Al-Khwarizmi in the context of "Completing the square"

Science in the medieval Islamic world was the science developed and practised during the Islamic Golden Age under the Abbasid Caliphate of Baghdad, the Umayyads of Córdoba, the Abbadids of Seville, the Samanids, the Ziyarids and the Buyids in Persia and beyond, spanning the period roughly between 786 and 1258. Islamic scientific achievements encompassed a wide range of subject areas, especially astronomy, mathematics, and medicine. Other subjects of scientific inquiry included alchemy and chemistry, botany and agronomy, geography and cartography, ophthalmology, pharmacology, physics, and zoology.

Medieval Islamic science had practical purposes as well as the goal of understanding. For example, astronomy was useful for determining the Qibla, the direction in which to pray, botany had practical application in agriculture, as in the works of Ibn Bassal and Ibn al-'Awwam, and geography enabled Abu Zayd al-Balkhi to make accurate maps. Islamic mathematicians such as Al-Khwarizmi, Avicenna and Jamshīd al-Kāshī made advances in algebra, trigonometry, geometry and Arabic numerals. Islamic doctors described diseases like smallpox and measles, and challenged classical Greek medical theory. Al-Biruni, Avicenna and others described the preparation of hundreds of drugs made from medicinal plants and chemical compounds. Islamic physicists such as Ibn Al-Haytham, Al-Bīrūnī and others studied optics and mechanics as well as astronomy, and criticised Aristotle's view of motion.

Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built upon syntheses of Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important developments of the period include extension of the place-value system to include decimal fractions, the systematised study of algebra and advances in geometry and trigonometry.

The medieval Islamic world underwent significant developments in mathematics. Muhammad ibn Musa al-Khwārizmī played a key role in this transformation, introducing algebra as a distinct field in the 9th century. Al-Khwārizmī's approach, departing from earlier arithmetical traditions, laid the groundwork for the arithmetization of algebra, influencing mathematical thought for an extended period. Successors like Al-Karaji expanded on his work, contributing to advancements in various mathematical domains. The practicality and broad applicability of these mathematical methods facilitated the dissemination of Arabic mathematics to the West, contributing substantially to the evolution of Western mathematics.

The Geography (Ancient Greek: Γεωγραφικὴ Ὑφήγησις, Geōgraphikḕ Hyphḗgēsis, lit. "Geographical Guidance"), also known by its Latin names as the Geographia and the Cosmographia, is a gazetteer, an atlas, and a treatise on cartography, compiling the geographical knowledge of the 2nd-century Roman Empire. Originally written by Claudius Ptolemy in Greek at Alexandria around 150 AD, the work was a revision of a now-lost atlas by Marinus of Tyre using additional Roman and Persian gazetteers and new principles.

Its translation into Arabic by al-Khwarismi in the 9th century was highly influential on the geographical knowledge and cartographic traditions of the Islamic world. Alongside the works of Islamic scholars—and the commentary containing revised and more accurate data by Alfraganus—Ptolemy's work was subsequently highly influential on Medieval and Renaissance Europe.

Abū al-ʿAbbās Abd Allāh ibn Hārūn al-Maʾmūn (Arabic: أبو العباس عبد الله بن هارون الرشيد, romanized: Abū al-ʿAbbās ʿAbd Allāh ibn Hārūn ar-Rashīd; 14 September 786 – 9 August 833), better known by his regnal name al-Ma'mun (Arabic: المأمون, lit. 'the Reliable'), was the seventh Abbasid caliph, who reigned from 813 until his death in 833. His leadership was marked by the power and prosperity of the Abbasid Caliphate, al-Ma'mun promoted the Graeco-Arabic translation movement, the flowering of learning and the sciences in Baghdad, and the publishing of al-Khwarizmi's book now known as "Algebra", making him one of the most important caliphs in the Islamic Golden Age. He is also known as a proponent of the rational Islamic theology of Mu'tazilism.

Al-Ma'mun succeeded his half-brother al-Amin after a civil war, much of his reign was spent on peace campaigns. His strong support for Mu'tazilism led him to imprison a Sunni Imam, Ahmad ibn Hanbal in an event that became known as mihna. Al-Ma'mun's foreign policy was due to his decision to continue war and diplomatic relations with the Byzantine Empire, the tension between conflict or diplomacy varying during his military campaigns.