# The father of algebra: Abu Jaafar Mohammad Ibn Mousa Al Khwarizmi

Although Babylonians invented algebra and Greek and Hindu mathematicians preceded the great Frenchman François Viète — who refined the discipline as we know it today — it was Abu Jaafar Mohammad Ibn Mousa Al Khwarizmi (AD780-850) who perfected it.

He used Al Jabr (algebra) in the title of a justifiably renowned study that became a classic textbook in leading universities for centuries: the “Hisab Al Jabr wal-Muqabalah” (The Book of Integration and Equation), introduced the use of Indo-Arabic numerals that, over time, came to be known as algorithms.

Indeed, algorithm is a Latin derivative of Al Khwarizmi’s name, and rather than attributing the collective work of many mathematicians to the scholar, it is safe to grant him grandfatherhood.

This would prevent excessive praise, though his introduction of the zero as a placeholder in equations paved the way for the development of the decimal system that, in this instance, was an exclusive and accurate attribution.

Undoubtedly one of the greatest mathematicians ever, Al Khwarizmi died in Baghdad before his 70th birthday, unaware that his work had changed history.

**Early life and times**

Though little is known of Al Khwarizmi’s life, his last name may suggest that he was from Khwarizm, in today’s Uzbekistan, which was under Persian control when he was born around AD780. Al Tabari refers to the mathematician as “Al Qutrubbulli”, which suggests that he may have been born in Qutrubbull, a district between the Tigris and Euphrates not far from Baghdad. Al Khwarizmi, whose ancestors may have come from Uzbekistan and may have been adherents of the old Zoroastrian religion, settled in Baghdad. Even if he had Persian roots, he became a pious orthodox Sunni, as he described himself in the introduction to his algebra.

During the reign of the sixth Abbasid ruler Al Ma’mun, a son of Harun Al Rashid, Baghdad became a genuine centre of learning. Al Ma’mun transformed the “Bayt Al Hikmah” (House of Wisdom) academy into a premier centre of scientific research and teaching by inviting several scholars, including Al Khwarizmi, to work there. The academy boasted a unique library of manuscripts that rivalled and even surpassed that of Alexandria, one that made available to Muslim scholars everything that was worth having from Byzantium. This delighted Al Khwarizmi, who first translated several Greek philosophical and scientific works, before delving into the study of geometry and astronomy.

Admittedly, his most significant innovations came with “Hisab Al Jabr wal-Muqabalah”, which defined the study of algebra. It must be acknowledged that while his primary interest was to facilitate the lives of men who required specific formulae to settle inheritance, to measure and partition lands, to dig canals, for geometrical computations, etc, his work left a far greater theoretical impact than he could imagine.

Moreover, his tracts on astronomy and geography, many of which were translated into European languages and Chinese, became standard texts. In AD830, a team of 70 geographers working under him produced the first map of the known world at the time, though all that remains of it are the various descriptions (see references below).

Based on Ptolemy’s seminal “Geography”, the book listed latitudes and longitudes, cities, mountains, seas, islands, geographical regions and rivers, and covered the Muslim world and leading Asian positions in addition to European and north African spots.

The result was known as the “Surat Al Ard” (The Image of Earth), which shocked European scientists who read its Latin translations — and Al Khwarizmi made his mark in the West in a rapidly growing field.

**Contributions and achievements**

The “Hisab Al Jabr wal-Muqabalah”, which was translated into Latin by both Gerard of Cremona and Robert of Chester in the 12th century, provided hundreds of simple quadratic equations by analysis as well as by geometrical examples.

Because of his emphasis on solving practical computational problems instead of what algebra evolved into (a highly theoretical discipline), Al Khwarizmi’s discussions of equations were limited to the first and second degrees, which were composed of units, roots and squares.

For example, a unit was presented as a number, while a root was x, and a square was x2. Still, Al Khwarizmi’s mathematics was done entirely in words rather than symbols, which may surprise contemporary scientists. The common equation (linear or quadratic) was reduced in his book to one of six standard forms:

1. Squares equal to roots. Example: ax2 = bx

2. Squares equal to numbers. Example: ax2 = b

3. Roots equal to numbers. Example: ax = b

4. Squares and roots equal to numbers. Examples: ax2 + bx = c or x2 + 10 x = 39.

5. Squares and numbers equal to roots. Examples: ax2 + c = bx or x2 + 21 = 10 x.

6. Roots and numbers equal to squares. Examples: ax2 = bx + c or 3x + 4 = x2.

This reduction process was carried out using the two operations of Al Jabr and Al Muqabalah, where Al Jabr meant completion and Al Muqabalah meant balancing.

The scientist’s innovation was in solving the six standard types of equations by using both algebraic and geometric methods. For example, to solve the equation x2 + 10x = 39 he wrote:

“... 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 concept of algorithm**

Al Khwarizmi wrote an equally influential treatise on Hindu-Arabic numerals, whose Latin translation survived under the title, “Algoritmi de numero Indorum” (Al Khwarizmi on the Hindu Art of Reckoning), which gave rise to the word algorithm.

The work describes the Hindu place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0. The mathematician’s added value came with his clever placement of the zero as a placeholder in positional base notation. Of course, algorithms are now used to do additions and long divisions, though the principles were first devised by Al Khwarizmi who, more than anyone else, was responsible for introducing the Arabic numbers to the West. Naturally, this set in motion a process that led to the use of the nine Arabic numerals, together with the zero sign.

**Insights on astronomy**

Al Khwarizmi wrote an important work on astronomy, covering calendars, calculation of true positions of the Sun, Moon and the planets, tables of sines and tangents, spherical astronomy, astrological tables and parallax and eclipse calculations, and focused on the visibility of the Moon.

His astronomical work, “Zij Al Sind wal Hind” (Astronomical Tables of Sind and Hind), consisted of 37 chapters on calendrical and astronomical calculations and 116 tables with calendrical, astronomical and astrological data, as well as a table of sine values.

Moreover, it contained tables for the movements of the Sun, Moon and the five planets that were known at the time. Parallel to this key addition, Al Khwarizmi also wrote a treatise on the Hebrew calendar, “Risala fi Istikhraj Tarikh Al Yahud” (Extraction of the Jewish Era), which described the 19-year intercalation cycle rules for determining on which day of the week the first day of the month fell.

Finally, it is important to mention that Al Khwarizmi made several important improvements to the theory and construction of astrolabes or sundials, which he inherited from his Indian and Hellenistic predecessors.

He made tables for these instruments that shortened the time needed to make specific calculations. In fact, his sundial was universal and could be used anywhere on the Earth, which was a significant improvement. From then on, sundials were frequently placed in mosques to determine the time of prayer.

The shadow square, an instrument used to determine the linear height of an object, in conjunction with the alidade for angular observations, was also invented by Al Khwarizmi.

**Legacy to Arabs and Muslims**

While a good deal of controversy lingered on his major contributions — as to whether they were the result of original research or based on Hindu and Greek sources — few can deny that beyond his ability to synthesise existing knowledge that the Greeks, Indians and others assembled.

Al Khwarizmi achieved unparalleled heights with his work on algebra. In the context of the times, his original work, which was briefly described above, secured his position among the greatest mathematicians of all times. In fact, it is fair to state that he composed the oldest works on arithmetic and algebra, which served both Eastern as well as Western scientific communities for over five centuries.

Without his use of the zero in the Hindu numbers that were introduced to Europe, the discipline may not have accomplished the kind of progress that gave rise to contemporary mathematics. Deservedly, a crater on the far side of the moon was named after Al Khwarizmi in 1973, which illustrated that he has been held in high esteem by the international scientific community and that his works have stood the test of time.

**List of works**

Al Khwarizmi was a prolific writer on Hindu-Arabic numerals. Among his numerous publications, the most distinguished are:

“Hisab Al Jabr wal-Muqabalah” [The Calculations (or Book) of Integration and Equation], in Frederic Rosen, trs, “Mohammad Ibn Musa Al Khwarizmi: Algebra”, London: The Oriental Translation Fund, 1831. This beautifully translated and edited work is available gratis online at http://ia700506.us.archive.org/19/items/algebraofmohamme00khuwuoft/algebraofmohamme00khuwuoft.pdf

“Kitab Al Jama‘ wal-Tafruq bil Hisab Al Hindi” (The Book of Addition and Subtraction According to the Hindu Calculation). The Latin version, “Algoritmi de numero Indorum” (Al Khwarizmi on the Hindu Art of Reckoning) is the only surviving text.

“Kitab Surat Al Ard” (The Image of the Earth). A single surviving copy of the text (no maps) has been kept at the Strasbourg University Library in France, while a Latin translation is available at the Biblioteca Nacional de España, in Madrid.

*Dr Joseph A. Kéchichian is an author, most recently of, “Legal and Political Reforms in Sa‘udi Arabia”, London: Routledge, 2013. *

*This article is the 16th of a series on Muslim thinkers who greatly influenced Arab societies across the centuries.*

Selected readings

André Allard, “La diffusion en occident des premières oeuvres latines issues de l’arithmétique perdue d’al-Khwarizmi”, Journal of Historical Arabic Sciences 9:1-2, 1991, pp 101-105.

Hughes Barnabas, “Gerard of Cremona’s Translation of al-Khwarizmi’s al-Jabr: A Critical Edition”, Mediaeval Studies 48, 1986, pp 211–263.

Carl B. Boyer, (Revised by Uta C. Merzbach), “The Arabic Hegemony,” in “A History of Mathematics”, 2nd ed, Hoboken, New Jersey: John Wiley & Sons, Inc, 1991, pp 225-245.

John N. Crossley and A.S. Henry, “Thus Spake al-Khwarizmi: A Translation of the Text of Cambridge University Library ms. Ii.vi.5”, Historia Mathematica 17:2, 1990, pp 103-131.

Solomon Gandz, “The Origin of the Term ‘Algebra’”, The American Mathematical Monthly 33:9, November 1926, pp 437–440.

Jonathan Lyons, “The House of Wisdom: How the Arabs Transformed Western Civilization”, New York: Bloomsbury Press, 2010.

K.H. Parshall, “The Art of Algebra from al-Khwarizmi to Viète: A Study in the Natural Selection of Ideas”, History of Sciences 26, 1988, pp 129-164.

Roshdi Rashed, “The Development of Arabic Mathematics: Between Arithmetic and Algebra”, London: Springer, 1994.

Roshdi Rashed, “Al-Khawrizmi’s Concept of Algebra”, in George N. Atiyeh and Ibrahim M. Oweiss, eds, “Arab Civilization: Challenges and Responses — Studies in Honor of Constantine K. Zurayk”, Albany, NY: State University of New York Press, 1988, pp 98-110.

B.A. Rosenfeld, “‘Geometric trigonometry’ in treatises of al-Khwarizmi, al-Mahani and ibn al-Haytham”, in “Vestigia Mathematica: Studies in Medieval and Early Modern Mathematics in Honour of H.L.L. Busard”, edited by M. Folkerts and J. P. Hogendijk, Amsterdam, The Netherlands: Rodopi, 1993, pp 305-308.