# Algebra

## Historical Development

### Abū ‘Abdullah Muhammad Ibn Mūsā Al-Khwārizmī

Al-Khwārizmī (ca. 800-847 CE) is called the father of algebra. The epithet "al-Khwārizmī" refers to his place of origin, Khwārizm or Khorezm, which is located south of the delta of the Amu Dar'ya River and the Aral Sea in central Asia. However, the historian al-Tabari adds the epithet "al-Qutrubbulli," indicating that al-Khwārizmī actually came from Qutrubull, near Baghdad between the Tigris and the Euphrates Rivers (Mohindi 50). Other sources state that his "stock" comes from Khwārizm so perhaps al-Khwārizmī's ancestors, rather than himself, come from Khwārizm (van der Waerden 3). Another interesting epithet added by al-Tabari is "al-Majusi," which would mean that al-Khwārizmī was an adherent of the Zoroastrian religion. However, Al-Khwārizmī's preface to his treatise on algebra shows beyond doubt that he was a devout Muslim; perhaps some of his ancestors or even al-Khwārizmī in his youth were Zoroastrian (Mohini 52).

Al-Khwārizmī grew up near Baghdad under the reign of Caliph al-Ma'mun (reign 813-833 CE), who was a great promoter of science. Al-Khwārizmī was offered a position at the Bayt al-Hikma (House of Wisdom) in Baghdad; most of his treatises are dedicated to the Caliph al-Ma'mun (Mohini 53). Most of al-Khwārizmī's treatises are in the field of astronomy. He was one of the developers of the astrolabe and also wrote about a hundred astronomical tables. One of these, *Zij al-sindhind*, is the first Arab astronomical work to survive in its entirety (Mohini 55-6). Al-Khwārizmī also produced work on the Jewish calendar, accurately describing the 19-year cycle, its 7 months, and the rules for determining which day of hte week the month of Tishri begins on (Mohini 58). His treatises on arithmetic, which survives only in its Latin translation possibly done by Adelard of Bath in the 12th century, introduced the decimal system and the Hindu-Arabic numerals 1-9 and 0 (van der Waerden 9). Al-Khwārizmī is probably responsible for the popularization of these numerals and especially of the important use of the number zero. "0" was actually used for about 250 years in the Islamic world after its introduction by al-Khwārizmī before the Western world ever knew of it (Mohini 61-2).

### Al-Khwārizmī's Mathematical Contribution: Algebra

In the historical development of algebra, the topic of most importance will be al-Khwārizmī's treatise *Kitab al-jabr wa'l-muqabalah*, or *The Book of Restoring and Balancing* (Boyer 256). The meanings of the words *al-jabr* and *al-muqabalah* are debated. *Al-jabr*, which comes to us in its form "algebra," probably meant something like "restoration" or "completion," referring to the transposition of subtracted terms to the other side of the equation or adding equal terms to both sides of the equation to eliminate negative terms (Boyer 257). *Al-muqabalah* probably means something like "restoration" or "balancing," referring to the cancellation of like terms on opposite sides of the equation, or reduction of positive terms by subtracting equal amounts from both sides of the equation(Boyer 257). Together, the two words *al-jabr wa'l-muqabalah* can mean the science of algebra. Al-Khwārizmī's treatise was the first book to use this title to designate algebra as a separate discipline.

### Algebraic Equations

*Kitab al-jabr wa'l-muqabalah* has three sections, the first of which states that all linear and quadratic equations can be reduced to one of six types: (van der Waerden 5).

ax^{2} = bx

ax^{2} = b

ax = b

ax^{2} + bx = c

ax^{2} + c = bx

ax^{2} = bx + c

He presents general solutions for all of these types. Looking at these six equations, it is apparent that al-Khwārizmī did not accept negative or zero coefficients.

Al-Khwārizmī's treatment of mixed quadratic equations ("roots and numbers equal to squares," "squares and numbers equal to roots," and "roots and numbers equal to squares") is best seen with an example of the first type of mixed quadratic equations.

### Completing the Square

In modern notation, one of al-Khwārizmī's example equations is x^{2} + 10x = 39. Al-Khwārizmī's solution is then:

(x+5)^{2} = 39 + 25 = 64

x + 5 = sqrt{64} = 8

x = 8 - 5 = 3

x^{2} = 9

Al-Khwārizmī demonstrates this solution with a square AB, the side of which is the desired root x. On each of the four sides, he constructs rectangles, each having 2.5 as their width. So, the square together with the four rectangles is equal to 39. To complete the square EH, al-Khwārizmī adds four times the square of 2.5, or 25. So the area of the large square EH is 64, and its side is 8. Thus, the side x of the original square AB is 8 - 5 = 3 (van der Waerden 8). *(See the figure below.)*

Al-Khwārizmī also presents a simpler, similar method which constructs rectangles of breadth 5 on two sides of the square AB. Then, the total area of the square EH is x^^{2} + 10x + 25 = 39 + 25 = 64, which yields the same result x = 3 or x^{2} = 9 (van der Waerden 8). *(See the figure below.)*

### Further Algebraic Contributions

Al-Khwārizmī also discusses methods of extracting the square root; this method may have been adapted from Hindu sources. Al-Khwārizmī's second chapter of *Algebra* is concerned with mensuration, which outlines rules for computing areas and volumes. The last chapter of *Algebra* is the largest, and it is concerned mainly with legacies. It consists entirely of problems and solutions involving simple arithmetic and linear equations. These problems are not going to be discussed here as they use the same algebra already mentioned and require an extensive knowledge of Islamic inheritance laws (van der Waerden 7).

*Please see the " Paper" section for further information on the mathematical contributions of Al-Khwārizmī.*

## Muslim Mathematicians

### Thābit ibn Qurra al-Harrānī

Thābit ibn Qurra (836-901 CE) followed al-Khwārizmī's general solutions; however, al-Khwārizmī presents his general proofs in conjunction with particular equations, whereas ibn Qurra presents his demonstrations in general. At this point, ibn Qurra had full access to Euclid's *Elements*, and freely used Euclid's theorems in his algebraic proofs. Ibn Qurra also correctly solved the quadratic equation x^{2} + px = q (Berggren). He follows his demonstrations with general proofs, following Euclid's examples of the definition-theorem-proof model.

### Abū Kāmil Sjujū‘ ibn Aslam ibn Muhammad ibn Shujā

Abū Kāmil (c. 850-930 CE) wrote a treatise titled *Algebra*, which was a commentary on al-Khwārizmī's work. His examples were later used by both the Muslim scholar al-Kharajī in the late 10th century and the Italian Leonardo of Pisa, or Fibonacci, in the late 12th century. Many of his examples are taken from al-Khwārizmī, and like al-Khwārizmī's work, the entire work is written out, including numbers. Abū Kāmil also discusses the geometrical proofs of equation solutions in terms of specific examples, like al-Khwārizmī, rather than using general proofs like ibn Qurra. Abū Kāmil does go beyond the algebra of either ibn Qurra and al-Khwārizmī by providing rules for manipulating the algebraic equation with both positive and negative coefficients. Abū Kāmil gives both algebraic and geometrical proofs for these equations (Berggren).

### Abū Bakr ibn Muhammad ibn al-Husayn al-Kharajī

Al-Kharajī (953 - 1029 CE) tends to apply arithmetic to algebra, in contrast to Abū Kāmil and ibn Qurra, both of whom apply geometry to algebra. Al-Kharajī wrote *The Marvellous*, in which he develops the algebra of expressions using high powers of the unknown. He uses "root," "side," or "thing," to denote x, "mal" for x^{2}, "cube" for x^{3}, "mal mal" for x^{4}, "mal cube" for x^{5}, and so on. He creates each power of the unknown by multiplication by the previous elements; this was in innovation which allowed al-Kharajī to treating equations such as x^{4} + 4x^{3} - 6 and 5x^{6} - (2x^{2} + 3).

### Ibn Yahyā al-Maghribī al-Samaw‘al

Al-Samaw‘al (1130-1180 CE) was born in Baghdad. Though born to a Jewish family, he converted to Islam in 1163 after he had a dream telling him to do so. He was a popular medical doctor, and traveled around modern-day Iran to care for his patients, which included princes. His *The Shining Book on Calculation* gives rules for signs, creating the concepts of positive (excess) and negative (deficiency) numbers. He then gives rules for subtracting powers, and rules for multiplying and dividing simple fractions. Al-Samaw'al also gives examples of the division of complex polynomials, which was a great development in algebra. His first example shows how to solve: 20x^{6} + 2x^{5} + 58x^{4} + 75x^{3} + 125x^{2} + 196x + 94 + 40x^{-1} + 50x^{-2} + 90x^{-3} + 20x^{-4} divided by 2x^{3} + 5x + 5 + 10x^{-1}. His discovery of the procedure for long division was a significant achievement in Islamic algebra.

*Please see the " Paper" section for further information on algebraic contributions of Muslim mathematicians.*