The Muslims were drawn to the study of mathematics from the very beginning to a large extent because of the "abstract" nature of the Islamic revelation and the love that Islam created in the minds of its followers for the doctrine of unity and for a vision of the universe understood mathematically as mathematics is comprehended in the traditional sense of the term. That is why the Muslims made remarkable contributions to many domains of mathematics.
In the field of arithmetic, the most important achievements of Muslims were perhaps the adoption of the Sanskrit numerals and the later transformation which they brought about upon it, creating as a final product what has come to be known as Arabic numerals in Europe and the use of the decimal system. The famous Arabic numerals appear in the Kitab al-hisab ("Treatise on Arithmetic") by Muhammad ibn Musa al-Khwarazmi, from Khwarazm, one of the Persian cities east of the Caspian Sea. This work was rendered into Latin thereby bringing the Arabic numerals to the West and causing one of the most important transformations within medieval Western civilization. In addition to the research in the field of number theory, decimal fractions and the computation of numerical series, the whole tradition in a sense culminating with Ghiyath al-Din Jamshid al-Kashani, the author of Miftah al-hisab ("The Key to Arithmetic") who lived in the eighth Islamic century. He is the real discover of decimal fractions and the maker of devices to carry out mathematical calculation.
As far as geometry is concerned, Muslim mathematicians began where the Greek mathematicians and geometers had left off. They further developed plane and solid geometry and systematized mathematical equations for the solution of many geometric problems, creating a relationship between algebra and geometry which was to be pursued later by Descartes and which became one of the most important elements in the development of modern mathematics. One of the problems which the Islamic geometers and mathematicians studied very carefully was the fifth postulate of Euclid concerning the fact that one and only one parallel line can be drawn to an existing line from a point outside of that line, a problem the proof of which was debated over the centuries. Both Khayyam and Tusi wrote treatises on the subject, indicating that this fact must be taken as a postulate and cannot be proven by Euclidean geometry itself. The criticism of the subject opened the door to a path which led finally to the development of non-Euclidean geometry in the West in the nineteenth century.
One of the fields of mathematics which was especially developed by Muslims is trigonometry. It was the Muslim mathematicians who for the first time systematized the six trigonometric functions which to this day bear the mark of their Arabic origin in Western languages, the word sine being a translation of jayb from Arabic (sinus in Latin meaning literary pocket or cavity which is what jayb means in Arabic). The first treatise on trigonometry which deals with the subject as an independent branch of mathematics is usually attributed to Nasir al-Din al-Tusi although in reality the first independent work on trigonometry as an independent field goes back to al-Biruni.
The science of algebra is also one in which the Muslims not only made contributions but in a sense they created a new field by drawing from certain elements of Greek mathematics as developed by Diophantus and also from certain ideas of Indian mathematics. The very word algebra is of course of Arabic origin, going back to another famous treatise of al-Khwarazmi entitled Jabr wa'l-muqabalah which was rendered into Latin and whsoe title became the origin of the word algebra. The name of the author al-Khwarazmi itself left its indelible mark upon Western languages by being used in certain languages such as Spanish for arithmetic or for number while it subsists to this day in the English language in the form of algorism. The Muslims developed algebra from its humble origins in the third Islamic century to the masterpiece of 'Umar Khayyam written in the sixth Islamic century, his work Algebra being perhaps the most perfect treatise on algebra written before modern times. In this book Khayyam dealt with equations up to the third degree and systematized the solution of quadratic equations.
Another branch of mathematics which one no longer associates directly with mathematics as such was music. Theoretical music was considered, following the teachings of Plato and Pythagoras, as a branch of mathematics by Muslims and appears often as such in the Islamic classifications of the sciences. Many of the great Islamic philosophers such as al-Farabi and Ibn Sina were interested in theoretical music and very elaborate treatises were written by them as well as by others like al-Urmawi and al-Mawsili dealing both with the theory of Arabic and Persian music and relating the study of music to other sciences and the study of mathematics to proportion and harmony which are the foundations of music. In the classical treatises dealing with mathematics, often a section is devoted to music as we see in the works of Ibn Sina and Qutb al-Din al-Shirazi.-Seyyed Hossein Nasr, A Young Muslims's Guide to the Modern World, pgs. 88-90.