Al-Khwarizmi: The Father of Algebra

Pioneers of Islamic History
By Dr. Ibrahim B. Syed

Many of his books were translated into Latin and used as a principle mathematical text book in European universities until the 16th. century

Abu Ja’far Muhammad ibn Musa al-Khwarizmi, (780 – 850 CE), was the grandfather of computer science and the father of Algebra.

He was the popularizer of Arabic numerals, adopter of zero (the symbol) and the decimal system, astronomer, cartographer, in brief an encyclopedic scholar.

Bayt Al-Hikmah (House of Wisdom)

In the year 832, the Abbasid Caliph Al Ma’mun (b. Baghdad, 786, d. Tarsus, Cilicia, August 833) founded the “House of Wisdom” in Baghdad, a center for study and research similar to the earlier Museum in Alexandria. Its most famous scholars were the mathematicians Muhammad ibn Musa Al-Khwarizmi and the Banu Musa (“sons of Moses”), three brothers who directed the translation of Greek works from Antiquity.

The modern word algorithm is derived from the name, al-Khwarizmi, the best mathematician of his age, thanks to his book, “al-Kitab al-mukhtasar fi Hisab al-jabr w’al-muqabala”, (a book showing how to solve equations and problems derived from ordinary life) which means “The Compendious Book on Calculation by Completion and Balancing”, which later evolved into algebra, was the first written text on the subject.

In al-Khwarizmi’s time, algebra was a practical system for solving all kinds of problems “in cases of inheritance, contracts, surveying, tax collection, 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.” Al-jabr was about removing the negative terms from an equation, while al-muqabala meant “balancing” the values of an equation across an equal sign.

It is the title of this text that gives us the word “algebra”. It is the first book to be written on algebra. In al-Khwarizmi’s own words, the purpose of the book was to teach what was easiest and most useful in arithmetic, such as what was constantly required 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 were concerned.

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 recognize as algebra. However it is important to realize 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 Islamic empire at that time.

After introducing the natural numbers, al Khwarizmi discusses the solution of equations. Al Khwarizmi's equations are linear or quadratic and are composed of units (numbers), roots (x) and squares (x2). He first reduces an equation to one of 6 standard forms, using the operations of addition and subtraction, and then shows how to solve these standard types of equations. He uses both algebraic methods of solution and the geometric method of completing the square.

a method to find square roots is known to have been in the Arabic original

The next part of al-Khwarizmi’s Algebra consists of applications and worked examples. He then goes on to look at rules for finding the area of figures such as the circle, and also finding the volume of solids such as the sphere, cone, and pyramid.

The text book of Algebra was intended to be highly practical and it was introduced to solve real life problems that were part of everyday life in the Islamic world at that time. Early in the book al-Khwarizmi wrote:

“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.”

Al-Khwarizmi also wrote a treatise on Hindu-Arabic numerals. The Arabic text is lost but a Latin translation, “Algoritmi de numero Indorum” in English “Al-Khwarizmi on the Hindu Art of Reckoning” gave rise to the word algorithm deriving from his name in the title as mentioned earlier.

Unfortunately the Latin translation (translated into English) is known to be much changed from al-Khwarizmi’s original text (of which even the title is unknown).

The work describes the Hindu place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. The first use of zero as a place holder in positional base notation was probably due to al-Khwarizmi in this work. Methods for arithmetical calculation are given, and a method to find square roots is known to have been in the Arabic original although it is missing from the Latin version.“… The decimal place-value system was a fairly recent arrival from India and … al-Khwarizmi’s work was the first to expound it systematically. Thus, although elementary, it was of seminal importance.”

Khwarizmi developed detailed trigonometric tables containing the sine functions which later included tangent functions.

Khwarizmi’s book on arithmetic was translated into Latin and published in Rome in 1857 by Prince Baldassare Boncompagni and appears as part 1 of a volume entitled “Tratti d’ aritmetica”. The book is titled as Algorithmi de numero indorum which means “Khwarizmi concerning the Hindu art of reckoning.”

Many of his books were translated into Latin and used as a principle mathematical text book in European universities until the 16th. century. Among them these two books had important place: “Kitab al-Jama wal-Tafreeq bil Hisab al-Hindi” and “Kitab al-Jabr wa al-muqabala.”

Khwarizmi’s contribution and influence are tremendous. Two important books on arithmetic, Carmen de Algorismo and Algorismus vulgaris which were written in 12th. and 13th. century respectively owe a lot to the Khwarizmi’s book and were used for several hundred years in Europe. Abu Kamil Shuja, an Islamic mathematician, whose work on mathematics was based on Khwarizmi’s works kept the influence of Khwarizmi on Leonardo of Pisa, a 13th. century scholar and up to Middle Ages and during the Renaissance.


Al-Khwarizmi wrote a major work on geography which gives latitudes and longitudes for 2,402 cities

Al Khwarizmi also wrote an important work on astronomy, covering calendars, calculating true positions of the sun, moon and planets, tables of sinus and tangents, spherical astronomy, astrological tables, parallax and eclipse calculations, and visibility of the moon. Although his astronomical work is based on that of the Indians, and most of the values from which he constructed his tables came from Hindu astronomers, al-Khwarizmi must have been influenced by Ptolemy’s work too.

Al-Khwarizmi performed detailed calculations of the positions of the sun, moon, and planets, and did a number of eclipse calculations. In addition to an important treatise on astronomy, Al-Khwarizmi wrote a book on astronomical tables, which were also translated into European languages and, later, into Chinese.


In geography, Al Khwarizmi wrote the book called Kitab Surat al- ard (Book of the Form of the Earth). His works differed from Ptolemy’s and he corrected Ptolemy’s views in detail. It is a description of a world (known world at that time) map and contains a list of the coordinates of the important places on it. He corrected the distortion that Ptolemy’s map had with regard to the length of the Mediterranean. It was much more accurate. However, it failed to replace the Ptolemaic geography used in Europe. He wrote many other books on topics such as clocks, sundials and astrolabes.

Al-Khwarizmi wrote a major work on geography which gives latitudes and longitudes for 2,402 cities and landmarks, forming the basis for a world map. The book, which is based on Ptolemy’s Geography, lists with latitudes and longitudes, cities, mountains, seas, islands, geographical regions, and rivers. The manuscript includes maps which on the whole are more accurate than those of Ptolemy.

A number of minor works were written by al-Khwarizmi on topics such as the astrolabe, on which he wrote two works, on the sundial, and on the Jewish calendar. He also wrote a political history containing horoscopes of prominent persons.

Al-Khwarizmi systematized and corrected Ptolemy’s research in geography and astronomy/astrology, using his own original findings. He supervised the work of 70 geographers to create a map of the then “known world”. He is also reported to have collaborated in the degree measurements ordered by Caliph Ma'mun al-Rashid. These were aimed at measuring the volume and circumference of the earth.

Al Khwarizmi’s Impact on Europe

In 1140 Robert of Chester (who read mathematics in Spain) translated Khwarizmi’s book into Latin as Liber algebrae et almucabala, then ultimately gave its name to the discipline of algebra. The Spanish Jew, John of Seville, produced another Latin version.

When Khwarizmi’s work became known in Europe through Latin translations, his influence made an indelible mark on the development of science in the West. His Algebra book introduced that discipline to Europe “unknown till then” and became the standard mathematical text at European universities until the 16th. century. In the 16th. century it is found in English as algiebar and almachabel and in various other forms but was finally shortened to algebra.

Al Khwarizmi is one of the Muslim scholars who laid the foundations for Europe’s Renaissance and the Scientific Revolution.

To the Muslims, Europe was backward, unorganized, carried no strategic importance

Several of Al-Khwarizmi’s books were translated into Latin in the early 12th. century by Adelard of Bath and Gerard of Cremona. The treatises on Arithmetic, Kitab al-Jam’a wal-Tafreeq bil Hisab al-Hindi, and the one on Algebra, Al-Maqala fi Hisab-al Jabr wa-al-Muqabilah, are known only from Latin translations. Introduction of Arabic numerals provided a pivotal advance over the cumbersome Roman numerals.

This development of a more convenient number system assisted progress in science, accounting and bookkeeping. Key to this was the use of the number zero, a concept unknown to the West. The use of this number system (Arabic numerals) spread throughout the Muslim world over the next two centuries, assisting the development of science.

The Arabic numeral system was first mentioned in Europe around 1200 CE, but Christian adherence to the Roman system hindered its use and introduction. It was only fully accepted in Europe after it was adopted by the Italian traders in the Renaissance of the 16th. century, who followed the practice of their Arab trading partners.

Muslim Impact on Europe

During the Middle Ages, the Islamic World had a very significant impact upon Europe, which in turn cleared the way for the Renaissance and the Scientific Revolution. One of the most important of these influences was the study of science.

Ever since Islam was born, Muslims had made immense leaps forward in the area of science. Cities like Baghdad, Damascus, Cairo and Cordoba were the centers of civilization. These cities were flourishing and Muslim scientists made tremendous progress in applied as well as theoretical Science and Technology.

In Europe, however, the situation was much different. Europe was in the Dark Ages. It had no infrastructure or central government. To the Muslims, Europe was backward, unorganized, carried no strategic importance and was essentially irrelevant.

This considering the time period was in fact true. Nevertheless, the Catholic Church, which at the time was the strongest institution in Europe, successfully convinced Christian Europe that the Muslims were infidels. This caused Europeans to think that Muslims were culturally inferior to Europe and thus Europe was unable to benefit from the new scientific discoveries being made in the Islamic lands before the 1100’s.

By doing this, Europe kept itself in the Dark Ages while from China to Spain Islamic Civilization prospered. During the Crusades there was limited contact between Muslims and Christians and not much was transferred. As Adrian Lewis explains: “The Crusaders were men of action, not men of learning”. The real exchange of ideas which lead to the scientific revolution and to the renaissance occurred in Muslim Spain.

Arabic Mathematics Worldwide

Muslim mathematicians had invented geometrical algebra and had taken it to advanced levels

In the 11th. century, the Arab mathematical foundation was one of the strongest in the world. The Muslim mathematicians had invented geometrical algebra and had taken it to advanced levels, capable of solving third and fourth degree equations. The world witnessed a new stage in the development of mathematical science, driven by the numerous translated works from Arabic into European languages.

Unquestionably, Al-Khwarizmi was very influential with his methods on arithmetic and algebra which were translated into much of southern Europe. Again, these translations became popular as “algorismi” – a term which is derived from the name of Al-Khwarizmi.

Not all went smoothly nonetheless. The Arabic numerals introduced by Al-Khwarizmi, like much of new mathematics, were not welcomed wholeheartedly. In fact, in 1299 there was a law in the commercial center of Florence (Italy) forbidding the use of such numerals. Initially, only universities dared use them, but later they became popular with merchants, and eventually became commonly used.

In time, Europe realized the great potential value of the Arab mathematical contributions and put into popular use all that seemed practical. The sciences, with mathematics as their essence, flourished and developed into the disciplines we know today. None would have been the same though, had it not been for that book on restoration, or had the zero not been invented, or had the Arabic numerals not made their way to Europe. That “fondness of science,” which inspired an early Arab mathematician to propose calculating by algebra and balancing, did much to make the world run as we know it today.

Number Zero

The 10th. millennium saw Muslim mathematical study concentrated in three main sub-disciplines. These were the ongoing progress in algebra, the development of arithmetic algorithms, and the increasing complexity in geometry. In addition, the introduction of the zero was destined to revolutionize mathematics as it allowed for key innovations. It was proposed by Muhammad Bin Ahmad in 967 CE. Zero arrived in the West much later, in 13th. century.

In the field of Mathematics the number Zero (0) and the decimal system was introduced to Europe, which became the basis for the scientific revolution. The Arabic numerals were also transferred to Europe, this made mathematical tasks much easier, problems that took days to solve could now be solved in minutes.

Many of his books were translated into European languages.

Al-Khwarizmi laid the ground work for algebra and found methods to deal with complex mathematical problems, such as square roots and complex fractions. He conducted numerous experiments, measured the height of the earth’s atmosphere and discovered the principle of the magnifying lens. Many of his books were translated into European languages.

Trigonometric work by Alkirmani of Toledo was translated into Latin (from which we get the sine and cosine functions) along with the Greek knowledge of Geometry by Euclid. Along with mathematics, masses of other knowledge in the field of physical science was transferred.

Breaking Boundaries

Certainly, the renaissance, the enlightenment and the industrial revolution were great achievements – and they occurred mainly in Europe and, later, in America. Yet many of these developments drew on the experience of the rest of the world, rather than being confined within the boundaries of a discrete Western civilization.

Our global civilization is a world heritage – not just a collection of disparate local cultures. When a modern mathematician in Boston invokes an algorithm to solve a difficult computational problem, he/she may not be aware that he/she is helping to commemorate the Arab mathematician Mohammad Ibn Musa-al-Khwarizmi, who flourished in the first half of the 9th. century.

The Square Root of Math Itself

There is a chain of intellectual relations that link Western mathematics and science to a collection of distinctly non-Western practitioners, of whom al-Khwarizmi was one.

Indeed, al-Khwarizmi is one of many non-Western contributors whose works influenced the European renaissance and, later, the Enlightenment and the Industrial Revolution. The West must get full credit for the remarkable achievements that occurred in Europe and Europeanized America, but the idea of an immaculate Western conception is an imaginative fantasy.

Modern prosperity, with all its improvement in welfare, has been delivered to humanity by science and technology. In the last two centuries especially, science has delivered better lives for people, longer lives, and for larger populations. The key to unlocking the source of these benefits was scientific method, the relentless search for truth through observation, theorizing and experimentation.

In the 13th. century the Muslim world, with its development of the culture of philosophy, science, mathematics, astronomy, physics, chemistry and medicine, led the world. The Muslim world once possessed in its hands the keys to the future prosperity that technology could deliver. Not only that, but with the invention of double entry bookkeeping, it possessed in its hands the blueprint of the plans for the modern corporation.

Eventually, after several hundred years, Europe was able to absorb this knowledge and overthrow the dark constraint of its own religion to unlock the mysteries of science and discover the path to prosperity. If the Muslim world had been able to continue on the Quranic commands on scientific research, the cause of human progress would have been advanced by about five hundred years.


In conclusion, algebra and algorithms are enabling the building of computers, and the creation of encryption. The modern technology industry would not exist without the contributions of Muslim mathematicians like Al-Khwarizmi.

Ms. Carly Fiorina, Hewlett-Packard’s Chairman and CEO delivered a speech in Minneapolis, Minnesota on September 26, 2001. The title of her speech was ‘Technology, Business and Our Way of Life: What’s Next”. She said:

“There was once a civilization that was the greatest in the world.” …….”And this civilization was driven more than anything, by invention. Its architects designed buildings that defied gravity. Its mathematicians created the algebra and algorithms that would enable the building of computers, and the creation of encryption. Its doctors examined the human body, and found new cures for disease. Its astronomers looked into the heavens, named the stars, and paved the way for space travel and exploration.”

“When other nations were afraid of ideas, this civilization thrived on them, and kept them alive. When censors threatened to wipe out knowledge from past civilizations, this civilization kept the knowledge alive, and passed it on others.”

“While modern Western civilization shares many of these traits, the civilization I’m talking about was the Islamic world from the year 800 to 1600, which included the Ottoman Empire and the courts of Baghdad, Damascus and Cairo, and enlightened rulers like Suleiman the Magnificent.”

“Although we are often unaware of our indebtedness to this other civilization, its gifts are very much part of our heritage. The technology industry would not exist without the contributions of Arab mathematicians.”

Works Cited:

Fiorina, Carly. “Technology, Business and our Way of Life: What’s Next?”. Minneapolis, Minnesota, September 26, 2001

Source: The History of Islam. This is a summary of the article published at http://www.historyofislam.com
First Published: October 2011
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