if a series of conclusions about al-Khwarizmi's personality, occasionally even the origins of his knowledge, had not been drawn. In his article ([1]) G J Toomer, with naive confidence, constructed an entire fantasy on the error which cannot be denied the merit of making amusing reading.

This is not the last disagreement that we shall meet in describing the life and work of al-Khwarizmi. However before we look at the few facts about his life that are known for certain, we should take a moment to set the scene for the cultural and scientific background in which al-Khwarizmi worked.

Harun al-Rashid became the fifth Caliph of the Abbasid dynasty on 14 September 786, about the time that al-Khwarizmi was born. Harun ruled, from his court in the capital city of Baghdad, over the Islam empire which stretched from the Mediterranean to India. He brought culture to his court and tried to establish the intellectual disciplines which at that time were not flourishing in the Arabic world. He had two sons, the eldest was al-Amin while the younger was al-Mamun. Harun died in 809 and there was an armed conflict between the brothers.

Al-Mamun won the armed struggle and al-Amin was defeated and killed in 813. Following this, al-Mamun became Caliph and ruled the empire from Baghdad. He continued the patronage of learning started by his father and founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated. He also built up a library of manuscripts, the first major library to be set up since that at Alexandria, collecting important works from Byzantium. In addition to the House of Wisdom, al-Mamun set up observatories in which Muslim astronomers could build on the knowledge acquired by earlier peoples.

Al-Khwarizmi and his colleagues the Banu Musa were scholars at the House of Wisdom in Baghdad. Their tasks there involved the translation of Greek scientific manuscripts and they also studied, and wrote on, algebra, geometry and astronomy. Certainly al-Khwarizmi worked under the patronage of Al-Mamun and he dedicated two of his texts to the Caliph. These were his treatise on algebra and his treatise on astronomy. The algebra treatise Hisab al-jabr w'al-muqabala was the most famous and important of all of al-Khwarizmi's works. It is the title of this text that gives us the word "algebra" and, in a sense that we shall investigate more fully below, it is the first book to be written on algebra.

Rosen's translation of al-Khwarizmi's own words describing the purpose of the book tells us that al-Khwarizmi intended to teach [11] (see also [1]):-

... what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.

Now this does not sound like the contents of an algebra text and indeed only the first part of the book is a discussion of what we would today recognise as algebra. However it is important to realise that the book was intended to be highly practical and that algebra was introduced to solve real life problems that were part of everyday life in the Islam empire at that time. Early in the book al-Khwarizmi describes the natural numbers in terms that are almost funny to us who are so familiar with the system, but it is important to understand the new depth of abstraction and understanding here [11]:-

When I consider what people generally want in calculating, I found that it always is a number. I also observed that every number is composed of units, and that any number may be divided into units. Moreover, I found that every number which may be expressed from one to ten, surpasses the preceding by one unit: afterwards the ten is doubled or tripled just as before the units were: thus arise twenty, thirty, etc. until a hundred: then the hundred is doubled and tripled in the same manner as the units and the tens, up to a thousand; ... so forth to the utmost limit of numeration.

Having introduced the natural numbers, al-Khwarizmi introduces the main topic of this first section of his book, namely the solution of equations. His equations are linear or quadratic and are composed of units, roots and squares. For example, to al-Khwarizmi a unit was a number, a root was x, and a square was x². However, although we shall use the now familiar algebraic notation in this article to help the reader understand the notions, Al-Khwarizmi's mathematics is done entirely in words with no symbols being used.

He first reduces an equation (linear or quadratic) to one of six standard forms:

1. Squares equal to roots.
2. Squares equal to numbers.
3. Roots equal to numbers.
4. Squares and roots equal to numbers; e.g. x²+ 10 x = 39.
5. Squares and numbers equal to roots; e.g. x²+ 21 = 10 x.
6. Roots and numbers equal to squares; e.g. 3 x + 4 = x².

The reduction is carried out using the two operations of al-jabr and al-muqabala. Here "al-jabr" means "completion" and is the process of removing negative terms from an equation. For example, using one of al-Khwarizmi's own examples, "al-jabr" transforms x²= 40 x - 4 x² into 5 x²= 40 x. The term "al-muqabala" means "balancing" and is the process of reducing positive terms of the same power when they occur on both sides of an equation. For example, two applications of "al-muqabala" reduces 50 + 3 x + x²= 29 + 10 x to 21 + x²= 7 x (one application to deal with the numbers and a second to deal with the roots).

Al-Khwarizmi then shows how to solve the six standard types of equations. He uses both algebraic methods of solution and geometric methods. For example to solve the equation x²+ 10 x = 39 he writes [11]:-

... a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square.

The geometric proof by completing the square follows. Al-Khwarizmi starts with a square of side x, which therefore represents x² (Figure 1). To the square we must add 10x and this is done by adding four rectangles each of breadth 10/4 and length x to the square (Figure 2). Figure 2 has area x²+ 10 x which is equal to 39. We now complete the square by adding the four little squares each of area 5/2 5/2 = 25/4. Hence the outside square in Fig 3 has area 4 25/4 + 39 = 25 + 39 = 64. The side of the square is therefore 8. But the side is of length 5/2 + x + 5/2 so x + 5 = 8, giving x = 3.

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The above is an extract from an article that can be read in Full with references at:

 http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khwarizmi.html