History of Algebra

In modern mathematics, algebra refers to a broad branch of mathematics that studies mathematical structures, such as groups, rings, fields, modules, and vector spaces, and their properties. Algebra is concerned with the manipulation and analysis of mathematical objects, such as variables, equations, and functions, using symbolic notation and mathematical operations. Algebraic techniques and methods are used in many areas of mathematics and science, including geometry, topology, number theory, cryptography, physics, and computer science. Algebraic concepts also have applications in the real world, such as in coding theory, data analysis, and engineering.

What is Algebra?

But what does the word “algebra” mean? This is a question whose answer guides us to the beginnings of this branch of mathematics. The word “algebra” (or “algèbre” in French) originates from the Arabic word “al-jabr”. It appeared in mathematics for the first time in the early ninth century CE in the title of an Arabic book entitled al-Mukhtaṣar fī Ḥisāb al-Jabr wa-al-Muqābala (The Compendious Book on Calculation by Completion and Balancing). It was written by a well-known mathematician and astronomer named Muḥammad ibn Mūsá al-Khwārazmī. “Al-jabr” was the title of one of the two techniques Khwārazmī used in his book to solve algebraic equations.

Where Algebra Comes From

The history of solving algebraic problems goes back several thousand years. Mathematicians from ancient Egypt and Mesopotamia were able to solve simple algebraic equations (and also some complicated ones in their special cases). Nevertheless, their solutions were based on practical numerical values, and no general solution was suggested to solve algebraic equations. No Greek mathematical work has reached us in which algebraic equations are discussed. Nevertheless, there are some geometrical methods in the works written by Apollonius and Euclid that have been called in early modern mathematics “geometric algebra.” Some historians of mathematics argue that these methods are the geometric representations of algebraic equations, and thus, they believe that knowledge of algebra existed prior to the composition of the works by Apollonius and Euclid. However, others argue that the relationship between the geometrical propositions in these works and the algebraic equations is too modern to be attributed to Euclid and Apollonius. The argument of this group of historians is based on the definition of basic concepts in mathematics such as number, length, area, etc.

Written Works on Algebra

There is another work, probably written in the third century CE by an Alexandrian mathematician named Diophantus, which can be related to algebra. His work was a collection of problems giving numerical solutions of equations. Diophantus never used general methods in his solutions. Although in some cases he used symbols in his solutions, which is a critical feature of modern algebra, Diophantus’ solutions are too primitive to be counted as an origin for algebra. Khwārazmī’s book is the earliest work dedicated independently to algebra. It laid down the foundation of algebra as a branch of mathematics for solving algebraic equations in their general forms. Soon after the composition of Khwārazmī’s book, many other mathematicians started writing on algebra. Thābit ibn Qurra (d. 901), Abū Kāmil Shujāʿ ibn Aslam (fl. c. 900), Abū Bakr Muḥammad ibn al-Ḥasan al-Karajī (fl. c. 1000), ʿUmar ibn Ibrāhīm al-Ḫayyāmī al‐Nīshābūrī (fl. 1100), Sharaf al-Dīn al-Ṭūsī (fl. 1170), and Samawʾal ibn Yaḥyá ibn ʿAbbās al‐Maghribī (d. 1174) are some of the major mathematicians who dedicated some of their works to algebra.

Through the Latin translations of Khwārazmī’s book on algebra, this branch of mathematics was introduced to European mathematicians. Additionally, the Arabic knowledge of algebra was transmitted to Europeans through works written by mathematicians like Leonardo Fibonacci (fl. c. 1230). Fibonacci wrote his famous Liber Abaci, in which he introduced the Arabic numerals to Europeans and explained their use in arithmetic and algebra.

Algebra Today

By the end of the Middle Ages, European mathematicians had absorbed the knowledge of algebra from Arabic sources, and algebra became a standard part of European mathematics. Algebra continued to develop throughout the Renaissance and the early modern period, with mathematicians such as François Viète (d. 1603), René Descartes (d. 1650), and Isaac Newton (d. 1727) making important contributions to the field.

Today, algebra remains a central branch of mathematics, with applications in many areas of science, engineering, and technology. Its concepts are used in designing algorithms for solving complex problems, in analyzing data and modeling systems, and in designing and analyzing experiments. Algebra is also essential for understanding and using many other areas of mathematics, including calculus, differential equations, and abstract algebra. As such, algebra continues to play a vital role in the development of modern mathematics and its applications.

This article is contributed by Sajjad Nikfahm Khubravan