Al-Khwarizmi and the Birth of Algebra in Abbasid Baghdad

Al-Khwarizmi and the birth of algebra belong to the scientific flowering of Abbasid Baghdad in the early 9th century. Around 820, the mathematician Muhammad ibn Musa al-Khwarizmi wrote a book on solving equations whose title gave the world the word algebra. His other works helped spread the Indian decimal number system, and his own name, in Latin form, became the word algorithm.

Background: science in Abbasid Baghdad

By the early 9th century Baghdad was the capital of an empire stretching from North Africa to Central Asia. The Abbasid caliphs, especially al-Ma’mun (r. 813–833), sponsored the translation of Greek, Persian and Indian scientific works into Arabic and supported scholars working at the court, an activity associated with the House of Wisdom.

Indian mathematics and astronomy were already known in Baghdad. Under the caliph al-Mansur, around the 770s, an Indian astronomical work based on Brahmagupta’s Brahmasphutasiddhanta (628) was translated into Arabic as the Sindhind. Greek works such as Euclid’s Elements and Ptolemy’s Almagest were translated in the decades that followed.

The government also had practical needs. Inheritance under Islamic law required precise division of estates among many heirs. Land surveys, taxation, canal digging and commerce all demanded calculation.

Who was al-Khwarizmi?

Very little is known about his life. He was probably born around 780 and died around 850. His name suggests a family from Khwarazm, the region south of the Aral Sea in modern Uzbekistan and Turkmenistan, though the historian al-Tabari also calls him al-Qutrubbulli, after a district near Baghdad, and scholars debate where he was born.

He worked in Baghdad under al-Ma’mun, to whom he dedicated his algebra book. Later sources connect him with the caliph’s scholarly circle, and he may have taken part in astronomical projects organised by al-Ma’mun. No portrait or detailed biography survives.

The book of al-jabr and al-muqabala

His most famous work is al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqabala, “The Compendious Book on Calculation by Completion and Balancing”, written around 820 (the usual range given is 813–833). In the introduction he says the caliph encouraged him to write a short work on calculation useful for inheritance, legacies, lawsuits, trade, land measurement and canal digging.

The two operations in the title describe how to simplify an equation:

• Al-jabr (“restoring” or “completing”) meant removing a subtracted quantity by adding it to both sides.
• Al-muqabala (“balancing” or “confronting”) meant cancelling equal quantities on both sides.

Al-Khwarizmi worked with three kinds of quantity: numbers, “roots” (the unknown, which we would call x) and “squares” (x²). He reduced every linear and quadratic equation to one of six standard forms, such as “squares and roots equal numbers”, and gave a rule for solving each one.

Because he did not use negative numbers or zero as coefficients, he needed six separate cases rather than the single general equation a modern student would write:

1. Squares equal roots (ax² = bx).
2. Squares equal numbers (ax² = c).
3. Roots equal numbers (bx = c).
4. Squares and roots equal numbers (ax² + bx = c).
5. Squares and numbers equal roots (ax² + c = bx).
6. Roots and numbers equal squares (bx + c = ax²).

The first three are simple; the last three are the true quadratic cases, and for each he gave a step-by-step procedure followed by a geometric justification. The fifth case can have two positive solutions, and he noted this, as well as the situation in which there is no solution.

A famous example

His best-known example is: a square and ten roots equal thirty-nine units. In modern notation, x² + 10x = 39. His rule: halve the number of roots (5), square it (25), add 39 to get 64, take the square root (8) and subtract 5. The answer is 3.

He then proved the rule geometrically, drawing a square and adding rectangles around it, a method we still call “completing the square”. The whole book is written in words; even the numbers are spelled out, and it uses no symbols. Like most mathematicians of his time, he accepted only positive solutions.

The second half of the book applies the methods to practical problems, especially the division of inheritances, which takes up a large part of the text.

Practical mathematics

Between the theory and the inheritance problems, al-Khwarizmi included a section on mensuration: rules for the areas of triangles, quadrilaterals and circles, and the volumes of solids such as pyramids and cones. For the circle he gave several approximations of the ratio we call pi, including 3 1/7, and noted that it was an approximation rather than an exact value.

The inheritance chapters show why the book was valued. Islamic law assigns fixed fractions of an estate to different relatives, and complications such as debts, bequests to non-relatives, or the freeing of slaves made the calculations difficult. Al-Khwarizmi showed how to set these up as equations in an unknown “thing” (shay’) and solve them.

Later algebra in the Islamic world

Al-Khwarizmi’s book started a tradition rather than ending one. Abu Kamil in Egypt worked with irrational coefficients and more complex systems of equations. In the 10th and 11th centuries al-Karaji developed the algebra of polynomials, and the Persian poet and mathematician Omar Khayyam classified cubic equations and solved them geometrically using intersecting conic sections.

The oldest surviving Arabic manuscript of al-Khwarizmi’s algebra is a copy dated 1342, kept at the Bodleian Library in Oxford. Its preservation, centuries after the original was written, reflects how long the book continued to be studied.

Hindu-Arabic numerals and the word algorithm

Al-Khwarizmi also wrote a book on arithmetic using Indian numerals. It explained the decimal place-value system with nine digits and a sign for zero, and showed how to add, subtract, multiply, divide and find square roots with it.

The Arabic original is lost. The work survives in a Latin version from the 12th century, preserved in a manuscript in Cambridge, which begins with the words Dixit Algorizmi, “al-Khwarizmi says”. European writers turned his name into algorismus, a term for calculation with the new numerals. From this came “algorism” and, later, “algorithm”, whose spelling was probably influenced by the Greek word arithmos, number. In modern computing an algorithm is any precise step-by-step procedure.

Astronomy and geography

Al-Khwarizmi’s astronomical tables, the Zij al-Sindhind, combined Indian, Persian and Greek methods for calculating the positions of the Sun, Moon and planets. A version revised in al-Andalus by Maslama al-Majriti was translated into Latin by Adelard of Bath in 1126.

He also wrote Kitab Surat al-Ard, “The Book of the Image of the Earth”, a list of coordinates for cities, mountains, seas and rivers. It was based on Ptolemy’s Geography but corrected and extended it, for example in its picture of the Mediterranean and of the Islamic lands.

How the ideas reached Europe

In the 12th century, scholars in Spain translated Arabic science into Latin. In 1145 Robert of Chester translated the algebra book as Liber algebrae et almucabola, the source of the word algebra in European languages. Gerard of Cremona made another translation.

The new numerals and methods spread further through the work of Leonardo of Pisa, known as Fibonacci, whose Liber Abaci of 1202 drew on Arabic mathematics and promoted the Hindu-Arabic numerals among Italian merchants (see Fibonacci’s Liber Abaci). Over the following centuries they replaced Roman numerals in Europe.

Key people

• Muhammad ibn Musa al-Khwarizmi (c. 780 – c. 850): mathematician, astronomer and geographer in Baghdad.
• Al-Ma’mun (r. 813–833): Abbasid caliph and patron of science, to whom the algebra was dedicated.
• Abu Kamil (c. 850 – c. 930): Egyptian mathematician who extended al-Khwarizmi’s algebra.
• Robert of Chester and Gerard of Cremona: 12th-century translators who brought the algebra into Latin.
• Leonardo Fibonacci (c. 1170 – c. 1250): Italian mathematician who spread Hindu-Arabic numerals in Europe.

Myths and facts

• Myth: al-Khwarizmi invented the numerals we use. The decimal place-value system with zero developed in India; the early history of zero is also traced through sources such as the Bakhshali manuscript. Al-Khwarizmi explained and transmitted it.
• Myth: nobody solved equations before him. Babylonian scribes solved quadratic problems more than 2,500 years earlier, and the Greek Diophantus and Indian mathematicians such as Brahmagupta also worked on equations. Al-Khwarizmi’s achievement was a general, systematic treatment with proofs.
• Fact: algebra and algorithm both come from his work. One from his book title, the other from his name.

Timeline

• c. 770s: Indian astronomical work based on Brahmagupta translated into Arabic in Baghdad.
• c. 780: Approximate birth of al-Khwarizmi.
• 813: Al-Ma’mun becomes caliph.
• c. 820: Al-Khwarizmi writes his book on al-jabr and al-muqabala.
• c. 850: Approximate death of al-Khwarizmi.
• 1126: Adelard of Bath translates his astronomical tables into Latin.
• 1145: Robert of Chester translates the algebra into Latin.
• 1202: Fibonacci’s Liber Abaci promotes Hindu-Arabic numerals in Europe.

Why it matters

Al-Khwarizmi turned the solving of equations into a field of its own, giving it a name, a method and proofs. His explanation of the Indian number system helped spread the numerals now used worldwide. His career also shows how the Abbasid court brought together Greek, Persian and Indian knowledge and passed the results on to medieval Europe. Every time we speak of algebra or algorithms, we echo the work of a scholar in 9th-century Baghdad.