Algebra is an Arabic word. To understand its origin, we must go back to the time and place of the Arabian Nights, during the Islamic Golden Age, when the Islamic Empire was transformed into a military and scientific world power. That was the time when Caliph Harun al-Rashid ruled in Baghdad, the capital of the empire. It was the age, so masterfully described in the tales of Scheherazade, the heroine of Arabian Nights, when the Arab world was the realm where reality and magic seemed to meet.
Article continues after advertisement
Before the Islamic Golden Age, the origins of mathematics go back to the first arithmetical knowledge—to the invention of numbers and the operations possible with them. Later, the Greeks developed geometry and the rudiments of symbolic manipulation in mathematics. It has been twenty-three centuries since the legendary Euclid of Alexandria compiled the arithmetical and geometrical knowledge of his time in his magnum opus, the Elements. Algebra, however, took longer to develop, since this discipline deals with numbers as abstract entities, variables that can take on different values.
While Europe groped through the darkness of the Middle Ages, the Arabs rescued the scientific legacy of the Greeks.
It was Diophantus, another great Greek mathematician, who dared to represent variables and complex equations with symbolic combinations. We do not know exactly when he was born, but some authors think he lived in the third century CE. With the thirteen books of his Arithmetica, Diophantus aspired to reach the same level of virtuosity as Euclid. And although the geometric knowledge of the Greeks was preserved, the algebraic tradition of Diophantus was lost in Europe for several centuries, only to be rediscovered and translated into Latin in the Renaissance.
While Europe groped through the darkness of the Middle Ages, the Arabs rescued the scientific legacy of the Greeks. During the Golden Age of Islam, Arab culture spread from Asia Minor to North Africa and the Iberian Peninsula. Over a period of six hundred years, from the eighth to the thirteenth century, the Arabs absorbed Egyptian, Babylonian, Greek, and Roman science and technology. With the establishment of the Abbasid Caliphate, great importance was given to science, medicine, and education. The capital of the empire was moved from Damascus to Baghdad. It was in this city that the House of Wisdom was founded, which at first was simply a library, but which evolved into a center where the greatest scholars of the time would meet and debate.
One such scholar was Muhammad ibn Mūsa al-Khwārizmī (ca. 780–850 CE), whose fame endures to this day and whom we evoke whenever we speak of algorithms, a word derived from his name. He is generally believed to have been born somewhere between Persia and Uzbekistan. He was a polymath who was as busy making astronomical observations and mapping and studying the geography of the empire as he was developing algebra.
This word is actually a fragment (al-jabr) of the title of al-Khwārizmī’s most famous book, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah, often translated as The Compendious Book on Calculation by Completion and Balancing. This book popularized the positional decimal system and contains an extensive and didactic exposition of practical methods for solving algebraic problems. Centuries later, in Italy, when they were solving numerical problems with the abacus or with paper and ink, they would say that they were using algorithms and decimal numbers.
Al-Khwārizmī’s book proceeds in a manner similar to that of many other later algebraic “recipe” books. It poses a particular problem and shows how to find the solution. For example, the problem might be to find a number that, when reduced by three units becomes two. What is important is the method of arriving at the result, which can then be extrapolated to new situations.
The book was intended for merchants and even judges, who had to distribute inheritances according to certain proportions. The style is that of a manual, not a work of research. In the case of algebraic equations, we proceed as when we have a scale to weigh and compare objects. If we move a weight—that is, a number—from one side of the scale to the other, we must be careful to maintain equality. That is why the word al-jabr in the title of al-Khwārizmī’s book is interpreted by many as meaning “to complete,” referring to the idea of completing mathematical expressions in order to maintain equilibrium. The Latin version of al-Khwārizmī’s book, translated in 1145, was titled Liber algebrae et almucabola. It was from this translation that the word algebra entered the European vocabulary.
All books have a dual history—that of their writing and that of their subsequent influence.
It might seem that the interpretation of algebra as meaning “to complete” has been universally accepted. But this is not so.
Nearly eighty years ago, the science historians Solomon Gandz and Otto Neugebauer tracked down the origin of the term al-jabr and came to a different conclusion. The two scholars analyzed the sources of al-Khwārizmī, who relied on Babylonian, Assyrian, and Sumerian texts. In particular, the Assyrian word gabru-maharu means “to contrast” or “to be equal.” The Arabs adopted the sound of the word but spelled it al-jabr. In addition, the Arabs had their own expression with the same meaning: al-muqa-bala. Hence, the title of al-Khwārizmī’s book (al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah) is actually redundant, because al-jabr and al-muqabalah—the Assyrian and Arabic terms, respectively—both denote an equation and refer, in short, to “the science of equations.”
But all books have a dual history—that of their writing and that of their subsequent influence. While al-Khwārizmī’s work did not reach Diophantus’s level of abstraction, it did have a more immediate direct impact. Diophantus could solve problems with several variables and even sixth powers, but his book could not be used as an algebraic handbook for most practically relevant problems. Al-Khwārizmī’s book, however, had an impact on everyday mathematics in Europe through those who popularized his work.
Many writers have been fascinated by classical Arabic literature. During a conference in Buenos Aires in 1977, Jorge Luis Borges said the following: “In the fifteenth century, a series of fables was collected in Alexandria, the city of Alexander the Great. These fables have a supposedly strange history. They were first told in India, then in Persia, then in Asia Minor, and finally, once written in Arabic, they were collected in Cairo. It is the book of A Thousand and One Nights.” Edgar Allan Poe wrote a kind of coda to the Arabian Nights with the satire “The Thousand-and-Second Tale of Scheherazade.” Some of us wish that Borges, so fond of mathematics, would have added to the Arabian Nights a magical story about al-Khwārizmī, Baghdad, and the mathematical books that transformed the world.
__________________________________
From The Language of Mathematics: The Stories Behind the Symbols by Raúl Rojas. Copyright © 2025. Available from Princeton University Press.
