Islamic mathematics refers to the mathematical advancements made in the Islamic world from 622-1600, flourishing particularly during the Abbasid Caliphate, where scholars translated classical texts and developed new concepts. Influential mathematicians such as Al-Khwārizmī established algebra and contributed to the understanding of numbers, geometry, and astronomy, affecting later developments in various fields. The integration of cultural and scientific exchange laid the groundwork for mathematical practices still relevant today.

1. Islamic Mathematics!!

2. Introduction to Islamic Mathematics Islamic mathematics refers to the mathematics developed in the Islamic world between 622-1600 (When Islam was the dominant religion). Islamic science and mathematics flourished under the Islamic Empire, established across the Middle East, Central Asia, North Africa, Sicily, the Iberian Peninsula, and in parts of France and India in the 8th century. The center of Islamic mathematics was located in Persia, but expanded to the west and east over time. Most scientists in this period were Muslims and Arabic was the dominant language. Arabic was used as the chosen written language of most scholars throughout the Islamic world at the time—contributions were made by people of different ethnic groups (Arabs, Persians, Berbers, Moors, Turks) and sometimes different religions (Muslims, Christians, Jews, etc.)

3. Origin & Influences During the 1 st century of the Islamic Empire, there were barely any mathematical achievements or knowledge since the other empires had no intellectual drive. In the 2 nd half of the 18 th century research in mathematics increased. The Muslim Abbasid caliph al-Mamun (809-833) supposedly had a vision where Aristotle appeared to him, and as a consequence al-Mamun ordered that Greek works, such as Ptolemy’s Almagest & Euclid’s Elements, be translated into Arabic. These works were given to the Muslims in the Byzantine Empire in exchange for peace between the two empires. Thabit ibn Qurra (826-901)translated most of these books (written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.) it is through the work of Islamic translators that many ancient Greek texts have survived throughout history. The “mathematics of practitioners” played an important role in mathematics brought about in medieval Islam. This included the applied mathematics of builders, artisans, tax and treasury officials, merchants, etc. This form of mathematics was universal in the Islamic world and was common to all.

4. Islam & Mathematics The Islamic law of inheritance served as an drive behind the development of algebra (Arabic: al-jabr ) by Muhammad ibn Mūsā al-Khwārizmī and other medieval Islamic mathematicians. Al-Khwārizmī's Hisab al-jabr w’al-muqabala had a chapter formulating the rules of inheritance as linear equations (& his knowledge of quadratic equations was unnecessary). Later mathematicians dedicated to the Islamic law of inheritance included Al-Hassār, who developed the modern symbol for fractions in the 12th century, and Abū al-Hasan ibn Alī al-Qalasādī, who developed an algebraic notation which affected the rise towards the introduction of algebraic symbols in the 15 th century.

5. The Use of Mathematics In order to observe holy days on the Islamic calendar (times determined by the phases of the moon), astronomers initially used Ptolemy's method to calculate the place of the moon and stars. Islamic months do not begin at the astronomical new moon, instead they begin when the thin crescent moon is first sighted in the western evening sky. The Qur'an says: "They ask you about the waxing and waning phases of the crescent moons, say they are to mark fixed times for mankind and Hajj." This led Muslims to find the phases of the moon in the sky, leading to new mathematical calculations. Predicting just when the crescent moon would become visible was a test for the Islamic mathematical astronomers. To predict the first visibility of the moon, it was essential to express its motion according to the horizon, and this problem demands pretty complicated spherical geometry. However, finding the direction of Mecca and knowing the specific times for prayer (by looking @ constellations/stars) motivated the Muslims to study and develop knowledge of spherical geometry.

6. Islamic Scholars & Their Contributions Al-Ḥajjāj ibn Yūsuf ibn Maṭar (786 – 833) translated Euclid's Elements into Arabic. Muḥammad ibn Mūsā al-Khwārizmī (c. 780 – c. 850) Persian mathematician, astronomer, astrologer and geographer; worked most of his life as a scholar in Baghdad; Algebra was the first book on linear and quadratic equations; introduced the decimal positional number system to the Western world in the 12th century; revised and updated Ptolemy's Geography as well as writing several works on astronomy and astrology. Al-’Abbās ibn Sa’id al-Jawharī (c. 800– c. 860) mathematician who worked @ House of Wisdom (Baghdad); most important work: Commentary on Euclid's Elements (contained 50 additional propositions and an attempted proof of the parallel postulate) ‘ Abd al-Hamīd ibn Turk (830) wrote a work on algebra (only 1 chapter of quadratic equations survived) Ya’qūb ibn Isḥāq al-Kindī (c. 801 – 873) contributions to mathematics include many works on arithmetic and geometry.

7. Islamic Scholars & Their Contributions Cont… Banū Mūsā (c. 800 – 873) three brothers in Baghdad; most famous mathematical treatise: The Book of the Measurement of Plane and Spherical Figures ; The eldest, Ja’far Muḥammad (c. 800) specialized in geometry and astronomy; Aḥmad (c. 805) specialized in mechanics and wrote On mechanics ; The youngest, al-Ḥasan (c. 810) specialized in geometry and wrote The elongated circular figure . Ikhwan al-Safa' (first half of 10th century) group wrote series 50+ letters on science, philosophy and theology. The first letter is on arithmetic and number theory, the second letter on geometry. Labana of Cordoba (Spain, ca. 10th century) Islamic female mathematicians & secretary of the Umayyad Caliph al-Hakem II; could solve the most complex geometrical and algebraic problems known in her time. Al-Hassar (ca.1100s) Developed the modern mathematical notation for fractions and the digits he uses for the ghubar numerals also cloesly resembles modern Western Arabic numerals. Ibn al-Yasamin (ca. 1100s) first to develop a mathematical notation for algebra Abū al-Hasan ibn Alī al-Qalasādī (1412-1482) Last major medieval Arab mathematician; Pioneer of symbolic algebra

8. Algebra The term algebra is derived from the Arabic term al-jabr in the title of Al-Khwarizmi's Al-jabr wa'l muqabalah . He originally used the term al-jabr to describe the method of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation. Before the fall of Islamic civilization, the Arabs used a fully abstract algebra, where the numbers were spelled out in words (four). They later replaced the words with Arabic numerals (4), but the Arabs never developed a symbolic algebra until the work of Ibn al-Banna al-Marrakushi (13th cent) & Abū al-Hasan ibn Alī al-Qalasādī (15 th cent) There were 4 stages in the development of Algebra : Geometric Stage : where the concepts of algebra are largely geometric Static equation-solving stage : find #s satisfying certain relationships Dynamic function : where motion is an primary idea Abstract Stage : where mathematical structure plays a essential role

9. Geometric Algebra Omar Khayyám (c. 1050-1123) wrote a book on Algebra that went beyond Al-Jabr. Omar Khayyám gave both arithmetic & geometric solutions for quadratic equations, but only gave geometric solutions for general cubic equations (he thought that arithmetic solutions were impossible). His method of solving cubic equations by using intersecting conics had been used by Menaechmus, Archimedes, and Alhazen.However, Omar was abut to generalize the method using only positive roots and didn’t go past the 3 rd degree. He also saw a strong relationship between Geometry and Algebra

10. Static Equation-Solving Algebra Muhammad ibn Mūsā al-Khwārizmī (c. 780-850) was a staff member of the "House of Wisdom“ in Baghdad (established by Al-Mamun). Al-Khwarizmi, wrote more than half a dozen mathematical and astronomical works. One of al-Khwarizmi's most famous books is entitled Al-jabr wa'l muqabalah ( The Compendious Book on Calculation by Completion and Balancing) , about solving polynomials up to the second degree. The book also introduced the fundamental method of "reduction" and "balancing", referring to the cancellation of like terms on opposite sides of the equation. This is the operation which Al-Khwarizmi originally described as al-jabr . 6 Chapters of Al-Jabr Squares equal its roots  (ax 2 = bx) Squares equal a number  (ax 2 = c) Roots equal a number  (bx = c) Squares & roots equal a number  (ax 2 + bx = c) Squares & numbers equal roots  (ax 2 + c = bx) Roots & numbers equal squares  (bx + c = ax 2 ) Arabic Mathematicians were the first to treat irrational #’s as algebraic objects

11. Dynamic Function Algebra 12th century, Sharaf al-Dīn al-Tūsī found algebraic and numerical solutions to cubic equations & was the first to discover the derivative of cubic polynomials. His Treatise on Equations had equations up to the third degree. The treatise did not follow the school of algebra, but was "an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry." The treatise had 25 types of equations, with 12 types of linear equations and quadratic equations, 8 types of cubic equations with positive solutions, & 5 types of cubic equations which may not have positive solutions. He understood the importance of the discriminate of the cubic equation and used an earlier formula to find algebraic solutions to certain types of cubic equations. Sharaf al-Din also developed the concept of a function.

12. Arithmetic Arabic Numerals In the Arab world (until early modern times) the Arabic numeral system was often only used by mathematicians Decimal Fractions decimal fractions were first used five centuries before by the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century. Real Numbers In Middle Ages acceptance of zero, negative, integral and fractional numbers, first by Indian and Chinese, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects, which was made possible by the development of algebra. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers, and they criticized Euclid's idea of ratios, developed the theory of composite ratios, and extended the concept of number to ratios of continuous magnitude

13. Arithmetic Cont… Number Theory Ibn al-Haytham solved problems involving congruences. In his Opuscula , Ibn al-Haytham considers the solution of a system of congruences, and gives two general methods of solution. His first method (canonical method) involved Wilson's theorem, while his second method involved a version of the Chinese remainder theorem. Another contribution to number theory is his work on perfect numbers. In his Analysis and synthesis , Ibn al-Haytham was the first to discover that every even perfect number is of the form 2 n −1(2 n − 1) where 2 n − 1 is prime, but he was not able to prove this result successfully (Euler later proved it in the 18th century). 14th century, Kamāl al-Dīn al-Fārisī made a number of important contributions to number theory. His most impressive work in number theory is on amicable numbers. In Tadhkira al-ahbab fi bayan al-tahabb introduced a major new approach to a whole area of number theory, introducing ideas about factorization and combinatorial methods. In fact, al-Farisi's approach is based on the unique factorization of an integer into powers of prime numbers.

14. Geometry Successors of Muhammad ibn Mūsā al-Khwārizmī (born 780) undertook a organized application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose. Al-Mahani (born 820) conceived the idea of reducing geometrical problems to problems in algebra. Al-Karajii (born 953) completely freed algebra & geometrical operations and replaced them with the arithmetical type of operations. Thabit ibn Qurra (born 836)  positive #s, real #s, intergral calculus, theorems in spherical trigonometry, analytic geometry, and non-Euclidean geometry. He also wrote a book on the composition of ratios. Thabit started a trend which led eventually to the generalization of the number concept. Thabit also made a generalization of the Pythagorean theorem, which he extended to all triangles in general Thabit was critical of the ideas of Plato & Aristotle (especially motion)

15. Calculus Around 1000 AD, Al-Karaji (using mathematical induction), found a proof for the sum of integral cubes. Al-Karaji was praised for being "the first who introduced the theory of algebraic calculus." Shortly afterwards, Ibn al-Haytham, an Iraqi mathematician, was the first to derive the formula for the sum of the fourth powers/degree, and came close to finding a general formula for the integrals of any polynomials. This was fundamental to the development of infinitesimal and integral calculus

16. The Power of Mathematics Mathematics helped bring about… Geometric Art & Architecture Astronomy Geography Physics Cryptography