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Encyclopedia > Islamic mathematics

In the history of mathematics, Islamic mathematics or Arabic mathematics refers to the mathematics developed by the Islamic civilization between 622 and 1600. While most scientists in this period were Muslims and Arabic was the dominant language, contributions were made by people of different ethnic groups (Arabs, Persians, Turks, Moors) and religions (Muslims, Christians, Jews, Zoroastrians).[1] The center of Islamic mathematics was located in present-day Iraq and Iran, but at its greatest extent stretched from Turkey, North Africa and Spain in the west, to the border of China in the east.[2] For a timeline of events in mathematics, see timeline of mathematics. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For people named Islam, see Islam (name). ... Events Hijra - Muhammad and his followers withdraw from Mecca to Medina - year one of the Islamic calendar. ... 1600 was a leap year starting on Saturday of the Gregorian calendar (or a leap year starting on Tuesday of the 10-day slower Julian calendar). ... There is also a collection of Hadith called Sahih Muslim A Muslim (Arabic: مسلم, Persian: Mosalman or Mosalmon Urdu: مسلمان, Turkish: Müslüman, Albanian: Mysliman, Bosnian: Musliman) is an adherent of the religion of Islam. ... Arabic redirects here. ... For other uses, see Arab (disambiguation). ... This article is about the Persian people, an ethnic group found mainly in Iran. ... For other uses, see moor. ... There is also a collection of Hadith called Sahih Muslim A Muslim (Arabic: مسلم, Persian: Mosalman or Mosalmon Urdu: مسلمان, Turkish: Müslüman, Albanian: Mysliman, Bosnian: Musliman) is an adherent of the religion of Islam. ... For other uses, see Christian (disambiguation). ... Zoroastrianism is the religion and philosophy based on the teachings ascribed to the prophet Zoroaster (Zarathustra, Zartosht). ...  Northern Africa (UN subregion)  geographic, including above North Africa or Northern Africa is the northernmost region of the African continent, separated by the Sahara from Sub-Saharan Africa. ...


Islamic science and mathematics flourished under the Islamic caliphate (also known as the Arab Empire or Islamic Empire) established across the Middle East, Central Asia, North Africa, Sicily, the Iberian Peninsula, and in parts of France and Pakistan (known as India at the time) in the 8th century. Although most Islamic texts on mathematics were written in Arabic, they were not all written by Arabs, since—much like Latin in Medieval Europe—Arabic was used as the written language of scholars throughout the Islamic world at the time. In particular, a large number of Islamic scientists in many disciplines, including mathematics, were Persians.[3] In the history of science, Islamic science refers to the science developed under the Islamic civilisation between the 8th and 15th centuries (the Islamic Golden Age). ... For main article see: Caliphate The Caliph (pronounced khaleef in Arabic) is the head of state in a Caliphate, and the title for the leader of the Islamic Ummah, an Islamic community ruled by the Sharia. ... The Arab Empire at its greatest extent The Arab Empire usually refers to the following Caliphates: Rashidun Caliphate (632 - 661) Umayyad Caliphate (661 - 750) - Successor of the Rashidun Caliphate Umayyad Emirate in Islamic Spain (750 - 929) Umayyad Caliphate of Córdoba in Islamic Spain (929 - 1031) Abbasid Caliphate (750-1258... Template:Islamic Empire infobox The Ottoman Empire (1299 - 29 October 1923) (Ottoman Turkish: Devlet-i Aliye-yi Osmaniyye; literally, The Sublime Ottoman State, modern Turkish: Osmanlı Ä°mparatorluÄŸu), is also known in the West as the Turkish Empire. ... A map showing countries commonly considered to be part of the Middle East The Middle East is a region comprising the lands around the southern and eastern parts of the Mediterranean Sea, a territory that extends from the eastern Mediterranean Sea to the Persian Gulf. ... Map of Central Asia showing three sets of possible boundaries for the region Central Asia located as a region of the world Central Asia is a vast landlocked region of Asia. ...  Northern Africa (UN subregion)  geographic, including above North Africa or Northern Africa is the northernmost region of the African continent, separated by the Sahara from Sub-Saharan Africa. ... Sicily ( in Italian and Sicilian) is an autonomous region of Italy and the largest island in the Mediterranean Sea, with an area of 25,708 km² (9,926 sq. ... The Iberian Peninsula, or Iberia, is located in the extreme southwest of Europe, and includes modern day Spain, Portugal, Andorra and Gibraltar. ... For other uses, see Arab (disambiguation). ... Latin was the language originally spoken in the region around Rome called Latium. ... The Middle Ages formed the middle period in a traditional schematic division of European history into three ages: the classical civilization of Antiquity, the Middle Ages, and modern times, beginning with the Renaissance. ... The Islamic world is the world-wide community of those who identify with Islam, known as Muslims, and who number approximately one-and-a-half billion people. ...


J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive: The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ...

"Recent research paints a new picture of the debt that we owe to Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the 16th, 17th, and 18th centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects, the mathematics studied today is far closer in style to that of Islamic mathematics than to that of Greek mathematics." For other uses, see Europe (disambiguation). ... Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. ...

R. Rashed wrote in The development of Arabic mathematics: between arithmetic and algebra:

"Al-Khwarizmi's successors undertook a systematic 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." (Arabic: ) was a Persian[1] mathematician, astronomer, astrologer and geographer. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ... This article is about the branch of mathematics. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... For other uses, see Euclid (disambiguation). ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... For other uses, see Geometry (disambiguation). ... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. ... Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ...

Contents

Origins and influences

The first century of the Islamic Arab Empire saw almost no scientific or mathematical achievements since the Arabs, with their newly conquered empire, had not yet gained any intellectual drive and research in other parts of the world had faded. In the second half of the eighth century Islam had a cultural awakening, and research in mathematics and the sciences increased.[4] The Muslim Abbasid caliph al-Mamun (809-833) is said to have had a dream where Aristotle appeared to him, and as a consequence al-Mamun ordered that Arabic translation be made of as many Greek works as possible, including Ptolemy's Almagest and Euclid's Elements. Greek works would be given to the Muslims by the Byzantine Empire in exchange for treaties, as the two empires held an uneasy peace.[4] Many of these Greek works were translated by Thabit ibn Qurra (826-901), who translated books written by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.[5] Historians are in debt to many Islamic translators, for it is through their work that many ancient Greek texts have survived only through Arabic translations. For people named Islam, see Islam (name). ... The Arab Empire at its greatest extent The Arab Empire usually refers to the following Caliphates: Rashidun Caliphate (632 - 661) Umayyad Caliphate (661 - 750) - Successor of the Rashidun Caliphate Umayyad Emirate in Islamic Spain (750 - 929) Umayyad Caliphate of Córdoba in Islamic Spain (929 - 1031) Abbasid Caliphate (750-1258... Mashriq Dynasties  Maghrib Dynasties  The Abbasid Caliphate Abbasid (Arabic: , ) is the dynastic name generally given to the caliph of Baghdad, the second of the two great Sunni dynasties of the Arab Empire, that overthrew the Umayyad caliphs from all but Spain. ... For main article see: Caliphate The Caliph (pronounced khaleef in Arabic) is the head of state in a Caliphate, and the title for the leader of the Islamic Ummah, an Islamic community ruled by the Sharia. ... Abu Jafar al-Mamun ibn Harun (786 - 833) (المأمون) was an Abbasid caliph who reigned from 813 until his death in 833. ... Byzantine redirects here. ... Abul Hasan Thabit ibn Qurra ibn Marwan al-Sabi al-Harrani, (826 – February 18, 901) was an Arab astronomer and mathematician. ... For other uses, see Euclid (disambiguation). ... For other uses, see Archimedes (disambiguation). ... Apollonius of Perga [Pergaeus] (ca. ... This article is about the geographer, mathematician and astronomer Ptolemy. ... Arabic can mean: From or related to Arabia From or related to the Arabs The Arabic language; see also Arabic grammar The Arabic alphabet, used for expressing the languages of Arabic, Persian, Malay ( Jawi), Kurdish, Panjabi, Pashto, Sindhi and Urdu, among others. ...


Greek, Indian, and Mesopotamian mathematics all played an important role in the development of early Islamic mathematics. The works of mathematicians such as Euclid, Apollonius, Archimedes, Diophantus, Aryabhata and Brahmagupta were all acquired by the Islamic world and incorporated into their mathematics. Perhaps the most influential mathematical contribution from India was the decimal place-value Indo-Arabic numeral system, also known as the Hindu numerals.[6] The Persian historian al-Biruni (c. 1050) in his book Tariq al-Hind states that the Abbasid caliph al-Ma'mun had an embassy in India from which was brought a book to Baghdad that was translated into Arabic as Sindhind. It is generally assumed that Sindhind is none other than Brahmagupta's Brahmasphuta-siddhanta.[7] The earliest translations from Sanskrit inspired several astronomical and astrological Arabic works, now mostly lost, some of which were even composed in verse.[8] This is an article about the ancient middle eastern region. ... For other uses, see Euclid (disambiguation). ... Apollonius of Perga [Pergaeus] (ca. ... For other uses, see Archimedes (disambiguation). ... Title page of the 1621 edition of Diophantus Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac. ... For other uses, see Aryabhata (disambiguation). ... Brahmagupta (ब्रह्मगुप्त) ( ) (589–668) was an Indian mathematician and astronomer. ... The place value system is a method of writing numbers with a base 10 numerical system. ... I like cream cheese, it tastes good on toast. ... This article is about the Hindu concept of numeration. ... This article is about the Persian people, an ethnic group found mainly in Iran. ... A statue of Biruni adorns the southwest entrance of Laleh Park in Tehran. ... Mashriq Dynasties  Maghrib Dynasties  The Abbasid Caliphate Abbasid (Arabic: , ) is the dynastic name generally given to the caliph of Baghdad, the second of the two great Sunni dynasties of the Arab Empire, that overthrew the Umayyad caliphs from all but Spain. ... For main article see: Caliphate The Caliph (pronounced khaleef in Arabic) is the head of state in a Caliphate, and the title for the leader of the Islamic Ummah, an Islamic community ruled by the Sharia. ... Abu Jafar al-Mamun ibn Harun (also spelled Almanon and el-Mâmoûn) (786 – October 10, 833) (المأمون) was an Abbasid caliph who reigned from 813 until his death in 833. ... The main work of Brahmagupta, Brahmasphutasiddhanta (The Opening of the Universe), written in 628, contains some remarkably advanced ideas, including a good understanding of the mathematical role of zero, rules for manipulating both positive and negative numbers, a method for computing square roots, methods of solving linear and some quadratic...


But Indian influences were soon overwhelmed by Greek mathematical and astronomical texts. It is not clear why this occurred but it may have been due to the greater availability of Greek texts in the region, the larger number of practitioners of Greek mathematics in the region, or because Islamic mathematicians favored the deductive exposition of the Greeks over the elliptic Sanskrit verse of the Indians. Regardless of the reason, Indian mathematics soon became mostly eclipsed by or merged with the "Graeco-Islamic" science founded on Hellenistic treatises.[8]


Biographies

Al-Ḥajjāj ibn Yūsuf ibn Maṭar (786 – 833)
Al-Ḥajjāj translated Euclid's Elements into Arabic.
Muḥammad ibn Mūsā al-Khwārizmī (c. 780 Khwarezm/Baghdad – c. 850 Baghdad)
Al-Khwārizmī was a mathematician, astronomer, astrologer and geographer. He worked most of his life as a scholar in the House of Wisdom in Baghdad. His Algebra was the first book on the systematic solution of linear and quadratic equations. Latin translations of his Arithmetic, on the Indian numerals, introduced the decimal positional number system to the Western world in the 12th century. He revised and updated Ptolemy's Geography as well as writing several works on astronomy and astrology.
Al-ʿAbbās ibn Saʿid al-Jawharī (c. 800 Baghdad? – c. 860 Baghdad?)
Al-Jawharī was a mathematician who worked at the House of Wisdom in Baghdad. His most important work was his Commentary on Euclid's Elements which contained nearly 50 additional propositions and an attempted proof of the parallel postulate.
ʿAbd al-Hamīd ibn Turk (fl. 830 Baghdad)
Ibn Turk wrote a work on algebra of which only a chapter on the solution of quadratic equations has survied.
Yaʿqūb ibn Isḥāq al-Kindī (c. 801 Kufah – 873 Baghdad)
Al-Kindī (or Alkindus) was a philosopher and scientist who worked as the House of Wisdom in Baghdad where he wrote commentaries on many Greek works. His contributions to mathematics include many works on arithmetic and geometry.
Hunayn ibn Ishaq (808 Al-Hirah – 873 Baghdad)
Hunayn (or Johannitus) was a translator who worked at the House of Wisdom in Baghdad. Translated many Greek works including those by Plato, Aristotle, Galen, Hippocrates, and the Neoplatonists.
Banū Mūsā (c. 800 Baghdad – 873+ Baghdad)
The Banū Mūsā where three brothers who worked at the House of Wisdom in Baghdad. Their most famous mathematical treatise is The Book of the Measurement of Plane and Spherical Figures, which considered similar problems as Archimedes did in his On the measurement of the circle and On the sphere and the cylinder. They contributed individually as well. The eldest, Jaʿfar Muḥammad (c. 800) specialised in geometry and astronomy. He wrote a critical revision on Apollonius' Conics called Premises of the book of conics. Aḥmad (c. 805) specialised in mechanics and wrote a work on pneumatic devices called On mechanics. The youngest, al-Ḥasan (c. 810) specialised in geometry and wrote a work on the ellipse called The elongated circular figure.
Al-Mahani
Ahmed ibn Yusuf
Thabit ibn Qurra (Syria-Iraq, 835-901)
Al-Hashimi (Iraq? ca. 850-900)
Muḥammad ibn Jābir al-Ḥarrānī al-Battānī (c. 853 Harran – 929 Qasr al-Jiss near Samarra)
Abu Kamil (Egypt? ca. 900)
Sinan ibn Tabit (ca. 880 - 943)
Al-Nayrizi
Ibrahim ibn Sinan (Iraq, 909-946)
Al-Khazin (Iraq-Iran, ca. 920-980)
Al-Karabisi (Iraq? 10th century?)
Ikhwan al-Safa' (Iraq, first half of 10th century)
The Ikhwan al-Safa' ("brethren of purity") were a (mystical?) group in the city of Basra in Irak. The group authored a series of more than 50 letters on science, philosophy and theology. The first letter is on arithmetic and number theory, the second letter on geometry.
Al-Uqlidisi (Iraq-Iran, 10th century)
Al-Saghani (Iraq-Iran, ca. 940-1000)
Abū Sahl al-Qūhī (Iraq-Iran, ca. 940-1000)
Al-Khujandi
Abū al-Wafāʾ al-Būzjānī (Iraq-Iran, ca. 940-998)
Ibn Sahl (Iraq-Iran, ca. 940-1000)
Al-Sijzi (Iran, ca. 940-1000)
Ibn Yunus (Egypt, ca. 950-1010)
Abu Nasr ibn `Iraq (Iraq-Iran, ca. 950-1030)
Kushyar ibn Labban (Iran, ca. 960-1010)
Al-Karaji (Iran, ca. 970-1030)
Ibn al-Haytham (Iraq-Egypt, ca. 965-1040)
Abū al-Rayḥān al-Bīrūnī (September 15, 973 in Kath, KhwarezmDecember 13, 1048 in Gazna)
Ibn Sina
al-Baghdadi
Al-Nasawi
Al-Jayyani (Spain, ca. 1030-1090)
Ibn al-Zarqalluh (Azarquiel, al-Zarqali) (Spain, ca. 1030-1090)
Al-Mu'taman ibn Hud (Spain, ca. 1080)
al-Khayyam (Iran, ca. 1050-1130)
Ibn Yaḥyā al-Maghribī al-Samawʾal (c. 1130 Baghdad – c. 1180 Maragha)
Sharaf al-Dīn al-Ṭūsī (Iran, ca. 1150-1215)
Ibn Mun`im (Maghreb, ca. 1210)
al-Marrakushi (Morocco, 13th century)
Naṣīr al-Dīn al-Ṭūsī (18 February 1201 in Tus, Khorasan26 June 1274 in Kadhimain near Baghdad)
Muḥyi al-Dīn al-Maghribī (c. 1220 Spain – c. 1283 Maragha)
Shams al-Dīn al-Samarqandī (c. 1250 Samarqand – c. 1310)
Ibn Baso (Spain, ca. 1250-1320)
Ibn al-Banna' (Maghreb, ca. 1300)
Kamal al-Din Al-Farisi (Iran, ca. 1300)
Al-Khalili (Syria, ca. 1350-1400)
Ibn al-Shatir (1306-1375)
Qāḍī Zāda al-Rūmī (1364 Bursa – 1436 Samarkand)
Jamshīd al-Kāshī (Iran, Uzbekistan, ca. 1420)
Ulugh Beg (Iran, Uzbekistan, 1394-1449)
Al-Umawi
Al-Qalasadi (Maghreb, 15th century)
Lotfi Asker Zadeh (Iran, 20th century)

(786–833) was an Arab mathematician who made the first translation of Euclids Elements from Greek into Arabic. ... For other uses, see Euclid (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... al-KhwārizmÄ« redirects here. ... After Islamic Conquest  Modern SSR = Soviet Socialist Republic Afghanistan  Azerbaijan  Bahrain  Iran  Iraq  Tajikistan  Uzbekistan  This box:      Khwarezm was a series of states centered on the Amu Darya river delta of the former Aral Sea, in modern Uzbekistan, extending across the Ust-Urt plateau and possibly as far west as... Baghdad (Arabic: ) is the capital of Iraq and of Baghdad Governorate. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... For other uses, see Astronomy (disambiguation). ... Hand-coloured version of the anonymous Flammarion woodcut (1888). ... A geographer is a crazy psycho whose area of study is geocrap, the pseudoscientific study of Earths physical environment and human habitat and the study of boring students to death. ... A scholar is either a student or someone who has achieved a mastery of some academic discipline, perhaps receiving financial support through a scholarship. ... The House of Wisdom (Arabic بيت الحكمة Bayt al-Hikma) was a library and translation institute in Abbassid-era Baghdad. ... Baghdad (Arabic: ) is the capital of Iraq and of Baghdad Governorate. ... Graph sample of linear equations A linear equation is an algebraic equation in which each term is either a constant or the product of a constant times the first power of a variable. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... For other uses, see Latin (disambiguation). ... India has produced many numeral systems. ... For other uses, see Decimal (disambiguation). ... A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system. ... Occident redirects here. ... This article is about the geographer, mathematician and astronomer Ptolemy. ... (c. ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems... This article is about the word proposition as it is used in logic, philosophy, and linguistics. ... Look up proof in Wiktionary, the free dictionary. ... a and b are parallel, the transversal t produces congruent angles. ... (Turkey/Iraq?; fl. ... This article is about the branch of mathematics. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... For the Christian theologian, see Abd al-Masih ibn Ishaq al-Kindi. ... Kufa (الكوفة al-Kufa in Arabic) is a city in Iraq, about 170 km south of Baghdad, and 10 km northeast of Najaf. ... A philosopher is a person who thinks deeply regarding people, society, the world, and/or the universe. ... This article is about the profession. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ... For other uses, see Geometry (disambiguation). ... Hunayn ibn Ishaq al-Ibadi (809—873) was Nestorian physician in the House of Wisdom. ... A manuscript from the 15th century describing the constructing of Al-Khornaq castle In Al-Hira,The Lakhmids capital city Al HÄ«ra (Arabic,الحيرة) was an ancient city located south of al-Kufah in south-central Iraq. ... For other uses, see Plato (disambiguation). ... For other uses, see Aristotle (disambiguation). ... For other uses, see Galen (disambiguation). ... For other uses, see Hippocrates (disambiguation). ... Wikipedia does not yet have an article with this exact name. ... It has been suggested that Ahmad ibn MÅ«sā ibn Shākir be merged into this article or section. ... For other uses, see Archimedes (disambiguation). ... It has been suggested that this article or section be merged into BanÅ« MÅ«sā. (Discuss) Ja‘far Muḥammad ibn MÅ«sā ibn Shākir (800 - 873) (Arabic: ) was a 9th century Persian astronomer, engineer, mathematician and physicist from Baghdad, the eldest of the BanÅ« MÅ«sā brothers. ... Apollonius of Perga [Pergaeus] (ca. ... It has been suggested that this article or section be merged into BanÅ« MÅ«sā. (Discuss) Drawing of Self trimming lamp in Ahmad ibn Musa ibn Shakirs treatise on mechanical devices. ... Pneumatics, from the Greek πνευματικός (pneumatikos, coming from the wind) is the use of pressurized air in science and technology. ... It has been suggested that this article or section be merged into BanÅ« MÅ«sā. (Discuss) Al-Hasan ibn MÅ«sā ibn Shākir (810–873) (Arabic: ) was a 9th century Persian mathematician and astronomer who lived in Baghdad. ... For other uses, see Ellipse (disambiguation). ... Abu Abdallah Mohammed ibn Isa al-Mahani, was a Persian mathematician and astronomer from Mahan, Kerman, Persia. ... Ahmed ibn Yusuf al-Misri (835 - 912) was a mathematician, like his father Yusuf ibn Ibrahim. ... Abul Hasan Thabit ibn Qurra ibn Marwan al-Sabi al-Harrani, (826 – February 18, 901) was an Arab astronomer and mathematician. ... Albategnius redirects here. ... Harran, also known as Carrhae, is a district of Åžanlıurfa Province in the southeast of Turkey, near the border with Syria, 24 miles (44 kilometres) southeast of the city of Åžanlıurfa, at the end of a long straight road across the roasting hot plain of Harran. ... Map showing Samarra near Baghdad Sāmarrā (سامراء) is a town in Iraq ( ). It stands on the east bank of the Tigris in the Salah ad Din Governorate, 125 km north of Baghdad and, in 2002, had an estimated population of 201,700. ... Abu Kamil Shuja ibn Aslam ibn Muhammad ibn Shuja (c. ... Sinan ibn Thabit ibn Qurra (Arabic,سنان بن ثابت بن قرة ) was an Arab physician, mathematician and astronomer. ... Abu-l-Abbas al-Fadl ibn Hatim al-Nairizi, Latin name: Anaritius, was a 9-10th century Persian mathematician and astronomer from Nayriz, a town near Shiraz, Fars, Iran. ... Ibrahim ibn Sinan ibn Thabit ibn Qurra (908, Baghdad – 946, Baghdad) was an Arab mathematician and astronomer who studied geometry and in particular tangents to circles. ... Abu Jafar Muhammad ibn al-Hasan Al-Khazini (900-971), was a Persian astronomer and mathematician from Khorasan. ... Arabic manuscript from the 12th century for Brethren of Purity (Arabic , Ikhwan al-Safa اخوان الصفا) The Brethren of Purity (Arabic اخوان الصفا Ikhwan al-Safa; also translated as Brethren of Sincerity) were an obscure and mysterious[1] organization[2] of Arab[3] philosophers in Basra, Iraq - which was then the seat of the... Abul Hasan Ahmad ibn Ibrahim Al-Uqlidisi was an Arab mathematician, possibly from Damascus He wrote the earliest surviving book on the Hindu place-value system, known in the west as Arabic numerals, around 952. ... Abu Hamid Ahmed ibn Mohammed al-Saghani al-Asturlabi, i. ... (sometimes ), was a Persian mathematician, physicist and astronomer. ... Abu Mahmud Hamid ibn al-Khidr Al-Khujandi was a Persian astronomer who lived in the late 10th century and helped build an observatory near in what is now Ray, Iran near Tehran. ... Abul Wáfa redirects here. ... Ibn Sahl - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... Abu Said Ahmed ibn Mohammed ibn Abd al-Jalil al-Sijzi (short for al-Sijistani) was a Persian mathematician. ... Ibn Yunus (Arabic: ابن يونس) (full name, Abu al-Hasan Ali abi Said Abd al-Rahman ibn Ahmad ibn Yunus al-Sadafi al-Misri) (c. ... Abu Nasr Mansur ibn Ali ibn Iraq (c. ... Abu-l-Hasan Kushayr ibn Labban ibn Bashahri al-Jili (971 - 1029), was a Persian mathematician, geographer, and astronomer from Jilan, a. ... Abu Bakr ibn Muhammad ibn al-Husayn Al-Karaji (953 - 1029), also known as Al-karkhi was a Persian mathematician and engineer. ... (Arabic: أبو علي الحسن بن الحسن بن الهيثم, Latinized: Alhacen or (deprecated) Alhazen) (965 – 1039), was an Arab[1] Muslim polymath[2][3] who made significant contributions to the principles of optics, as well as to anatomy, astronomy, engineering, mathematics, medicine, ophthalmology, philosophy, physics, psychology, visual perception, and to science in general with his introduction of the... Al-Biruni redirects here. ... is the 258th day of the year (259th in leap years) in the Gregorian calendar. ... Events Edgar of England is crowned king by Saint Dunstan Births September 15 - Al_Biruni, mathematician († 1048) Abu al-Ala al-Maarri, poet Deaths May 7 - Otto I, Holy Roman Emperor Categories: 973 ... After Islamic Conquest  Modern SSR = Soviet Socialist Republic Afghanistan  Azerbaijan  Bahrain  Iran  Iraq  Tajikistan  Uzbekistan  This box:      Khwarezm was a series of states centered on the Amu Darya river delta of the former Aral Sea, in modern Uzbekistan, extending across the Ust-Urt plateau and possibly as far west as... is the 347th day of the year (348th in leap years) in the Gregorian calendar. ... Events The city of Oslo is founded by Harald Hardråde of Norway. ... , Ghazni City (Persian: - ÄžaznÄ«) is a city in eastern Afghanistan, with an approximate population of 141,000 people. ... This article needs cleanup. ... Abu Mansur Abd al-Qahir ibn Tahir ibn Muhammad ibn Abdallah al-Tamimi al-Shaffi al-Baghdadi (Arabic:أبو منصور عبدالقاهر ابن طاهر بن محمد بن عبدالله التميمي الشافعي البغدادي) was an Arab mathematician who is best known for his treatise al-Takmila fil-Hisab. ... Abu lHasan Ali ibn Ahmad Al-Nasawi, also spelled Nasavi, (1010 - 1075), was a Persian mathematician from Khurasan, Iran. ... Abu Abd Allah Muhammad ibn Muadh Al-Jayyani (Al-Jayyani; 989, Cordoba, Spain - 1079, Jaen, Spain) was an Arabic mathematician from present-day Spain. ... Arzachel redirects here. ... Yusuf ibn Ahmad al-Mutaman ibn Hud was an Arab mathematician and a member of the Banu Hud family, al-Mutamin ruled Zaragoza from 1082 to 1085. ... For other people, places or with similar names of Khayam, see Khayyam (disambiguation). ... مغربي، السموءل بن يحي، also known as Samaual al-Maghribi [1] (c. ... Events February 13 - Innocent II is elected pope An antipope schism occurs when Roger II of Sicily supports Anacletus II as pope instead of Innocent II. Innocent flees to France and Anacletus crowns Roger King. ... Baghdad (Arabic: ) is the capital of Iraq and of Baghdad Governorate. ... Events April 13 - Frederick Barbarossa issues the Gelnhausen Charter November 18 - France Emperor Antoku succeds Emperor Takakura as emperor of Japan Afonso I of Portugal is taken prisoner by Ferdinand II of Leon Artois is annexed by France Prince Mochihito amasses a large army and instigates the Genpei War between... Maragheh or Maraghah is a town in the East Azarbaijan Province of Iran, on the Safi River. ... (1135 - 1213) was a Persian mathematician of the Islamic Golden Age (during the Middle Ages). ... For the lunar crater, see Al-Marrakushi (crater). ... For other uses, see Tusi. ... is the 49th day of the year in the Gregorian calendar. ... // The town of Riga was chartered as a city. ... Categories: Iran geography stubs | Cities in Iran ... Map showing the pre-2004 Khorasan Province in Iran Khorasan (Persian: خراسان) (also transcribed as Khurasan and Khorassan, anciently called Traxiane during Hellenistic and Parthian times is currently a region located in north eastern Iran, but historically referred to a much larger area east and north-east of the Persian Empire... is the 177th day of the year (178th in leap years) in the Gregorian calendar. ... Events May 7 - In France the Second Council of Lyons opens to consider the condition of the Holy Land and to agree to a union with the Byzantine church. ... Kazimain (Arabic: ‎; BGN: Al Kāz̧imÄ«yah; also spelled Al Kadhimiya (Kurdish: Qasimîyê)) is an old town located in Iraq that is now a neighbourhood of Baghdad, located in the northern area of the city about 5 km from the center of the city. ... Baghdad (Arabic: ) is the capital of Iraq and of Baghdad Governorate. ... , (Arabic,محي الدين المغربي) (c. ... Maragheh or Maraghah is a town in the East Azarbaijan Province of Iran, on the Safi River. ... (c. ... Samarkand (Samarqand or Самарқанд in Uzbek) (population 400,000) is the second-largest city in Uzbekistan, capital of the Samarkand region (Samarqand Wiloyati). ... Ibn al-Banna al-Murrakushi al-Azdi (Arabic: ابن البنّا) [c. ... Kamal al-Din Abul-Hasan Muhammad Al-Farisi (1260-1320) (Arabic: ) (Tabriz, Iran) was a prominent Persian Muslim mathematician and physicist. ... (born 1320 possibly in Damascus, Syria - died 1380 possibly in Damascus, Syria). ... Ibn al-Shatir (or Ibn ash-Shatir) (1304–1375) was a Muslim astronomer of Damascus. ... (1364 in Bursa, Turkey – 1436 in Samarkand, Uzbekistan), whose actual name was Salah al-Din Musa Pasha ( means son of the judge), was an astronomer and mathematician who worked at the observatory in Samarkand. ... For other uses, see Bursa (disambiguation). ... A stamp issued 1979 in Iran commemorating al-KāshÄ«. (or ) (c. ... Ulugh Beg, here depicted on a Soviet stamp, was one of Islams greatest astronomers during the Middle Ages. ... Abul Hasan ibn Ali al Qalasadi (1412-1486) was an Arab mathematician known for being one of the most influential voice in algebraic notation since antiquity. ... Lotfali Askar Zadeh (born February 4, 1921) is a mathematician and computer scientist, and a professor of computer science at the University of California, Berkeley. ...

Fields

Algebra

Further information: History of algebra

There are three theories about the origins of Arabic Algebra. The first emphasizes Hindu influence, the second emphasizes Mesopotamian or Persian-Syriac influence and the third emphasizes Greek influence. Many scholars believe that it is the result of a combination of all three sources.[9] // The word Algebra is derived from the Arabic word Al-Jabr, and this comes from the treatise written in 820 by the Persian mathematician entitled, in Arabic, كتاب الجبر والمقابلة, or , which can be translated as The Compendious Book on Calculation by Completion and Balancing. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... A page from the book (Arabic for The Compendious Book on Calculation by Completion and Balancing), also known under a shorter name spelled as Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala and other transliterations) is a mathematical book written approximately 820 AD by the Persian...


Throughout their time in power, before the fall of Islamic civilization, the Arabs used a fully rhetorical algebra, where sometimes even the numbers were spelled out in words. The Arabs would eventually replace spelled out numbers (eg. twenty-two) with Arabic numerals (eg. 22), but the Arabs never adopted or developed a syncopated or symbolic algebra.[5] For other uses, see Arabic numerals (disambiguation). ...


The Muslim[10] Persian mathematician Muhammad ibn Mūsā al-khwārizmī was a faculty member of the "House of Wisdom" (Bait al-hikma) in Baghdad, which was established by Al-Mamun. Al-Khwarizmi, who died around 850 A.D., wrote more than half a dozen mathematical and astronomical works; some of which were based on the Indian Sindhind.[4] One of al-Khwarizmi's most famous books is entitled Al-jabr wa'l muqabalah or The Compendious Book on Calculation by Completion and Balancing, and it gives an exhaustive account of solving polynomials up to the second degree.[11] A stamp issued September 6, 1983 in the Soviet Union, commemorating al-KhwārizmÄ«s (approximate) 1200th anniversary. ... A page from the book (Arabic for The Compendious Book on Calculation by Completion and Balancing), also known under a shorter name spelled as Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala and other transliterations) is a mathematical book written approximately 820 AD by the Persian...


Al-Jabr is divided into six chapters, each of which deals with a different type of formula. The first chapter of Al-Jabr deals with equations whose squares equal its roots (ax² = bx), the second chapter deals with squares equal to number (ax² = c), the third chapter deals with roots equal to a number (bx = c), the fourth chapter deals with squares and roots equal a number (ax² + bx = c), the fifth chapter deals with squares and number equal roots (ax² + c = bx), and the sixth and final chapter deals with roots and number equal to squares (bx + c = ax²).[12]


'Abd al-Hamid ibn-Turk authored a manuscript entitled Logical Necessities in Mixed Equations, which is very similar to al-Khwarzimi's Al-Jabr and was published at around the same time as, or even possibly earlier than, Al-Jabr.[13] The manuscript gives the exact same geometric demonstration as is found in Al-Jabr, and in one case the same example as found in Al-Jabr, and even goes beyond Al-Jabr by giving a geometric proof that if the determinant is negative then the quadratic equation has no solution.[13] The similarity between these two works has led some historians to conclude that Arabic algebra may have been well developed by the time of al-Khwarizmi and 'Abd al-Hamid.[13]


Al-Karkhi was the successor of Abu'l-Wefa and he was the first to discover the solution to equations of the form ax2n + bxn = c.[14] Al-Karkhi only considered positive roots.[14] (or ) (c. ...


Omar Khayyám (c. 1050-1123) wrote a book on Algebra that went beyond Al-Jabr to include equations of the third degree.[15] Omar Khayyám provided both arithmetic and geometric solutions for quadratic equations, but he only gave geometric solutions for general cubic equations since he mistakenly believed that arithmetic solutions were impossible.[15] His method of solving cubic equations by using intersecting conics had been used by Menaechmus, Archimedes, and Alhazen, but Omar Khayyám generalized the method to cover all cubic equations with positive roots.[15] He only considered positive roots and he did not go past the third degree.[15] He also saw a strong relationship between Geometry and Algebra.[15]


In the 12th century, Sharaf al-Din al-Tusi found algebraic and numerical solutions to cubic equations and was the first to discover the derivative of cubic polynomials.[16] Sharafeddin Muzzafar-i Tusi (1135 - 1213) was a Persian mathematician of the Middle Ages. ... Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... This article is about derivatives and differentiation in mathematical calculus. ... Polynomial of degree 3 In mathematics, a cubic function is a function of the form where b is nonzero; or in other words, a polynomial of degree three. ...


J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive: The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ...

"Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before." In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ...

Arithmetic

Main article: Arabic numerals

The Indian numeral system came to be known to both the Persian mathematician Al-Khwarizmi, whose book On the Calculation with Hindu Numerals written circa 825, and the Arab mathematician Al-Kindi, who wrote four volumes, On the Use of the Indian Numerals (Ketab fi Isti'mal al-'Adad al-Hindi) circa 830, are principally responsible for the diffusion of the Indian system of numeration in the Middle-East and the West [1]. In the 10th century, Middle-Eastern mathematicians extended the decimal numeral system to include fractions using decimal point notation, as recorded in a treatise by Syrian mathematician Abu'l-Hasan al-Uqlidisi in 952-953. For other uses, see Arabic numerals (disambiguation). ... India has produced many numeral systems. ... The Persians of Iran (officially named Persia by West until 1935 while still referred to as Persia by some) are an Iranian people who speak Persian (locally named Fârsi by native speakers) and often refer to themselves as ethnic Iranians as well. ... A stamp issued September 6, 1983 in the Soviet Union, commemorating al-Khwārizmīs (approximate) 1200th anniversary. ... Events Egbert of Wessex defeats Beornwulf of Mercia at Ellandun. ... For other uses, see Arab (disambiguation). ... For the Christian theologian, see Abd al-Masih ibn Ishaq al-Kindi. ... Events Christian missionary Ansgar visits Birka, trade city of the Swedes. ... The traditional Middle East and the G8s Greater Middle East Political & transportation map of the traditional Middle East today The Middle East is a historical and political region of Africa-Eurasia with no clear definition. ... As a means of recording the passage of time, the 10th century was that century which lasted from 901 to 1000. ... The traditional Middle East and the G8s Greater Middle East Political & transportation map of the traditional Middle East today The Middle East is a historical and political region of Africa-Eurasia with no clear definition. ... For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ... The decimal separator is a symbol used to mark the boundary between the integral and the fractional parts of a decimal numeral. ... Abul Hasan Ahmad ibn Ibrahim Al-Uqlidisi was an Arab mathematician, possibly from Damascus He wrote the earliest surviving book on the Hindu place-value system, known in the west as Arabic numerals, around 952. ... Events Hugh Capet marries Adelaide of Aquitaine Deaths Emperor Suzaku of Japan Hugh, Duke of Burgundy Categories: 952 ... Events First time that Póvoa de Varzim, Portugal appeared in a Roman map. ...


In the Arab world—until modern times—the Arabic numeral system was used only by mathematicians. Muslim scientists used the Babylonian numeral system, and merchants used the Abjad numerals. A distinctive "West Arabic" variant of the symbols begins to emerge in ca. the 10th century in the Maghreb and Al-Andalus, called the ghubar ("sand-table" or "dust-table") numerals. Arab States redirects here. ... Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. ... The Abjad numerals are a decimal numeral system which was used in the Arabic-speaking world prior to the use of the Hindu-Arabic numerals from the 8th century, and in parallel with the latter until Modern times. ... As a means of recording the passage of time, the 10th century was that century which lasted from 901 to 1000. ... This article or section does not cite any references or sources. ... Al-Andalus is the Arabic name given the Iberian Peninsula by its Muslim conquerors; it refers to both the Caliphate proper and the general period of Muslim rule (711–1492). ...


The first mentions of the numerals in the West are found in the Codex Vigilanus of 976 [2]. From the 980s, Gerbert of Aurillac (later, Pope Silvester II) began to spread knowledge of the numerals in Europe. Gerbert studied in Barcelona in his youth, and he is known to have requested mathematical treatises concerning the astrolabe from Lupitus of Barcelona after he had returned to France. The people of Toledo. ... Events January 10 - Basil II becomes Eastern Roman Emperor, see Byzantine Emperors. ... Centuries: 9th century - 10th century - 11th century Decades: 930s - 940s _ 950s - 960s - 970s - 980s - 990s - 1000s - 1010s - 1020s - 1030s Years: 980 981 982 983 984 985 986 987 988 989 Events Kievan Rus accepts Eastern Orthodox Christianity Categories: 980s ... Gerbert of Aurillac, later known as pope Silvester II, (or Sylvester II), (ca. ... Gerbert of Aurillac, later known as pope Silvester II, (or Sylvester II), (ca. ... Location Coordinates : Time Zone : CET (GMT +1) - summer: CEST (GMT +2) General information Native name Barcelona (Catalan) Spanish name Barcelona Nickname Ciutat Comtal (City of Counts) Postal code 08001–08080 Area code 34 (Spain) + 93 (Barcelona) Website http://www. ... A 16th century astrolabe. ... Lupitus of Barcelona, identified with a Christian archdeacon called Sunifred, was an astronomer in late 10th century Barcelona, then part of the Marca Hispanica between Islamic Al-Andalus and Christian France (in 985 changing from Christian back into Muslim hands by the conquest of Al-Mansur). ...


Al-Khwārizmī, the Persian scientist, wrote in 825 a treatise On the Calculation with Hindu Numerals, which was translated into Latin in the 12th century, as Algoritmi de numero Indorum, where "Algoritmi", the translator's rendition of the author's name gave rise to the word algorithm (Latin algorithmus) with a meaning "calculation method". Soviet postage stamp commemorating the 1200th anniversary of Muhammad al‑Khwarizmi in 1983. ... This article is about the Persian people, an ethnic group found mainly in Iran. ... Events Egbert of Wessex defeats Beornwulf of Mercia at Ellandun. ... The 12th century saw a major search by European scholars for new learning, which led them to the Arabic fringes of Europe, especially to Spain and Sicily. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ...


Calculus

Around 1000 AD, Al-Karaji, using mathematical induction, found a proof for the sum of integral cubes.[17] The historian of mathematics, F. Woepcke,[18] praised Al-Karaji for being "the first who introduced the theory of algebraic calculus." Shortly afterwards, Ibn al-Haytham (known as Alhazen in the West), an Iraqi mathematician working in Egypt, was the first mathematician to derive the formula for the sum of the fourth powers. In turn, he developed a method for determining the general formula for the sum of any integral powers, which was fundamental to the development of integral calculus.[19] Europe in 1000 The year 1000 of the Gregorian Calendar was the last year of the 10th century as well as the last year of the first millennium. ... Abu Bakr ibn Muhammad ibn al-Husayn Al-Karaji (953 - 1029), also known as Al-karkhi was a Persian mathematician and engineer. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... This article is about the concept of integrals in calculus. ... y=x³, for integer values of 1≤x≤25. ... This article is about the occupation of studying history. ... The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ... This article is about the branch of mathematics. ... For other uses, see Calculus (disambiguation). ... (Arabic: أبو علي الحسن بن الحسن بن الهيثم, Latinized: Alhacen or (deprecated) Alhazen) (965 – 1039), was an Arab[1] Muslim polymath[2][3] who made significant contributions to the principles of optics, as well as to anatomy, astronomy, engineering, mathematics, medicine, ophthalmology, philosophy, physics, psychology, visual perception, and to science in general with his introduction of the... In mathematics and elsewhere, the adjective quartic means fourth order, such as the function . ... “Exponent” redirects here. ... This article is about the concept of integrals in calculus. ... This article is about the concept of integrals in calculus. ...


Analytic geometry, an important part of calculus, began with Omar Khayyám, a poet-mathematician in 11th century Persia, who applied it to his general geometric solution of cubic equations.[20] In the 12th century, the Persian mathematician Sharaf al-Din al-Tusi was the first to discover the derivative of cubic polynomials, an important result in differential calculus.[16] Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... For other people, places or with similar names of Khayam, see Khayyam (disambiguation). ... Kelileh va Demneh Persian manuscript copy dated 1429, from Herat, depicts the Jackal trying to lead the Lion astray. ... edit Geographical extent of Iranian influence in the 1st century BCE. The Parthian Empire (mostly Western Iranian) is shown in red, other areas, dominated by Scythia (mostly Eastern Iranian), in orange. ... Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ... This article is about the Persian people, an ethnic group found mainly in Iran. ... Sharafeddin Muzzafar-i Tusi (1135 - 1213) was a Persian mathematician of the Middle Ages. ... This article is about derivatives and differentiation in mathematical calculus. ... Polynomial of degree 3 In mathematics, a cubic function is a function of the form where b is nonzero; or in other words, a polynomial of degree three. ... The differential dy In calculus, a differential is an infinitesimally small change in a variable. ...


Cryptography

In the 9th century, al-Kindi was a pioneer in cryptanalysis and cryptology. He gave the first known recorded explanation of cryptanalysis in A Manuscript on Deciphering Cryptographic Messages. In particular, he is credited with developing the frequency analysis method whereby variations in the frequency of the occurrence of letters could be analyzed and exploited to break ciphers (i.e. crypanalysis by frequency analysis).[21] This was detailed in a text recently rediscovered in the Ottoman archives in Istanbul, A Manuscript on Deciphering Cryptographic Messages, which also covers methods of cryptanalysis, encipherments, cryptanalysis of certain encipherments, and statistical analysis of letters and letter combinations in Arabic.[22] For the Christian theologian, see Abd al-Masih ibn Ishaq al-Kindi. ... Cryptanalysis (from the Greek kryptós, hidden, and analýein, to loosen or to untie) is the study of methods for obtaining the meaning of encrypted information, without access to the secret information which is normally required to do so. ... Cryptology is an umbrella term for cryptography and cryptanalysis. ... Cryptanalysis (from the Greek kryptós, hidden, and analýein, to loosen or to untie) is the study of methods for obtaining the meaning of encrypted information, without access to the secret information which is normally required to do so. ... A typical distribution of letters in English language text. ... Encrypt redirects here. ...


Ahmad al-Qalqashandi (1355-1418) wrote the Subh al-a 'sha, a 14-volume encyclopedia which included a section on cryptology. This information was attributed to Taj ad-Din Ali ibn ad-Duraihim ben Muhammad ath-Tha 'alibi al-Mausili who lived from 1312 to 1361, but whose writings on cryptology have been lost. The list of ciphers in this work included both substitution and transposition, and for the first time, a cipher with multiple substitutions for each plaintext letter. Also traced to Ibn al-Duraihim is an exposition on and worked example of cryptanalysis, including the use of tables of letter frequencies and sets of letters which can not occur together in one word. Shihab al-Din abu l-Abbas Ahmad ben Ali ben Ahmad Abd Allah al-Qalqashandi (1355 or 1356 – 1418) was a medieval Egyptian writer born in a village in the Nile Delta. ... In cryptography, a substitution cipher is a method of encryption by which units of plaintext are substituted with ciphertext according to a regular system; the units may be single letters (the most common), pairs of letters, triplets of letters, mixtures of the above, and so forth. ... In classical cryptography, a transposition cipher changes one character from the plaintext to another (to decrypt the reverse is done). ... This article is about cryptography. ... The factual accuracy of this article is disputed. ...


Fuzzy mathematics

Main article: Fuzzy Mathematics

In 1965, Lotfi Asker Zadeh founded fuzzy set theory as an extension of the classical notion of set and he founded the field of Fuzzy Mathematics. Later in 1973, Zadeh founded the field of fuzzy logic. // Fuzzy mathematics (not to be confused with the term fuzzy math), form a branch of mathematics related to fuzzy logic. ... Lotfali Askar Zadeh (born February 4, 1921) is a mathematician and computer scientist, and a professor of computer science at the University of California, Berkeley. ... Fuzzy sets are an extension of classical set theory and are used in fuzzy logic. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... // Fuzzy mathematics (not to be confused with the term fuzzy math), form a branch of mathematics related to fuzzy logic. ... Fuzzy logic is derived from fuzzy set theory dealing with reasoning that is approximate rather than precisely deduced from classical predicate logic. ...


Geometry

Further information: History of geometry
An engraving by Albrecht Dürer featuring Mashallah, from the title page of the De scientia motus orbis (Latin version with engraving, 1504). As in many medieval illustrations, the compass here is an icon of religion as well as science, in reference to God as the architect of creation
An engraving by Albrecht Dürer featuring Mashallah, from the title page of the De scientia motus orbis (Latin version with engraving, 1504). As in many medieval illustrations, the compass here is an icon of religion as well as science, in reference to God as the architect of creation

The successors of Muhammad ibn Mūsā al-Khwārizmī (born 780) undertook a systematic 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. Table of Geometry, from the 1728 Cyclopaedia. ... Image File history File links Durer_astronomer. ... Image File history File links Durer_astronomer. ... Albrecht Dürer (pronounced /al. ... For other uses, see Mashallah (disambiguation). ... a compass In drafting, a compass (or pair of compasses) is an instrument]] used by mathematicians and craftsmen in for drawing or inscribing a circle or arc. ... (Arabic: ) was a Persian[1] mathematician, astronomer, astrologer and geographer. ... Events Constantine VI becomes Byzantine Emperor with Irene as guardian. ... For other uses, see Euclid (disambiguation). ...


Al-Mahani (born 820) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Al-Karaji (born 953) completely freed algebra from geometrical operations and replaced them with the arithmetical type of operations which are at the core of algebra today. Abu Abdallah Mohammed ibn Isa al-Mahani, was a Persian mathematician and astronomer from Mahan, Kerman, Persia. ... Tahir, the son of a slave, is rewarded with the governorship of Khurasan because he have supported the caliphate. ... Abu Bakr ibn Muhammad ibn al-Husayn Al-Karaji (953 - 1029), also known as Al-karkhi was a Persian mathematician and engineer. ... Events First time that Póvoa de Varzim, Portugal appeared in a Roman map. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ...


Although Thabit ibn Qurra (known as Thebit in Latin) (born 836) contributed to a number of areas in mathematics, where he played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-Euclidean geometry. An important geometrical aspect of Thabit's work was his book on the composition of ratios. In this book, Thabit deals with arithmetical operations applied to ratios of geometrical quantities. The Greeks had dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of arithmetic could be applied. By introducing arithmetical operations on quantities previously regarded as geometric and non-numerical, Thabit started a trend which led eventually to the generalization of the number concept. Another important contribution Thabit made to geometry was his generalization of the Pythagorean theorem, which he extended from special right triangles to all triangles in general, along with a general proof.[23] (836 in Harran, Mesopotamia – February 18, 901 in Baghdad) was an Arab astronomer and mathematician, who was known as Thebit in Latin. ... For other uses, see Latin (disambiguation). ... Events Abbasid caliph al-Mutasim establishes new capital at Samarra, Iraq. ... In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... This article deals with the concept of an integral in calculus. ... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... For other uses, see Geometry (disambiguation). ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... Two types of special right triangles appear commonly in geometry, the angle based and the side based triangles. ... A triangle. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...


Ibrahim ibn Sinan (born 908), who introduced a method of integration more general than that of Archimedes, and al-Quhi (born 940) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. These mathematicians, and in particular Ibn al-Haytham, studied optics and investigated the optical properties of mirrors made from conic sections. Ibrahim ibn Sinan ibn Thabit ibn Qurra (908, Baghdad – 946, Baghdad) was an Arab mathematician and astronomer who studied geometry and in particular tangents to circles. ... Events Battle of Belach Mugna Births Deaths Categories: 908 ... This article is about the concept of integrals in calculus. ... For other uses, see Archimedes (disambiguation). ... (sometimes ), was a Persian [1] mathematician, physicist and astronomer. ... Events Births Brian Boru, high king of Ireland Abul-Wafa, iranian mathematician Deaths ar-Radi (Caliph of Baghdad) Athelstan, who was succeeded by his half-brother, Edmund Categories: 940 ... (Arabic: أبو علي الحسن بن الحسن بن الهيثم, Latinized: Alhacen or (deprecated) Alhazen) (965 – 1039), was an Arab[1] Muslim polymath[2][3] who made significant contributions to the principles of optics, as well as to anatomy, astronomy, engineering, mathematics, medicine, ophthalmology, philosophy, physics, psychology, visual perception, and to science in general with his introduction of the... For the book by Sir Isaac Newton, see Opticks. ... Wikibooks has more on the topic of Conic section Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ...


Astronomy, time-keeping and geography provided other motivations for geometrical and trigonometrical research. For example Ibrahim ibn Sinan and his grandfather Thabit ibn Qurra both studied curves required in the construction of sundials. Abu'l-Wafa and Abu Nasr Mansur both applied spherical geometry to astronomy. Abul Hasan Thabit ibn Qurra ibn Marwan al-Sabi al-Harrani, (826 – February 18, 901) was an Arab astronomer and mathematician. ... Abul Wafa Muhammad Ibn Muhammad Ibn Yahya Ibn Ismail al-Buzjani (940 – 997 or 998) was a Persian mathematician and astronomer. ... Abu Nasr Mansur ibn Ali ibn Iraq (c. ... Spherical geometry is the geometry of the two-dimensional surface of a sphere. ...


Omar Khayyám (born 1048) was a Persian mathematician, as well as a poet. Along with his fame as a poet, he was also famous during his lifetime as a mathematician, well known for inventing the general method of solving cubic equations by intersecting a parabola with a circle. In addition he discovered the binomial expansion, and authored criticisms of Euclid's theories of parallels which made their way to England, where they contributed to the eventual development of non-Euclidean geometry. Omar Khayyam also combined the use of trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means. His work marked the beginnings of algebraic geometry[24][25] and analytic geometry.[20] Khayyam also made the first attempt at formulating a non-Euclidean postulate as an alternative to the Euclidean parallel postulate,[26] and he was the first to consider the cases of elliptical geometry and hyperbolic geometry, though he excluded the latter.[27] For other people, places or with similar names of Khayam, see Khayyam (disambiguation). ... Events The city of Oslo is founded by Harald Hardråde of Norway. ... This article is about the Persian people, an ethnic group found mainly in Iran. ... Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ... For other uses, see Euclid (disambiguation). ... a and b are parallel, the transversal t produces congruent angles. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... In mathematics, approximation theory is concerned with how functions can be approximated with other, simpler, functions, and with characterising in a quantitative way the errors introduced thereby. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... This article or section does not adequately cite its references or sources. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... a and b are parallel, the transversal t produces congruent angles. ... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... Lines through a given point P and asymptotic to line l. ...


Persian mathematician Sharafeddin Tusi (born 1135) did not follow the general development that came through al-Karaji's school of algebra but rather followed Khayyam's application of algebra to geometry. He wrote a treatise on cubic equations, which represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the study of algebraic geometry. Sharafeddin Muzzafar-i Tusi, was a Persian mathematician of the middle ages, (1135 - 1213). ... Events January - Byland Abbey founded Stephen of Blois succeeds King Henry I. Empress Maud, daughter of Henry I and widow of Henry V opposed Stephen and claims the throne as her own Owain Gwynedd of Wales defeats the Normans at Crug Mawr. ... Abu Bakr ibn Muhammad ibn al-Husayn Al-Karaji (953 - 1029), also known as Al-karkhi was a Persian mathematician and engineer. ...


In 1250, Nasīr al-Dīn al-Tūsī, in his Al-risala al-shafiya'an al-shakk fi'l-khutut al-mutawaziya (Discussion Which Removes Doubt about Parallel Lines), wrote detailed critiques of the Euclidean parallel postulate and on Omar Khayyám's attempted proof a century earlier. Nasir al-Din attempted to derive a proof by contradiction of the parallel postulate.[19] He was one of the first to consider the cases of elliptical geometry and hyperbolic geometry, though he ruled out both of them.[27] For other uses, see Muhammad Nasir-al-din. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... a and b are parallel, the transversal t produces congruent angles. ... For other people, places or with similar names of Khayam, see Khayyam (disambiguation). ... Reductio ad absurdum (Latin for reduction to the absurd, traceable back to the Greek ἡ εις το αδυνατον απαγωγη, reduction to the impossible, often used by Aristotle) is a type of logical argument... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... Lines through a given point P and asymptotic to line l. ...


His son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), wrote a book on the subject in 1298, based on al-Tusi's later thoughts, which presented one of the earliest arguments for a non-Euclidean hypothesis equivalent to the parallel postulate.[19][28] Sadr al-Din's work was published in Rome in 1594 and was studied by European geometers. This work marked the starting point for Giovanni Girolamo Saccheri's work on the subject, and eventually the development of modern non-Euclidean geometry.[19] A proof from Sadr al-Din's work was quoted by John Wallis and Saccheri in the 17th and 18th centuries. They both derived their proofs of the parallel postulate from Sadr al-Din's work, while Saccheri also derived his Saccheri quadrilateral from Sadr al-Din, who himself based it on his father's work.[29] Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... For other uses, see Rome (disambiguation). ... Giovanni Girolamo Saccheri (September 5, 1667 - October 25, 1733) was an Italian Jesuit priest and mathematician. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... John Wallis John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. ... A Saccheri Quadrilateral A Saccheri quadrilateral is a four-sided figure. ...


The theorems of Ibn al-Haytham (Alhacen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were the first theorems on elliptical geometry and hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works marked the beginning of non-Euclidean geometry and had a considerable influence on its development among later European geometers, including Witelo, Levi ben Gerson, Alfonso, John Wallis, and Giovanni Girolamo Saccheri.[30] (Arabic: أبو علي الحسن بن الحسن بن الهيثم, Latinized: Alhacen or (deprecated) Alhazen) (965 – 1039), was an Arab[1] Muslim polymath[2][3] who made significant contributions to the principles of optics, as well as to anatomy, astronomy, engineering, mathematics, medicine, ophthalmology, philosophy, physics, psychology, visual perception, and to science in general with his introduction of the... Tomb of Omar Khayam, Neishapur, Iran. ... Tusi couple from Vat. ... This article is about the geometric shape. ... A Lambert quadrilateral A Lambert quadrilateral is a four sided figure. ... A Saccheri Quadrilateral A Saccheri quadrilateral is a four-sided figure. ... Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p. ... Lines through a given point P and asymptotic to line l. ... a and b are parallel, the transversal t produces congruent angles. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... Witelo - also known as Erazmus Ciolek Witelo, Witelon, Vitellio, Vitello, Vitello Thuringopolonis, Erazm Ciołek, (born ca. ... Levi ben Gerson (1288-1344)wrote Book of Numbers in 1321 dealing with arithmetical operations, including extraction of roots. ... Alfonso (Italian and Spanish), Alfons (Catalan and German), Afonso (Portuguese), Affonso (Ancient Portuguese), Alphonse (French and English), Alphons (Dutch), or Alphonso (English and Filipino) is a masculine name, originally from the Gothic language. ... John Wallis John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. ... Giovanni Girolamo Saccheri (September 5, 1667 - October 25, 1733) was an Italian Jesuit priest and mathematician. ...


Mathematical induction

The first known proof by mathematical induction was introduced in the al-Fakhri written by Al-Karaji around 1000 AD, who used it to prove arithmetic sequences such as the binomial theorem, Pascal's triangle, and the sum formula for integral cubes.[31][32] His proof was the first to make use of the two basic components of an inductive proof, "namely the truth of the statement for n = 1 (1 = 13) and the deriving of the truth for n = k from that of n = k - 1."[33] Look up proof in Wiktionary, the free dictionary. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... Abu Bakr ibn Muhammad ibn al-Husayn Al-Karaji (953 - 1029), also known as Al-karkhi was a Persian mathematician and engineer. ... Europe in 1000 The year 1000 of the Gregorian Calendar was the last year of the 10th century as well as the last year of the first millennium. ... In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ... The first five rows of Pascals triangle In mathematics, Pascals triangle is a geometric arrangement of the binomial coefficients in a triangle. ... This article is about the concept of integrals in calculus. ... y=x³, for integer values of 1≤x≤25. ... Time Saving Truth from Falsehood and Envy, François Lemoyne, 1737 For other uses, see Truth (disambiguation). ...


Shortly afterwards, Ibn al-Haytham (Alhazen) used the inductive method to prove the sum of fourth powers, and by extension, the sum of any integral powers, which was an important result in integral calculus. He only stated it for particular integers, but his proof for those integers was by induction and generalizable.[34][35] (Arabic: أبو علي الحسن بن الحسن بن الهيثم, Latinized: Alhacen or (deprecated) Alhazen) (965 – 1039), was an Arab[1] Muslim polymath[2][3] who made significant contributions to the principles of optics, as well as to anatomy, astronomy, engineering, mathematics, medicine, ophthalmology, philosophy, physics, psychology, visual perception, and to science in general with his introduction of the... In mathematics, the fourth powers are given by the expression a4 = a × a × a × a The sequence of fourth powers of integers goes: 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, ... They are also formed by multiplying a number by its cube. ... “Exponent” redirects here. ... This article is about the concept of integrals in calculus. ... For other uses, see Calculus (disambiguation). ...


Ibn Yahyā al-Maghribī al-Samaw'al came closest to a modern proof by mathematical induction in pre-modern times, which he used to extend the proof of the binomial theorem and Pascal's triangle previously given by al-Karaji. Al-Samaw'al's inductive argument was only a short step from the full inductive proof of the general binomial theorem.[36] (c. ...


Number theory

In number theory, Ibn al-Haytham solved problems involving congruences using what is now called Wilson's theorem. In his Opuscula, Ibn al-Haytham considers the solution of a system of congruences, and gives two general methods of solution. His first method, the 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 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).[37] Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... (Arabic: أبو علي الحسن بن الحسن بن الهيثم, Latinized: Alhacen or (deprecated) Alhazen) (965 – 1039), was an Arab[1] Muslim polymath[2][3] who made significant contributions to the principles of optics, as well as to anatomy, astronomy, engineering, mathematics, medicine, ophthalmology, philosophy, physics, psychology, visual perception, and to science in general with his introduction of the... In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ... In mathematics, Wilsons theorem (also known as Al-Haythams theorem) states that p > 1 is a prime number if and only if (see factorial and modular arithmetic for the notation). ... Several related results in number theory and abstract algebra are known under the name Chinese remainder theorem. ... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ...


Recreational mathematics

In recreational mathematics, magic squares were known to Arab mathematicians, possibly as early as the 7th century, when the Arabs got into contact with Indian or South Asian culture, and learned Indian mathematics and astronomy, including other aspects of combinatorial mathematics. It has also been suggested that the idea came via China. The first magic squares of order 5 and 6 appear in an encyclopedia from Baghdad circa 983 AD, the Rasa'il Ihkwan al-Safa (the Encyclopedia of the Brethern of Purity); simpler magic squares were known to several earlier Arab mathematicians.[38] Recreational mathematics includes many mathematical games, and can be extended to cover such areas as logic and other puzzles of deductive reasoning. ... In recreational mathematics, a magic square of order n is an arrangement of n² numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. ... For other uses, see Arab (disambiguation). ... Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with counting the objects in those collections (enumerative combinatorics) and with deciding whether certain optimal objects exist (extremal combinatorics). ... Baghdad (Arabic: ) is the capital of Iraq and of Baghdad Governorate. ... The Encyclopedia of the Brethren of Purity (also variously known as the Epistles of the Brethren of Sincerity, the Epistles of the Brethren of Purity or Epistles of the Brethren of Purity and Loyal Friends; Arabic:رسائل أخوان الصفا و خلان الوفا Rasail ikhwan as-safa wa khillan al-wafa ) was a large encyclopedia [1...


The Arab mathematician Ahmad al-Buni, who worked on magic squares around 1200 AD, attributed mystical properties to them, although no details of these supposed properties are known. There are also references to the use of magic squares in astrological calculations, a practice that seems to have originated with the Arabs.[38] Ahmad ibn ‘Ali ibn Yusuf al-Buni (Arabic:أحمد البوني) (d. ...


Trigonometry

The early Indian works on trigonometry were translated and expanded in the Muslim world by Arab and Persian mathematicians. Muhammad ibn Mūsā al-Khwārizmī produced tables of sines and tangents, and also developed spherical trigonometry. By the 10th century, in the work of Abū al-Wafā' al-Būzjānī, Muslim mathematicians were using all six trigonometric functions, and had sine tables in 0.25° increments, to 8 decimal places of accuracy, as well as tables of tangent values. Abū al-Wafā' also developed the trigonometric formula sin 2x = 2 sin x cos x. This article is under construction. ... Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... Nations with a Muslim majority appear in green, while nations that are approximately 50% Muslim appear yellow. ... A 9th century picture of Arab scientists working in Baghdad, Iraq. ... Photo taken from medieval manuscript by Qotbeddin Shirazi. ... (Arabic: ) was a Persian[1] mathematician, astronomer, astrologer and geographer. ... Sine redirects here. ... For other uses, see tangent (disambiguation). ... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... (940 – 997/8) was a Persian mathematician and astronomer. ... Sine redirects here. ... In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ...


Omar Khayyam solved cubic equations using approximate numerical solutions found by interpolation in trigonometric tables. Al-Jayyani, an Arabic mathematician in Islamic Spain, wrote the first treatise on spherical trigonometry in 1060. Tomb of Omar Khayam, Neishapur, Iran. ... Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ... Abu Abd Allah Muhammad ibn Muadh Al-Jayyani (Al-Jayyani; 989, Cordoba, Spain - 1079, Jaen, Spain) was an Arabic mathematician from present-day Spain. ... Al-Andalus is the Arabic name given the Iberian Peninsula by its Muslim conquerors; it refers to both the Caliphate proper and the general period of Muslim rule (711–1492). ... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ...


All of these earlier works on trigonometry treated it mainly as an adjunct to astronomy; perhaps the first treatment as a subject in its own right was by Bhaskara II and Nasir al-Din al-Tusi (13th century). Nasir al-Din al-Tusi stated the law of sines and provided a proof for it, and also listed the six distinct cases of a right angled triangle in spherical trigonometry. Bhāskara (1114-1185), also called Bhāskara II and Bhāskarācārya (Bhaskara the teacher) was an Indian mathematician. ... Tusi couple from Vat. ... In trigonometry, the law of sines (or sine law, sine formula) is a statement about arbitrary triangles in the plane. ...


Ghiyath al-Kashi (14th century) gives trigonometric tables of values of the sine function to four sexagesimal digits (equivalent to 8 decimal places) for each 1° of argument with differences to be added for each 1/60 of 1°. Ulugh Beg (14th century) also gives accurate tables of sines and tangents correct to 8 decimal places. Kashani, dubbed, the Second Ptolemy, was an outstanding Persian mathematician of the middle ages. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... Ulugh Beg, here depicted on a Soviet stamp, was one of Islams greatest astronomers during the Middle Ages. ...


The method of triangulation, which was unknown in the Greco-Roman world, was also first developed by Muslim mathematicians, who applied it to practical uses such as surveying.[39] Triangulation can be used to find the distance from the shore to the ship. ... The Greco-Roman period of history refers to the culture of the peoples who were incorporated into the Roman Republic and Roman Empire. ... Surveyor at work with a leveling instrument. ...


See also

A Muslim mathematicians is a person that professes Islam and engaged in the mathematicians aspect of Islamic science. ... The 12th century saw a major search by European scholars for new learning, which led them to the Arabic fringes of Europe, especially to Spain and Sicily. ... In the history of science, Islamic science refers to the science developed under the Islamic civilisation between the 8th and 15th centuries (the Islamic Golden Age). ... During the Islamic Golden Age, usually dated from the 8th century to the 13th century,[1] engineers, scholars and traders of the Islamic world contributed enormously to the arts, agriculture, economics, industry, literature, navigation, philosophy, sciences, and technology, both by preserving and building upon earlier traditions and by adding many... A significant number of inventions were produced in the Muslim world, many of them with direct implications for Fiqh related issues. ...

Notes

  1. ^ Hogendijk 1999
  2. ^ O'Connor 1999
  3. ^ The Persistence of Cultures in World History: Persia/Iran by Dr. Laina Farhat-Holzman
  4. ^ a b c Boyer (1991). "The Arabic Hegemony", , 227. “The first century of the Muslim empire had been devoid of scientific achievement. This period (from about 650 to 750) had been, in fact, perhaps the nadir in the development of mathematics, for the Arabs had not yet achieved intellectual drive, and concern for learning in other parts of the world had faded. Had it not been for the sudden cultural awakening in Islam during the second half of the eighth century, considerably more of ancient science and mathematics would have been lost. [...] It was during the caliphate of al-Mamun (809-833), however, that the Arabs fully indulged their passion for translation. The caliph is said to have had a dream in which Aristotle appeared, and as a consequence al-Mamun determined to have Arabic versions made of all the Greek works that he could lay his hands on, including Ptolemy's Almagest and a complete version of Euclid's Elements. From the Byzantine Empire, with which the Arabs maintained an uneasy peace, Greek manuscripts were obtained through peace treaties. Al-Mamun established at Baghdad a "House of Wisdom" (Bait al-hikma) comparable to the ancient Museum at Alexandria. Among the faculty members was a mathematician and astronomer, Mohammed ibn-Musa al-Khwarizmi, whose name, like that of Euclid, later was to become a household word in Western Europe. The scholar, who died sometime before 850, wrote more than half a dozen astronomical and mathematical works, of which the earliest were probably based on the Sindhad derived from India.” 
  5. ^ a b Boyer (1991). "The Arabic Hegemony", , 234. “but al-Khwarizmi's work had a serious deficiency that had to be removed before it could serve its purpose effectively in the modern world: a symbolic notation had to be developed to replace the rhetorical form. This step the Arabs never took, except for the replacement of number words by number signs. [...] Thabit was the founder of a school of translators, especially from Greek and Syriac, and to him we owe an immense debt for translations into Arabic of works by Euclid, Archimedes, Apollonius, Ptolemy, and Eutocius.” 
  6. ^ Berggren, J. Lennart (2007). "Mathematics in Medieval Islam", The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press, 516. ISBN 9780691114859. “The mathematics, to speak only of the subject of interest here, came principally from three traditions. The first was Greek mathematics, from the great geometrical classics of Euclid, Apollonius, and Archimedes, through the numerical solutions of indeterminate problems in Diophantus's Arithmatica, to the practical manuals of Heron. But, as Bishop Severus Sebokht pointed out in the mid-seventh century, "there are also others who know something." Sebokht was referring to the Hindus, with their in genius arithmetic system based on only nine signs and a dot for an empty place. But they also contributed algebraic methods, a nascent trigonometry, and methods from solid geometry to solve problems in astronomy. The third tradition was what one may call the mathematics of practitioners. Their numbers included surveyors, builders, artisans, in geometric design, tax and treasury officials, and some merchants. Part of an oral tradition, this mathematics transcended ethnic divisions and was common heritage of many of the lands incorporated into the Islamic world.” 
  7. ^ Boyer (1991). "The Arabic Hegemony", , 226. “By 766 we learn that an astronomical-mathematical work, known to the Arabs as the Sindhind, was brought to Baghdad from India. It is generally thought that this was the Brahmasphuta Siddhanta, although it may have been the Surya Siddhanata. A few years later, perhaps about 775, this Siddhanata was translated into Arabic, and it was not long afterwards (ca. 780) that Ptolemy's astrological Tetrabiblos was translated into Arabic from the Greek.” 
  8. ^ a b Plofker, Kim (2007). , 434. “The early translations from Sanskrit inspired several other astronomical/astrological works in Arabic; some even imitated the Sanskrit practice of composing technical treatises in verse. Unfortunately, the earliest texts in this genre have now mostly been lost, and are known only from scattered fragments and allusions in later works. They reveal that the emergent Arabic astronomy adopted many Indian parameters, cosmological models, and computational techniques, including the use of sines.
    These Indian influences were soon overwhelmed - although it is not completely clear why - by those of the Greek mathematical and astronomical texts that were translated into Arabic under the Abbasid caliphs. Perhaps the greater availability of Greek works in the region, and of practitioners who understood them, favored the adoption of the Greek tradition. Perhaps its prosaic and deductive expositions seemed easier for foreign readers to grasp than elliptic Sanskrit verse. Whatever the reasons, Sanskrit inspired astronomy was soon mostly eclipsed by or merged with the "Graeco-Islamic" science founded on Hellenistic treatises.”
     
  9. ^ Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 230. ISBN 0471543977. 

    "Al-Khwarizmi continued: "We have said enough so far as numbers are concerned, about the six types of equations. Now, however, it is necessary that we should demonsrate geometrically the truth of the same problems which we have explained in numbers." The ring of this passage is obviously Greek rather than Babylonian or Indian. There are, therefore, three main schools of thought on the origin of Arabic algebra: one emphasizes Hindu influence, another stresses the Mesopotamian, or Syriac-Persian, tradition, and the third points to Greek inspiration. The truth is probably approached if we combine the three theories." Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... For other uses, see Euclid (disambiguation). ... For other uses, see Archimedes (disambiguation). ... Apollonius of Perga [Pergaeus] (ca. ... This article is about the geographer, mathematician and astronomer Ptolemy. ... Eutocius of Ascalon (ca. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ...

  10. ^ Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 228-229. ISBN 0471543977. 

    "the author's preface in Arabic gave fulsome praise to Mohammed, the prophet, and to al-Mamun, "the Commander of the Faithful"." Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ...

  11. ^ Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 228. ISBN 0471543977. 

    "The Arabs in general loved a good clear argument from premise to conclusion, as well as systematic organization - respects in which neither Diophantus nor the Hindus excelled." Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ...

  12. ^ Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 229. ISBN 0471543977. “in six short chapters, of the six types of equations made up from the three kinds of quantities: roots, squares, and numbers (that is x, x2, and numbers). Chapter I, in three short paragraphs, covers the case of squares equal to roots, expressed in modern notation as x2 = 5x, x2/3 = 4x, and 5x2 = 10x, giving the answers x = 5, x = 12, and x = 2 respectively. (The root x = 0 was not recognized.) Chapter II covers the case of squares equal to numbers, and Chapter III solves the cases of roots equal to numbers, again with three illustrations per chapter to cover the cases in which the coefficient of the variable term is equal to, more than, or less than one. Chapters IV, V, and VI are mor interesting, for they cover in turn the three classical cases of three-term quadratic equations: (1) squares and roots equal to numbers, (2) squares and numbers equal to roots, and (3) roots and numbers equal to squares.” 
  13. ^ a b c Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 234. ISBN 0471543977. “The Algebra of al-Khwarizmi usually is regarded as the first work on the subject, but a recent publication in Turkey raises some questions about this. A manuscript of a work by 'Abd-al-Hamid ibn-Turk, entitled "Logical Necessities in Mixed Equations," was part of a book on Al-jabr wa'l muqabalah which was evidently very much the same as that by al-Khwarizmi and was published at about the same time - possibly even earlier. The surviving chapters on "Logical Necessities" give precisely the same type of geometric demonstration as al-Khwarizmi's Algebra and in one case the same illustrative example x2 + 21 = 10x. In one respect 'Abd-al-Hamad's exposition is more thorough than that of al-Khwarizmi for he gives geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution. Similarities in the works of the two men and the systematic organization found in them seem to indicate that algebra in their day was not so recent a development as has usually been assumed. When textbooks with a conventional and well-ordered exposition appear simultaneously, a subject is likely to be considerably beyond the formative stage. ... Note the omission of Diophantus and Pappus, authors who evidently were not at first known in Arabia, although the Diophantine Arithmetica became familiar before the end of the tenth century.” 
  14. ^ a b Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 239. ISBN 0471543977. “Abu'l Wefa was a capable algebraist as well as a trigonometer. ... His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus - but without Diophantine analysis! ... In particular, to al-Karkhi is attributed the first numerical solution of equations of the form ax2n + bxn = c (only equations with positive roots were considered),” 
  15. ^ a b c d e Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 241-242. ISBN 0471543977. 

    Omar Khayyam (ca. 1050-1123), the "tent-maker," wrote an Algebra that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the sixteenth century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved." Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ...

  16. ^ a b J. L. Berggren (1990). "Innovation and Tradition in Sharaf al-Din al-Tusi's Muadalat", Journal of the American Oriental Society 110 (2), p. 304-309.
  17. ^ Victor J. Katz (1998). History of Mathematics: An Introduction, p. 255-259. Addison-Wesley. ISBN 0321016181.
  18. ^ F. Woepcke (1853). Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi. Paris.
  19. ^ a b c d Victor J. Katz (1995). "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3), p. 163-174.
  20. ^ a b Glen M. Cooper (2003). "Omar Khayyam, the Mathmetician", The Journal of the American Oriental Society 123.
  21. ^ Simon Singh. The Code Book. p. 14-20
  22. ^ Al-Kindi, Cryptgraphy, Codebreaking and Ciphers (HTML). Retrieved on 2007-01-12.
  23. ^ Aydin Sayili (1960). "Thabit ibn Qurra's Generalization of the Pythagorean Theorem", Isis 51 (1), p. 35-37.
  24. ^ R. Rashed (1994). The development of Arabic mathematics: between arithmetic and algebra. London.
  25. ^ O'Connor, John J; Edmund F. Robertson "Arabic mathematics: forgotten brilliance?". MacTutor History of Mathematics archive.  
  26. ^ Victor J. Katz (1998), History of Mathematics: An Introduction, p. 270, Addison-Wesley, ISBN 0321016181:

    "In some sense, his treatment was better than ibn al-Haytham's because he explicitly formulated a new postulate to replace Euclid's rather than have the latter hidden in a new definition." Pearson can mean Pearson PLC the media conglomerate. ... This article is about the capital of France. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era in the 21st century. ... is the 12th day of the year in the Gregorian calendar. ... Isis is an academic journal published by the University of Chicago devoted to the history of science, history of medicine, and the history of technology, as well as their cultural influences, featuring both original research articles as well as extensive book reviews and review essays. ... This article is about the capital of England and the United Kingdom. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... Pearson can mean Pearson PLC the media conglomerate. ...

  27. ^ a b Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447-494 [469], Routledge, London and New York:

    "Khayyam's postulate had excluded the case of the hyperbolic geometry whereas al-Tusi's postulate ruled out both the hyperbolic and elliptic geometries." The Encyclopedia of the History of Arabic Science is a three-volume encyclopedia covering the history of Arabic contributions to science, mathematics and technology which had a tremendous influence on the rise of the European Renaissance. ... Routledge is an imprint for books in the humanities part of the Taylor & Francis Group, which also has Brunner-Routledge, RoutledgeCurzon and RoutledgeFalmer divisions. ...

  28. ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447-494 [469], Routledge, London and New York:

    "In Pseudo-Tusi's Exposition of Euclid, [...] another statement is used instead of a postulate. It was independant of the Euclidean postulate V and easy to prove. [...] He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements." The Encyclopedia of the History of Arabic Science is a three-volume encyclopedia covering the history of Arabic contributions to science, mathematics and technology which had a tremendous influence on the rise of the European Renaissance. ... Routledge is an imprint for books in the humanities part of the Taylor & Francis Group, which also has Brunner-Routledge, RoutledgeCurzon and RoutledgeFalmer divisions. ...

  29. ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447-494 [469], Routledge, London and New York:

    "His book published in Rome considerably influenced the subsequent development of the theory of parallel lines. Indeed, J. Wallis (1616-1703) included a Latin translation of the proof of postulate V from this book in his own writing On the Fifth Postulate and the Fifth Definition from Euclid's Book 6 (De Postulato Quinto et Definitione Quinta lib. 6 Euclidis, 1663). Saccheri quited this proof in his Euclid Cleared of all Stains (Euclides ab omni naevo vindicatus, 1733). It seems possible that he borrowed the idea of considering the three hypotheses about the upper angles of the 'Saccheri quadrangle' from Pseudo-Tusi. The latter inserted the exposition of this subject into his work, taking it from the writings of al-Tusi and Khayyam." The Encyclopedia of the History of Arabic Science is a three-volume encyclopedia covering the history of Arabic contributions to science, mathematics and technology which had a tremendous influence on the rise of the European Renaissance. ... Routledge is an imprint for books in the humanities part of the Taylor & Francis Group, which also has Brunner-Routledge, RoutledgeCurzon and RoutledgeFalmer divisions. ...

  30. ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447-494 [470], Routledge, London and New York:

    "Three scientists, Ibn al-Haytham, Khayyam and al-Tusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the ninteenth century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between tthis postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investiagtions of their European couterparts. The first European attempt to prove the postulate on parallel lines - made by Witelo, the Polish scientists of the thirteenth century, while revising Ibn al-Haytham's Book of Optics (Kitab al-Manazir) - was undoubtedly prompted by Arabic sources. The proofs put forward in the fourteenth century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that Pseudo-Tusi's Exposition of Euclid had stimulated borth J. Wallis's and G. Saccheri's studies of the theory of parallel lines." The Encyclopedia of the History of Arabic Science is a three-volume encyclopedia covering the history of Arabic contributions to science, mathematics and technology which had a tremendous influence on the rise of the European Renaissance. ... Routledge is an imprint for books in the humanities part of the Taylor & Francis Group, which also has Brunner-Routledge, RoutledgeCurzon and RoutledgeFalmer divisions. ... The title page of a 1572 Latin manuscript of Ibn al-Haythams Book of Optics The Book of Optics (Arabic: Kitab al-Manazir, Latin: De Aspectibus or Perspectiva) was a seven volume treatise on optics written by the Iraqi Muslim scientist Ibn al-Haytham (Latinized as Alhacen or Alhazen...

  31. ^ Victor J. Katz (1998), History of Mathematics: An Introduction, p. 255-259, Addison-Wesley, ISBN 0321016181:

    "Another important idea introduced by al-Karaji and continued by al-Samaw'al and others was that of an inductive argument for dealing with certain arithmetic sequences. Thus al-Karaji used such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] Al-Karaji did not, however, state a general result for arbitary n. He stated his theorem for the particular integer 10 [...] His proof, nevertheless, was clearly designed to be extendable to any other integer. Pearson can mean Pearson PLC the media conglomerate. ... Abu Bakr ibn Muhammad ibn al-Husayn Al-Karaji (953 - 1029), also known as Al-karkhi was a Persian mathematician and engineer. ... (c. ... For other uses, see Aryabhata (disambiguation). ...

  32. ^ O'Connor, John J; Edmund F. Robertson "Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji". MacTutor History of Mathematics archive.  

    "Al-Karaji also uses a form of mathematical induction in his arguments, although he certainly does not give a rigorous exposition of the principle." The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ...

  33. ^ Katz (1998), p. 255:

    "Al-Karaji's argument includes in essence the two basic components of a modern argument by induction, namely the truth of the statement for n = 1 (1 = 13) and the deriving of the truth for n = k from that of n = k - 1. Of course, this second component is not explicit since, in some sense, al-Karaji's argument is in reverse; this is, he starts from n = 10 and goes down to 1 rather than proceeding upward. Nevertheless, his argument in al-Fakhri is the earliest extant proof of the sum formula for integral cubes."

  34. ^ Victor J. Katz (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3), p. 163-174:

    "The central idea in ibn al-Haytham's proof of the sum formulas was the derivation of the equation [...] Naturally, he did not state this result in general form. He only stated it for particular integers, [...] but his proof for each of those k is by induction on n and is immediately generalizable to any value of k." (Arabic: أبو علي الحسن بن الحسن بن الهيثم, Latinized: Alhacen or (deprecated) Alhazen) (965 – 1039), was an Arab[1] Muslim polymath[2][3] who made significant contributions to the principles of optics, as well as to anatomy, astronomy, engineering, mathematics, medicine, ophthalmology, philosophy, physics, psychology, visual perception, and to science in general with his introduction of the...

  35. ^ Katz (1998), p. 255-259.
  36. ^ Katz (1998), p. 259:

    "Like the proofs of al-Karaji and ibn al-Haytham, al-Samaw'al's argument contains the two basic components of an inductive proof. He begins with a value for which the result is known, here n = 2, and then uses the result for a given integer to derive the result for the next. Although al-Samaw'al did not have any way of stating, and therefore proving, the general binomial theorem, to modern readers there is only a short step from al-Samaw'al's argument to a full inductive proof of the binomial theorem."

  37. ^ O'Connor, John J; Edmund F. Robertson "Abu Ali al-Hasan ibn al-Haytham". MacTutor History of Mathematics archive.  
  38. ^ a b Swaney, Mark. History of Magic Squares.
  39. ^ Donald Routledge Hill (1996), "Engineering", in Roshdi Rashed, Encyclopedia of the History of Arabic Science, Vol. 3, p. 751-795 [769].

The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... Donald Routledge Hill (1922–1994) was an engineer and historian of science. ...

References and further reading

  • Berggren, J. Lennart (1986). Episodes in the Mathematics of Medieval Islam. New York: Springer-Verlag. ISBN 0-387-96318-9. 
  • Berggren, J. Lennart (2007). "Mathematics in Medieval Islam", in Victor J. Katz: The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. ISBN 9780691114859. 
  • Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second Edition, John Wiley & Sons, Inc. ISBN 0471543977. 
  • Cooke, Roger (1997). "Islamic Mathematics", The History of Mathematics: A Brief Course. Wiley-Interscience. ISBN 0471180823. 
  • Daffa', Ali Abdullah al- (1977). The Muslim contribution to mathematics. London: Croom Helm. ISBN 0-85664-464-1. 
  • Daffa, Ali Abdullah al- (1984). Studies in the exact sciences in medieval Islam. New York: Wiley. ISBN 0471903205. 
  • Joseph, George Gheverghese (2000). The Crest of the Peacock: Non-European Roots of Mathematics, 2nd Edition, Princeton University Press. ISBN 0691006598. 
  • Kennedy, E. S. (1984). Studies in the Islamic Exact Sciences. Syracuse Univ Press. ISBN 0815660677. 
  • Plofker, Kim (2007). "Mathematics in India", in Victor J. Katz: The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. ISBN 9780691114859. 
  • O'Connor, John J; Edmund F. Robertson "Arabic mathematics: forgotten brilliance?". MacTutor History of Mathematics archive.  
  • Rashed, Roshdi (2001). The Development of Arabic Mathematics: Between Arithmetic and Algebra, Transl. by A. F. W. Armstrong, Springer. ISBN 0792325656. 
  • Sánchez Pérez, José A (1921). Biografías de Matemáticos Árabes que florecieron en España. Madrid: Estanislao Maestre. 
  • Sezgin, Fuat (1997). Geschichte Des Arabischen Schrifttums (in German). Brill Academic Publishers. ISBN 9004020071. 
  • Suter, Heinrich (1900). Die Mathematiker und Astronomen der Araber und ihre Werke, Abhandlungen zur Geschichte der Mathematischen Wissenschaften Mit Einschluss Ihrer Anwendungen, X Heft. 
  • Youschkevitch, Adolf P.; Boris A. Rozenfeld (1960). Die Mathematik der Länder des Ostens im Mittelalter.  Sowjetische Beiträge zur Geschichte der Naturwissenschaft pp. 62-160.
  • Youschkevitch, Adolf P. (1976). Les mathématiques arabes: VIIIe-XVe siècles, translated by M. Cazenave and K. Jaouiche, Paris: Vrin. ISBN 978-2-7116-0734-1. 

Carl Benjamin Boyer (November 3, 1906 - April 26, 1976) was a historian of mathematics. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ...

External links


  Results from FactBites:
 
History of mathematics - Wikipedia, the free encyclopedia (5411 words)
Mathematical study in Egypt later continued under the Islamic Caliphate as part of Islamic mathematics, when Arabic became the written language of Egyptian scholars.
Greek mathematics studied from the time of the Hellenistic period (from 323 BC) refers to all mathematics of those who wrote in the Greek language, since Greek mathematics was now not only written by Greeks but also non-Greek scholars throughout the Hellenistic world, which was spread across the Eastern end of the Mediterranean.
The Islamic Caliphate (Islamic Empire) established across the Middle East, North Africa, Iberia, and in parts of India (in Pakistan) in the 8th century preserved and translated much of the Hellenistic mathematics (from Greek to Arabic) that was largely forgotten in Europe at the time.
  More results at FactBites »

 
 

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