The history of Arabic and Islamic culture and the history of the development of theories and practices in mathematics and science during the 8th-12th centuries greatly overlap or are the same in many instances. These areas of learning and accomplishment that had been the purview of the Greeks and then, later, the more practical culture of the Romans, were taken over to much of a degree by the Arabs who came into Southern Europe and Northern Africa from the east, led and unified by, among others, the prophet Mohammed. The Arabs settled down rather quickly and, unlike many conquering nations, proceeded to learn about and build on the works of the nations they had conquered, those being the Greeks and the Romans. Partially because of this, the Arabs made many notable contributions to mathematics and the sciences in a period of a few hundred years.
The list of noted mathematicians and scientists in the developed world in the period of history, 8th-12th centuries, is extensive, as are their contributions to mathematics and science. To decrease the amount of confusion, it is somewhat helpful to consider the meanings and connotations of some terms. The following is probably over-simplistic but should suffice for the purpose of the following discussion. Arab here refers to anyone or thing associated with a number of countries where Arabic is the official language or one of the official “state” languages. This area includes several, but not all, countries in southeastern Europe, Eastern Asia, and Northern Africa. The list varies from source to source and depends on the related definition of Arab. “Arabia” is not considered synonymous with "Middle East." The latter phrase, while widely used, is troublesome due to it implying the location of the area with respect to Europe.
The identifiers “Islam” and “Islamic” refer to culture related to the “five pillars of Islam.” The majority of followers of Islam or the descendants of such followers, Muslims, do not live in the Middle East. But Islam was founded in Mecca in what is now Saudi Arabia. Most scholars date its origin to the 7th century and agree that the Arab nations were where the religion gained popularity in the 8th century.
As indicated above, the people who moved into what we call the Arab countries adopted the culture and known works of the nations they had conquered. But it is important to keep in mind that they did not just preserve this. Instead, they built on it. It is believed that they were inspired by the Islamic view of nature that mankind has a duty to “study nature in order to discover God and to use nature for the benefit of mankind.” They contributed many original works to mathematics and the different fields of science.
Information on just two of the better-known Arab scholars is presented here, the first of whom is undoubtedly the best known because of his works, not only in mathematics but also in astronomy and geography. Some consider his name as the basis for the term algebra. Muḥammad ibn Mūsā al-Khwārizmī (780-850), who is usually known as al-Khwarizmi (there are several forms of his name), was a Persian polymath whose treatise on algebra (The Compendious Book on Calculation by Completion and Balancing, c. 813–833) presented the first systematic solution of linear and quadratic equations. He provided geometric justifications for much of his work. Because he was the first to treat algebra as an independent discipline and introduced the methods of adding and subtracting terms to both sides of the equation, he has been described as the founder of algebra.
Abu Kamil Shuja’ ibn Aslam (850--930) was a prominent Egyptian Muslim mathematician of the Islamic Golden age (generally considered the 8th to the 13th or 14th centuries). He was also known as al-Hasib al-Misri, the Egyptian Calculator (or Reckoner). His full name was Abu Kamil Shujaʿ ibn Aslam ibn Muhammad ibn Shuja. He is regarded as the first mathematician to use irrational numbers as solutions and coefficients to equations methodically. He worked on algebraic equations with powers higher than two and solved non-linear simultaneous equations with three unknown variables.
He always computed the general solution to all of the problems he worked on rather than particular solutions. One of his strengths was to state the problem and the solutions rhetorically rather than use mathematical notation. This made his innovations and techniques understandable to more people. The mathematical techniques of Shuja became better known when centuries later, they were adopted by Fibonacci. Fibonacci (1170-1240?) was an Italian mathematician who produced the first European work on Indian and Arabian mathematics. This introduced Hindu-Arabic numerals to Europe and greatly contributed to the spread of knowledge of Islamic mathematics to Europe.
The contributions of the Arab world to mathematics and the physical and biological sciences are well documented and remarkable. Some of the references below might be considered a starting place.
Creighton, N. (2011). Abu, Kamil Shuja. African American Studies Center, 850–930. https://doi.org/10.1093/acref/9780195301731.013.48128
Faruqi, Yasmeen. (2006). Contributions of Islamic scholars to the scientific enterprise. International Education Journal. 7.
Gies, F. Carney (2021, January 22). Fibonacci. Encyclopedia Britannica. https://www.britannica.com/biography/Fibonacci
H. (2020, May 6). Top 10 Arab Mathematicians. Steamdaily. https://Steamdaily.com/top-10-arab-mathematicians/
Wikipedia contributors. (2021, April 10). Glossary of Islam. Wikipedia. https://en.wikipedia.org/wiki/Glossary_of_Islam