The Science of Restoring and Balancing – The Science of Algebra
|Figure 1. Some geometrical shapes from Suhayl al-Quhî's book "Fî istihraci mesaha al-muhassama al-maqafî or Risala-i abu Sahl". Suleymaniya Library, Ayasofya 4832.|
There are many occasions where we are confronted by problems in which we are concerned with determining the value of an unknown quantity. Often, these problems are problems of geometry: for instance, given a line segment AB, we wish to divide it into segments AG and GB, such that the rectangle whose sides are AB and GB is equal to the square whose side is AG. The unknown item here is the segment AG. Such puzzles were very popular amongst the Greeks, many of them stated and solved in Euclid's celebrated work, the Elements. Problems involving unknowns are not limited to geometry, however; the famous mathematician of antiquity, Diophantus solved the following purely numerical problem in his Arithmetica: To find three numbers so that the product of any two added to the third gives a square. Nowadays, from an early age, students are taught to treat problems like the above in a unified way using the tools and techniques of algebra. This unified approach was developed and placed on firm foundations by Muslim scholars, one of the earliest and well-known being Mohammed bin Musa Al-Khwarizmi. It is due to Al-Khwarizmi that we have acquired the name algebra, transformed from the Arabic word al-Jabr appearing in the title of his most famous treatise, Kitab al-Jabr Wa l-Muqabala, literally meaning, "The book of restoring and balancing".
Confident in their own values and traditions, Muslim mathematicians benefited from their encounters with great civilisations, often integrating their ideas and techniques within a broader, more general framework. This was certainly the case with algebra. On the one hand, Muslim scholars were thoroughly versed with the work of the Greeks in geometry, having translated and produced critical commentary on crucial works such as Euclid's Elements and Archimedes' Sphere and Cylinder. The numerical and arithmetic work of the Babylonians also came under the scrutiny of the curious Muslim intellect. Of special interest to Muslim scholars were the investigations carried out by Hindu mathematicians as early as the late fifth century AD. For instance, Brahmagupta in the first half of the seventh century AD is interested, like the Babylonians, in what we today know as quadratic equations, and gave numerical procedures for obtaining their solutions. Recognising the effectiveness of numerical methods of the Hindus and Babylonians and the certainty provided by the axiomatic approach based on proof from the Greeks, the Muslims drew together these two strands to produce the new science of Algebra.
|Figure 2. An artistic impression of Nasir al-Tusi on a Iranian stamp.|
Al-Khwarizmi's main concern was with quadratic equations possessing positive roots, which he noted can be encountered in one of three standard forms. These equations involve three kinds of quantities: simple numbers, the root (which is the unknown, x) and wealth, known as Mal in Arabic and is the square of the root. The labels indicate the real world motivation that often drove such enquiries within Muslim civilisation. Al-Khwarizmi then proceeded to describe in detail the numerical procedures that solve particular examples of equations drawn from one of the three standard types. The formula that is recorded is nothing more than a verbal description of the standard quadratic formula that we learnt at school. The distinguishing feature of Al-Khwarizmi's work, and indeed of his successors, is the proof that is provided for the validity of the numerical procedure using the axioms and theorems of geometry. Thabit bin Al-Qurra extended Al-Khwarizmi's contributions by demonstrating the validity of the formula for the unknown of general classes of quadratic equations. He undertook this by first stating basic theorems of geometry from Euclid; the various entities in the equations, including the unknowns are related to the corresponding geometric quantities, namely line segments and areas; finally using this geometric interpretation for the terms of the equation, Al-Qurra was able to show the correspondence between the geometric and algebraic solutions.
The full "arithmetisation of algebra" and extension of the study of equations to include higher order unknowns, was ushered in by Al-Karaji, who conducted his work in Baghdad around AD 1000. It was Al-Karaji's view that unknowns need not be limited to roots and their squares, whether geometric magnitudes or absolute numbers. More generally, unknowns could appear as cubes, x3, mal mal, x4, mal cube, x5, and so on. Thus was he able to manipulate polynomial expressions, such as x4 + 4 x3 – 6, employing rules based on the ordinary arithmetic rules for adding, subtracting, multiplying, dividing and extracting square roots. However, Al-Karaji did not quite complete the arithmetisation of algebra; the matter had to wait 70 years for another brilliant scholar, al-Samaw'al bin Yahya bin Yahuda al-Maghribi, to add the finishing touches. The remaining step rested on fully incorporating negative numbers into the theory. Although al-Karaji had managed to discover rules such as a – (– b) = a + b, he hadn't quite encountered the related identity, – a – (– b) = – (a + b). Such identities involving negative entities are not as trivial as they seem, particularly when they must be developed or discovered for the first time. As Berggren consoles :
"Students who have struggled with the law of signs may find comfort in learning that at one time the discovery of these rules taxed the ingenuity of the best mathematicians, and that the discovery of much of our elementary (pre-calculus) mathematics was a matter of considerable labor and many false starts".
|Figure 3. The miniature of Ala al-Din al-Aswad from Tarjama-i Shakaik al-Numaniya, TSMK, H 1263.|
A contemporary scholar, Ruth McNeill, reminisces on how such rules led her to abandon mathematics:
"What did me in was the idea that a negative number times a negative number comes out to a positive number. This seemed (and still seems) inherently unlikely – counterintuitive, as mathematicians say. I wrestled with the idea for what I imagine to be several weeks, trying to get a sensible explanation from my teacher, my classmates, my parents, anybody."
This, then, makes al-Samaw'al's statement of the missing relation all the more remarkable. The statement appears in his exotically entitled work, Al-Bahir fi'l – Hasib (The Shining Book on Calculation), which he wrote when he was only nineteen: " … if we subtract a deficient number from a deficient number larger than it, there remains the difference [eg – 5 – (– 2) = – (5 – 2)], deficient; but in the other case there remains their difference, excess. [eg – 2 – (– 5) = + (5 – 2)]."
Al-Samaw'al's personal life makes for interesting reading. He was actually born into a Jewish family and was forced on his own to complete the study of the remaining volumes of Euclid's Elements. This was on account of not finding a sufficiently competent teacher of Mathematics in Baghdad at the time. He proceeded to study, again by himself, the work of Al-Karaji, which he then elaborated and extended. His conversion to Islam, according to his autobiography, was inspired by a dream he had in 1163. He spent his life traveling as a medical doctor, treating Princes on occasion, and died in Maragha, northern Iran, around 1180. In total, Al-Samaw'al's encyclopedic achievements spanning mathematics, astronomy, medicine and theology, fill eighty-five works, only a few of which have survived. Along with the rules relating to manipulating negative numbers described above, the law of exponents and division of polynomials are all considered in one of Al-Samawa'al's surviving mathematical studies, The Shining. What we would express today in modern notation as x-3 x-4 = x-7, Al-Samaw'al records in the language of his time as in this excerpt:
"Opposite [above] the order of part of cube is 3 and opposite part of mal mal is 4. We add them to obtain 7 and opposite it is the order of part of mal mal cube."
Such excursions in the world of exponents assisted Al-Samaw'al as he applied his sharp mind to the problem of dividing one polynomial by another. The details of the procedure need not concern us here; it suffices to reproduce Berggren's summary:
"… the discovery of this procedure of long division, which is in all its computation precisely our present-day one, is a fine contribution to the history of mathematics, and it seems to be a joint accomplishment of al-Karaji and al-Samaw'al".
Umar al-Khayyami, born in Nishapur around the year 1048 is renowned and admired in popular circles more for his poetry, especially the Rub‘ayat, than for his extensive and extraordinary accomplishments in mathematics. Before we examine his contributions to algebra, it is worth noting that his insights into the ratios of magnitudes (for instance, resulting in or ) "amounted to the introduction of positive real numbers," as noted by Berggren, which was communicated to European mathematicians via Nasir al-Din al-Tusi. Umar al-Khayyami was especially keen to classify and solve cubic equations. He notes in his introduction to Algebra that he intends to pursue an algebraic treatment of problems hitherto not given the same kind of attention, until modern writers such as Abu ‘Abdullah al-Mahani. The kind of problems that were of interest to the Muslims is exemplified by Archimedes' problem of cutting a sphere by a plane so that the volumes of the two sections of the sphere are related to one another by a given ratio. The problem leads to equations of the form x3 + m = nx2, a particular class of cubic. Al-Khayyam remarks that neither Al-Mahani nor Thabit bin al-Qurra could solve the equation; it would be conquered by a mathematician of the next generation, Abu Ja‘far Al-Khazin, who solved it employing intersecting conic sections. Following Abu Ja‘far, various attempts were made to solve special cases of cubic equations; Al-Khayyam's aim in Algebra was to enumerate all possible equations of the above type and then to solve them all. Although al-Khayyami used geometric arguments, he viewed his work as a contribution to algebra, beginning the first chapter as follows:
"Algebra. By the help of God and with His precious assistance, I say that algebra is a scientific art. The objects with which it deals are absolute numbers and (geometrical) magnitudes which, though themselves unknown, are related to things which are known."
** Dr Mahbub Ghani is Lecturer at King's College, University London.
by: Mahbub Ghani, Fri 05 January, 2007