Little is known of Abu Ja'far Muhammad Bin Musa Al-Khwarizmi's life. One unfortunate effect of this lack of knowledge seems to be guesses based on very little evidence. But the famous 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.

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. His son, Al-Mamun became Caliph after the father. He continued the patronage of learning started by his father and founded an academy called the House of Wisdom (Beit Al Hikmah), 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 scholars.

Al-Khwarizmi and his colleagues worked 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. Al-Khwarizmi worked under the patronage of Al-Mamun and 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 below, it is the first book ever to be written on algebra.

Al-Khwarizmi's own words describing the purpose of the book tell us that he intended to teach "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.

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:

"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 described the main topic of the 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, 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.

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:-

"... 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."

Al-Khwarizmi continued his study of algebra in Hisab Al-jabr w'al-muqabala by examining how the laws of arithmetic extend to an arithmetic for his algebraic objects. For example he shows how to multiply out expressions such as:

(a + b x) (c + d x)

although again we should emphasise that Al-Khwarizmi uses only words to describe his expressions, and no symbols are used. One researcher sees a remarkable depth and novelty in these calculations by Al-Khwarizmi which appear to us, when examined from a modern perspective, as relatively elementary:

"Al-Khwarizmi's concept of algebra can now be grasped with greater precision: it concerns the theory of linear and quadratic equations with a single unknown, and the elementary arithmetic of relative binomials and trinomials. ... The solution had to be general and calculable at the same time and in a mathematical fashion, that is, geometrically founded. ... The restriction of degree, as well as that of the number of unsophisticated terms, is instantly explained. From its true emergence, algebra can be seen as a theory of equations solved by means of radicals, and of algebraic calculations on related expressions..."

Sarton describes Al-Khwarizmi as:

"... the greatest mathematician of the time, and if one takes all the circumstances into account, one of the greatest of all time...."

While Gandz gives this opinion of Al-Khwarizmi's algebra:

"Al-Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences. In a sense, Al-Khwarizmi is more entitled to be called "the father of algebra" because Al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake."

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. This section on mesuration certainly has more in common with Hindu and Hebrew texts than it does with any Greek work. The final part of the book deals with the complicated Islamic rules for inheritance but require little from the earlier algebra beyond solving linear equations.

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. Unfortunately the Latin translation 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

Another important work by Al-Khwarizmi was his work Sindhind zij on astronomy. The work is based in Indian astronomical works. The Indian text on which Al-Khwarizmi based his treatise was one, which had been given to the court in Baghdad around 770 as a gift from an Indian political mission. There are two versions of Al-Khwarizmi's work, which he wrote in Arabic but both are lost. In the tenth century Al-Majriti made a critical revision of the shorter version and this was translated into Latin. There is also a Latin version of the longer version and both these Latin works have survived. The main topics covered by Al-Khwarizmi in the Sindhind zij are 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. A related manuscript, attributed to Al-Khwarizmi, on spherical trigonometry exists.

Al-Khwarizmi wrote a major work on geography, which gives latitudes and longitudes for 2402 localities as a 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 does include maps, which on the whole are more accurate than those of Ptolemy. In particular it is clear that where more local knowledge was available to Al-Khwarizmi such as the regions of Islam, Africa and the Far East then his work is considerably more accurate than that of Ptolemy, but for Europe Al-Khwarizmi seems to have used Ptolemy's data.

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.