ISLAMIC MATHEMATICS. Buket Çökelek. 23 October 2017

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1 ISLAMIC MATHEMATICS Buket Çökelek 23 October 2017 Islamic Mathematics is the term used to refer to the mathematics done in the Islamic world between the 8th and 13th centuries CE. Mathematics from the medieval Middle East is very important to the mathematics we use today. While Europe endured its Dark Ages, the Middle East preserved and expanded the arithmetic, geometry, trigonometry, and astronomy from the ancient Greek philosophers, such as Euclid. The most important contribution may be the invention of algebra, which originated in Baghdad in the House of Wisdom (bayt alhikma). The House of Wisdom The House of Wisdom was primarily a library and a place for translation and research. Scholars would work here in translating Greek and Hindu treatises to Arabic, and also conducted their own research and wrote original treatises. The House of Wisdom was established in the early 9th century, by Caliph alrashid. His son, Caliph al-ma mun, was the ruler who made the House of Wisdom so important. Al-Ma mun had a dream in which Aristotle appeared to him; after this dream, alma mun wanted to translate as many Greek manuscripts as he could! He commissioned scholars to begin translating Greek, Hindu, Syriac-Persian, and Hebrew texts into Arabic. Most of these texts dealt with philosophy or mathematics and science. The concepts of zero Mūsā al-khawarizmi real issue remains with the number zero. It cannot be proven to exist using math. The old Indian texts insist zero divided by zero equals zero. But al-khawarizmi knows that any division by zero is impossible. Eventually he comes to the conclusion that the zero must simply be accepted without being proven. Furthermore, he reports to the Caliph al- Ma mun that belief in Allah is the same: it cannot be proven using science, but must be accepted on faith in the religion. Al-Khawarizmi was as much a philosopher as he was a mathematician. Algebraic Operations

2 Al-Khwārizmī wrote a treatise entitled Kitab aljabr wa l-muqabalah. The treatise actually had a very practical reason behind it: the longest chapter of the treatise teaches people how to apply algebra to Islamic inheritance laws! The words al-jabr and al-muqabalah were operations used by Al-Khwārizmī, much like addition, subtraction, multiplication, and division. Al-jabr means something like restoration or completion, and was the operation used to add equal terms to both sides of an equation to get rid of a negative term. Geometry Ibn Sinān Ibrāhīm ibn Sinān (d. 946) is the grandson of Thābit ibn Qurra, the famous mathematician and translator of Archimedes. His treatment of the area of a segment of a parabola is the simplest construction from the time before the Renaissance. He wrote that he invented the proof out of necessity, to save his family's scientific reputation after hearing accusations that his grandfather's method was too long-winded! He also was more concerned with general methods and theories than with particular problems. Geometry in Art Much of the geometry done in the Islamic world was concerned with flat, 2-dimensional, figures, and was primarily concerned with the practical uses of geometry in art. Because Islamic art created flat designs and patterns, in opposition to the perspective art which emerged in the West in the renaissance period, Islamic geometry did not develop to the modern geometry which describes surfaces and 3-dimentional figures. Islamic art used a lot of geometry, so mathematicians such as Abu Nasr al-farabi wrote treatises on how to solve geometrical common problems for artists. Abu Nasr al-farabi taught philosophy in both Baghdad and Apello (in northern Syria), and was killed by highway robbers outside Damascus in 950 CE. He wrote a treatise called A Book of Spiritual Crafts and Natural Secrets in the Details of Geometrical Figures, in which he taught artists how to solve the following three problems, which helped artist produce infinite patterns in their designs and to fit their patterns into specific areas. 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

3 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. 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 Number Theory Ibn al-haytham solved problems involving congruences. In his Opuscula, he 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, was the first to discover that every even perfect number is of the form 2n 1(2n 1) where 2n 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. Number Theory 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

4 methods. In fact, al-farisi s approach is based on the unique factorization of an integer into powers of prime numbers. Al-Khwārizmī Muhammad ibn Mūsā al-khwārizmī( ) Al-Khwārizmī Muhammad ibn Mūsā al-khwārizmī is probably the most famous Muslim mathematician. He lived about CE. AlKhwārizmī was born in Qutrubull, an area near Baghdad between the Tigris and Euphrates rives, but was brought to work at the House of Wisdom by the Caliph al-ma mun. He popularized a number of mathematical concepts, including the use of HinduArabic numbers and the number zero, algebra, and the use of geometry to demonstrate and prove algebraic results. Many of his works deal with astronomy, but he also wrote about the Jewish calendar, arithmetic, and algebra. Thabit ibn Qurra( ) Thabit ibn Qurra was an Arab Sabian mathematician, physician, astronomer, and translator who lived in Baghdad in the second half of the ninth century during the time of Abbasid Caliphate. In mathematics, Thabit discovered an equation for determining amicable numbers. He also wrote on the theory of numbers, and extended their use to describe the ratios between geometrical quantities, a step which the Greeks did not take.he is known for having calculated the solution to a chessboard problem involving an exponential series. He also described a Pythagoras theorem. (Amicable Numbers: Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive factor of that number other than the number itself.) Omar Khayyam( ) Khayyam s most famous works include his highly influential mathematical treatise called Treatise on Demonstration of Problems of Algebra which he completed in This treatise highlighted the basic algebraic principles that were ultimately shifted to Europe. He

5 laid the foundation of the Pascal s triangle with his work on triangular array of binomial coefficients. In 1077 another major work was written by Khayyam namely Sharh ma ashkala min musadarat kitab Uqlidis meaning Explanations of the Difficulties in the Postulates of Euclid. It was published in English as On the Difficulties of Euclid s Definitions. In this book he contributed to non-euclidean geometry even though this was not his original plan. It is said that Omar Khayyam was originally trying to prove the parallels postulate when he proven the properties of figures in the non-euclidean geometry. His geometrical work consisted of his efforts on the theory of proportion and geometrical algebra topics such as cubic equations. Khayyam was the first mathematican to consider the Saccheri quadrilateral in the 11th century. It was mentioned in his book the Explanations of the difficulties in the postulates of Euclid. It wasn t until 6 centuries later when another mathematician, Giordano Vitale made further advances on Khayyam s theory. Other books by Khayyam include his book named Problems of Arithmetic, a book on music and algebra. Khayyam, like the other Persian mathematicians of the time was also an astronomer. The Sultan Jalal ud Din Malik Shah Saljuqi requested him to build an observatory with a team of scientists. He was part of the team that made several reforms to the Iranian calendar which was made the official Persian calendar to be followed by the Sultan on March 15th The Jalali Calendar became the base for other calendars and is also known to be more accurate than the Gregorian calendar. References [1] The Center for South Asian and Middle Eastern Studies, University of Illinois at Urbana- Champaign < > [2] Mathematics in the islamic worlds

6 < [3] Islamic Mathematics - The Story of Mathematics <

Islamic Mathematics. Aybars Hallik. 23 October, 2017

Islamic Mathematics. Aybars Hallik. 23 October, 2017 Islamic Mathematics Aybars Hallik 23 October, 2017 Islamic Mathematics is the term used to refer to the mathematics done in the Islamic world between the 8th and 13th centuries CE. Mathematics from the