One of the Islamic world's most important contributions to European mathematics was undoubtedly the introduction of the positional decimal numeral system, with nine new symbols and zero. Originating in India, this system is thought to have been introduced into the Islamic world by al-Khwārizmī and to have reached Europe through Latin translations of his text (see the article), as well as through the influence of Arabic works on commercial arithmetic upon, among others, the Pisan mathematician Leonardo Pisano, better known by his nickname Fibonacci (13th century). Compared with Roman numerals, this numeral system made it easier not only to write large numbers but also to carry out algorithms such as multiplication, division and square-root extraction.
Several Arabic treatises offered practical methods for solving complex arithmetic problems, notably through the use of fractions, powers and square roots. New computational algorithms were developed to suit the new numeral system and the use of ink and paper, as al-Uqlīdisī (10th century) explains, for example, in Sections on Indian Arithmetic (al-Fusūl fī al-Hisāb al-Hindī). The methods of simple and double false position were thus introduced as new problem-solving procedures, as set out in the Book of Increase and Decrease (Liber augmenti et diminutionis), probably a translation by Gerard of Cremona of an unidentified Arabic text (see the book review here).
Trigonometry and geometry ------------------------------
Arabic mathematicians such as al-Battānī (c. 858–929) and al-Farghani (c. 800–870) developed trigonometric tables and methods for calculating angles in spherical triangles. Al-Battānī in particular improved on the work of the ancient Greeks by introducing the concept of the sine, which originated in India, and developing more accurate methods for calculating the positions of celestial bodies.
The art or science of mensuration (sinā’at al-misāha) saw original and significant developments. Several texts on the subject were translated into Latin, including the Book of the Three Brothers (Liber trium fratrum) by the Bānū Mūsā brothers and the Book of Measurements (Liber mensurationum) by a certain Abū Bakr, who has yet to be identified. Among other things, these works enabled Fibonacci to write his Practica geometriae.