After the golden age of Greek geometry, scientific learning diffused over Europe and Asia, particularly after the Emperor Justinian closed the pagan academies in 529ad. Meanwhile in India, the decimal system for whole numbers and the sine function were being developed. After the Prophet Muhammad returned from exile in 630ad, the Muslim world began its expansion and soon Greek, Persian and Indian mathematical texts were translated into Arabic. A great flowering of mathematical thought was underway.
The caliph al-Ma'Mun built the House of Wisdom, a research and translation centre in Baghdad in the 9th century and much important work originated there. For example, Muhammad ibn Musa al-Khwarizmi wrote the Book of Addition and Subtraction According to the Hindu Calculation, beginning a tradition of arithmetic books which led to decimal fractions (complete with point) within a century. The word algorithm, from "al-gorismi," is a corruption of the author's name. Al-Khwarizmi also gave the world algebra from his Book of Restoring and Balancing; al-jabr being the Arabic for "restoring".
Like many of his successors, Al-Khwarizmi's work had a strong religious purpose, being used to calculate prayer times and Islamic inheritance.
In the 10th century, Islamic mathematicians developed three complete numeration systems, including adopting the Babylonian astronomical geometry numeration to the base 60 and the decimal system we use today. Omar Khayy m (1048-1131) developed the Greek and Hindu system of extracting square and cube-roots to extract roots of any desired degree.
Meanwhile, algebra flourished and extended, to some extent together with arithmetic. As-Samaw'al in the 12th century gave a rule for finding the co-efficients of the successive powers of 1x. By the 14th century algebraic symbolism was in use. Names to conjure with include Abu Kamil,al-Karaji and al-Khazin who worked on Diophantine equations and also part of what is now known as Fermat's last theorem - that there are no rational solutions to x cubed plus y cubed = z cubed.
Geometry zoomed forward, too, Alhazen analysing conic sections, including areas and volumes of plane and solid figures formed from them; others worked on the curves needed for sundials and geometric methods for artisans. Applied geometry expanded into the realms of astronomy and geography and calculation of latitude and longitude.
Omar Khayy m, a poet and mathematician lived in Iran and developed many of Euclid's theories, including the parallel postulate and the definition of ratios. Omar Khayy m argued that ratios should be regarded as "ideal numbers" and he conceived a much broader system of numbers than the Greek one of positive real numbers.
While Islamic mathematicians continued during the Dark Ages in Europe to develop algorithms and algebra others such as Sharaf ad-Din at-Tusi also expanded the Islamic contribution to astronomy, inventing a device called the linear astrolabe which took its place in a library of astrolabes invented by Islamic scientists. Astrolabes use overlapping plates to enable complicated calculations about heavenly bodies. Kamal ad-Din Farisi, using Alhazen's great work on Optics, provided the first satisfactory mathematical explanation of the rainbow.
Muslim theorists also developed those tables of trigonometrical functions which dominated every maths student's life until the advent of the calculator, culminating in the work of Jamshid al-Kashi (who died in 1429) whose work The Reckoner's Key gave a masterful summary of the mathematical knowledge of the time, including his computation of 2pi. His work ends the golden age of Muslim maths.