Ислам Әдемі Заманындағы Математиканың Дамуы

Overview of the role of mathematics in the Golden Age of Islam

Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important progress was made, such as full development of the decimal place value system to include decimal fractions, the first 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. Arabic mathematical knowledge spread through various channels during the medieval era, driven by the practical applications of al Khwārizmī's methods. This dissemination was influenced not only by economic and political factors but also by cultural exchanges, exemplified by events such as the Crusades and the translation movement. The Islamic Golden Age, spanning from the 8th to the 14th century, marked a period of considerable advancements in various scientific disciplines, attracting scholars from medieval Europe seeking access to this knowledge. Trade routes and cultural interactions played a crucial role in introducing Arabic mathematical ideas to the West. The translation of Arabic mathematical texts, along with Greek and Roman works, during the 14th to 17th century, played a pivotal role in shaping the intellectual landscape of the Renaissance.

Arabic mathematics, particularly algebra, developed significantly during the medieval period. Muhammad ibn Musa al Khwārizmī's (محمد بن موسى الخوارزمي; c. 780 – c. 850) work between AD 813 and 833 in Baghdad was a turning point. He introduced the term "algebra" in the title of his book, "Kitab al jabr wa al muqabala," marking it as a distinct discipline. He regarded his work as "a short work on Calculation by (the rules of) Completion and Reduction, confining it to what is easiest and most useful in arithmetic". Later, people commented his work was not just a theoretical treatise but also practical, aimed at solving problems in areas like commerce and land measurement. Al Khwārizmī's approach was groundbreaking in that it did not arise from any previous "arithmetical" tradition, including that of Diophantus. He developed a new vocabulary for algebra, distinguishing between purely algebraic terms and those shared with arithmetic. Al Khwārizmī noticed that the representation of numbers is crucial in daily life. Thus, he wanted to find or summarize a way to simplify the mathematical operation, so called later, the algebra. Al Khwārizmī's method, which involved completing the square, not only provided a practical solution for equations of this type but also introduced an abstract and generalized approach to mathematical problems. His work, encapsulated in his seminal text "Al Kitab al Mukhtasar fi Hisab al Jabr wal Muqabala" (The Compendious Book on Calculation by Completion and Balancing), was translated into Latin in the 12th century. This translation played a pivotal role in the transmission of algebraic knowledge to Europe, significantly influencing mathematicians during the Renaissance and shaping the evolution of modern mathematics. Al Khwārizmī's algebra was an autonomous discipline with its historical perspective, eventually leading to the "arithmetization of algebra". His successors expanded on his work, adapting it to new theoretical and technical challenges and reorienting it towards a more arithmetical direction for abstract algebraic calculation. Arabic mathematics, epitomized by al Khwārizmī's work, was crucial in shaping the mathematical landscape. Its spread to the West was driven by its practical applications, the expansion of mathematical concepts by his successors, and the translation and adaptation of these ideas into the Western context. This spread was a complex process involving economics, politics, and cultural exchange, greatly influencing Western mathematics. The period known as the Islamic Golden Age (8th to 14th century) was characterized by significant advancements in various fields, including mathematics. Scholars in the Islamic world made substantial contributions to mathematics, astronomy, medicine, and other sciences. As a result, the intellectual achievements of Islamic scholars attracted the attention of scholars in medieval Europe who sought to access this wealth of knowledge. Trade routes, such as the Silk Road, facilitated the movement of goods, ideas, and knowledge between the East and West. Cities like Baghdad, Cairo, and Cordoba became centers of learning and attracted scholars from different cultural backgrounds. Therefore, mathematical knowledge from the Islamic world found its way to Europe through various channels. Meanwhile, the Crusades connected Western Europeans with the Islamic world. While the primary purpose of the Crusades was military, there was also cultural exchange and exposure to Islamic knowledge, including mathematics. European scholars who traveled to the Holy Land and other parts of the Islamic world gained access to Arabic manuscripts and mathematical treatises. During the 14th to 17th century, the translation of Arabic mathematical texts, along with Greek and Roman ones, played a crucial role in shaping the intellectual landscape of the Renaissance. Figures like Fibonacci, who studied in North Africa and the Middle East, helped introduce and popularize Arabic numerals and mathematical concepts in Europe.

The study of algebra, the name of which is derived from the Arabic word meaning completion or "reunion of broken parts", flourished during the Islamic golden age. Muhammad ibn Musa al Khwarizmi, a Persian scholar in the House of Wisdom in Baghdad was the founder of algebra, is along with the Greek mathematician Diophantus, known as the father of algebra. In his book The Compendious Book on Calculation by Completion and Balancing, Al Khwarizmi deals with ways to solve for the positive roots of first and second degree (linear and quadratic) polynomial equations. He introduces the method of reduction, and unlike Diophantus, also gives general solutions for the equations he deals with. Al Khwarizmi's algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the algebraic work of Diophantus, which was syncopated, meaning that some symbolism is used. The transition to symbolic algebra, where only symbols are used, can be seen in the work of Ibn al Banna' al Marrakushi and Abū al Ḥasan ibn ʿAlī al Qalaṣādī. Several other mathematicians during this time period expanded on the algebra of Al Khwarizmi. Abu Kamil Shuja' wrote a book of algebra accompanied with geometrical illustrations and proofs. He also enumerated all the possible solutions to some of his problems. Abu al Jud, Omar Khayyam, along with Sharaf al Dīn al Tūsī, found several solutions of the cubic equation. Omar Khayyam found the general geometric solution of a cubic equation.

Omar Khayyam (c. 1038/48 in Iran – 1123/24) wrote the Treatise on Demonstration of Problems of Algebra containing the systematic solution of cubic or third order equations, going beyond the Algebra of al Khwārizmī. Khayyám obtained the solutions of these equations by finding the intersection points of two conic sections. This method had been used by the Greeks, but they did not generalize the method to cover all equations with positive roots. Sharaf al Dīn al Ṭūsī (? in Tus, Iran – 1213/4) developed a novel approach to the investigation of cubic equations—an approach which entailed finding the point at which a cubic polynomial obtains its maximum value. For example, to solve the equation , with a and b positive, he would note that the maximum point of the curve occurs at , and that the equation would have no solutions, one solution or two solutions, depending on whether the height of the curve at that point was less than, equal to, or greater than a. His surviving works give no indication of how he discovered his formulae for the maxima of these curves. Various conjectures have been proposed to account for his discovery of them.

The earliest implicit traces of mathematical induction can be found in Euclid's proof that the number of primes is infinite (c. 300 BCE). The first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique (1665). In between, implicit proof by induction for arithmetic sequences was introduced by al Karaji (c. 1000) and continued by al Samaw'al, who used it for special cases of the binomial theorem and properties of Pascal's triangle.

The Greeks had discovered irrational numbers, but were not happy with them and only able to cope by drawing a distinction between magnitude and number. In the Greek view, magnitudes varied continuously and could be used for entities such as line segments, whereas numbers were discrete. Hence, irrationals could only be handled geometrically; and indeed Greek mathematics was mainly geometrical. Islamic mathematicians including Abū Kāmil Shujāʿ ibn Aslam and Ibn Tahir al Baghdadi slowly removed the distinction between magnitude and number, allowing irrational quantities to appear as coefficients in equations and to be solutions of algebraic equations. They worked freely with irrationals as mathematical objects, but they did not examine closely their nature. In the twelfth century, Latin translations of Al Khwarizmi's Arithmetic on the Indian numerals introduced the decimal positional number system to the Western world. His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations. In Renaissance Europe, he was considered the original inventor of algebra, although it is now known that his work is based on older Indian or Greek sources. He revised Ptolemy's Geography and wrote on astronomy and astrology. However, C. A. Nallino suggests that al Khwarizmi's original work was not based on Ptolemy but on a derivative world map, presumably in Syriac or Arabic.

The spherical law of sines was discovered in the 10th century: it has been attributed variously to Abu Mahmud Khojandi, Nasir al Din al Tusi and Abu Nasr Mansur, with Abu al Wafa' Buzjani as a contributor. Ibn Muʿādh al Jayyānī's The book of unknown arcs of a sphere in the 11th century introduced the general law of sines. The plane law of sines was described in the 13th century by Nasīr al Dīn al Tūsī. In his On the Sector Figure, he stated the law of sines for plane and spherical triangles and provided proofs for this law.

In the 9th century, Islamic mathematicians were familiar with negative numbers from the works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid. Al Khwarizmi did not use negative numbers or negative coefficients. Al Karaji wrote in his book al Fakhrī that "negative quantities must be counted as terms".