Ancient Indian mathematicians (800 BC-1200 AD) were basically number theorists, and their mathematics an appendage to Astronomy. The difference between Hindu and Greek mathematics was also greater in its focus. While Hindu mathematics was more arithmetic oriented, Greek mathematics was geometric centered. The growth of Hindu mathematics falls into two periods: The SULVASUTRA period starting roughly from 800 BC to 200 AD Astronomical and Mathematical period starting from 400 AD to 1200 AD

Brahmagupta Brahamagupta was the foremost Indian mathematician of his time. He was born possibly in 598 in Ujjain, India. Brahmagupta's well known book was Brahmasphutasiddhanta (The Opening of the Universe) which he wrote in 628. The book has 25 chapters and Brahmagupta tells us in the text that he wrote it at Bhillamala which today is the city of Bhinmal. This was the capital of the lands ruled by the Gurjara dynasty. Brahmagupta became the head of an astronomical observatory at Ujjain and his second work on mathematics and astronomy was the Khandakhadyaka. Brahmagupta's understanding of the number systems went far beyond that of others of the period. He made advances in astronomy and most importantly in number systems including algorithms for square roots and the solution of quadratic equation.

In the Brahmasphutasiddhanta he defined zero as the result of subtracting a number from itself. He gave some properties as follows: When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero. He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers): A debt minus zero is a debt. A fortune minus zero is a fortune. Zero minus zero is a zero. A debt subtracted from zero is a fortune. A fortune subtracted from zero is a debt. The product of zero multiplied by a debt or fortune is zero. The product of zero multiplied by zero is zero. The product or quotient of two fortunes is one fortune. The product or quotient of two debts is one fortune. He wrote about quadratic equations, lunar eclipses, planetary conjunctions, and the determination of the positions of the planets. He died in the year 670 AD.

The Muslim Empire

Development and Spread of Hindu-Arabic Numbers A numeration system using base 10, the zero symbol and powerful arithmetic techniques is developed by the Hindus, approx. 150 B.C. to 800 A.D. The Hindu numeration system is adopted by the Arabs and spread throughout their sphere of influence (approx. 700 A.D. to 1250 A.D.). Preservation of Greek Mathematics Arab scholars translated and studied ancient Greek Mathematical works, principally in Baghdad. C. Development of Algebra and Trigonometry Arab mathematicians find methods of solution for quadratic, cubic and higher degree polynomial eqns. The English word “algebra” is derived from the title of an Arabic book describing these methods. Hindu trigonometry, especially sine tables, is improved and advanced by Arab mathematicians

The Islamic Empire established across Persia, the Middle East, Central Asia, North Africa, Iberia, and in parts of India in the 8th century made significant contributions towards mathematics. Although most Islamic texts on mathematics were written in Arabic, most of them were not written by Arabs, since much like the status of Greek in the Hellenistic world, Arabic was used as the written language of non-Arab scholars throughout the Islamic world at the time. Persians contributed to the world of Mathematics alongside Arabs.

In the late 11th century, Omar Khayyam wrote Discussions of the Difficulties in Euclid, a book about what he perceived as flaws in Euclid's Elements, especially the parallel postulate. He was also the first to find the general geometric solution to cubic equations. In the 13th century, Nasir al-Din Tusi (1201-1274) made advances in trigonometry. He also wrote influential work on Euclid's parallel postulate. In the 15th century, al-Kashi computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating nth roots, which was a special case of the methods given many centuries later by Ruffini and Horner. Other achievements of Muslim mathematicians during this period include the addition of the decimal point notation to the Arabic numerals, the discovery of all the modern trigonometric functions besides the sine,, the development of analytic geometry by Ibn al-Haytham, the beginning of algebraic geometry by Omar Khayyam .

Baghdad and the House of Wisdom About the middle of the ninth century Bait Al Hikma, the "House of Wisdom" was founded in Baghdad which combined the functions of a library, academy, and translation bureau. Baghdad attracted scholars from the Islamic world and became a great center of learning.

Baghdad at that time was at cultural crossroads, and, under the patronage of the Abbasid caliphs, the so called House of Wisdom at Baghdad, produced a Golden Age of Arabic science and mathematics. In Baghdad, scholars encountered and built upon the ideas of ancient Greek and Indian mathematicians. There, Al-Khwarizmi encountered the Indian numeral system (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and he wrote a treatise on what we call Arabic numerals. It was translated into Latin in the twelfth century as Algoritmi de numero Indorum (that is, Al-Khwarizmi on the Hindu Art of Reckoning) and was crucial in the introduction of Arabic numerals to medieval Europe. It may well represent the first use of zero as a positional place holder. From that title, we have the word "algorithm."

The Great Mosque of Cordoba The Great Mosque, Cordoba During the Middle Ages Cordoba was the greatest center of learning in Europe, second only to Baghdad in the Islamic world. SEE: VIDEO 1 The Great Mosque of Cordoba

Arabic Translation of Apollonius’ Conic Sections.

Arabic Translation of Ptolemy’s Almagest Pages from a 13th century Arabic edition of Ptolemy’s Almagest.

A page from Al-Khwarizmi’s kitab “Hisab al-jabr w'al”. Textbook of Algebra A page from Al-Khwarizmi’s kitab “Hisab al-jabr w'al”. We’ll discuss him separately.                                                                                                                                                                           

Abu Abdullah Muhammad bin Musa al-Khwarizmi, 800 AD was a Persian mathematician, scientist, and author. He worked in Baghdad and wrote all his works in Arabic. He developed the concept of an algorithm in mathematics. The words "algorithm" and "algorism" derive ultimately from his name. His systematic and logical approach to solving linear and quadratic equations gave shape to the discipline of algebra, a word that is derived from the name of his book on the subject, Hisab al-jabr wa al-muqabala (“al-jabr” became “algebra”).He was also instrumental in promoting the Hindu-Arabic numeration system. Al-Khwarizmi

Nasir al-Din al-Tusi (1201-1274) He was born in Tus in Khurasan (now Iran). In 1256, the invading (istila) Mongols led by Hulagu Khan, grandson of Genghis Khan, captured Alamut. Quite interested in the sciences, Hulagu treated Tusi with respect and appointed him one of his ministers.

Later, while serving as an administrator of Auqaf, Al-Tusi persuaded Hulagu to build an observatory at Meragha, with al-Tusi as its director. The observatory was equipped with the best instruments from Baghdad and other Islamic centers of learning. Al-Tusi wrote some 150 books in Arabic, Persian, and Turkish, with sixty-four treatises known to have survived. Many were consolidated accounts of what others had previously written. But he also made many original contributions, most particularly in mathematics.

He was the first to treat trigonometry as a separate science, rather than just a set of tools for astronomy. He edited the definitive Arabic versions of the works of Euclid, Archimedes, Ptolemy and Theodosius. He was familiar with the properties of the so-called “Pascal’s Triangle” long before Pascal’s birth. His most famous astronomical work is the four-volume Al-Zij-Ilkhani (Astronomic Tables of Ilkhan), dedicated to Ilkhan (Hulagu Khan).

Ulugh Bey (1394-1449) Ulugh Beg was also notable for his work in astronomy-related mathematics, such as trigonometry and spherical geometry. He built the great Ulugh Beg Observatory in Samarkand between 1424 and 1429. It was considered by scholars to have been one of the finest observatories in the Islamic world at the time and the largest in Central Asia.

Ulugh Bey was the grandson of the conqueror, Timur (1336–1405). Between 1417 and 1420, he built a madrasa ("university" or "institute") on Registan Square in Samarkand (currently in Uzbekistan), and he invited numerous Islamic astronomers and mathematicians to study there. The madrasa building still survives. Ulugh Beg's most famous pupil in astronomy was Ali Qushchi (Kuşçu) (died in 1474). In mathematics, Ulugh Beg wrote accurate trigonometric tables of sine and tangent values correct to at least eight decimal places. SEE: VIDEO 2

Period of Transmission (1000 AD – 1500 AD) A. Discovery of Greek and Hindu-Arab mathematics Greek mathematics texts are translated from Arabic into Latin; Greek ideas about logic, geometrical reasoning, and a rational view of the world are re-discovered. Arab works on algebra and trigonometry are also translated into Latin and disseminated throughout Europe. B. Spread of the Hindu-Arabic numeration system Hindu-Arabic numerals slowly spread over Europe Pen and paper arithmetic algorithms based on Hindu-Arabic numerals replace the use the abacus.

Leonardo of Pisa(Fibonacci) (1175-1240 AD) From Leonardo of Pisa’s famous book Liber Abaci -The Book of Calculation (1202 A.D.): "These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated." Leonardo of Pisa(Fibonacci) (1175-1240 AD)

Fibonacci, or more correctly Leonardo da Pisa, was born in Pisa in 1175 AD. He was the son of a Pisan merchant who also served as a customs officer in North Africa. He travelled widely in Barbary (Algeria) and was later sent on business trips to Egypt, Syria, Greece, Sicily and Provence. In 1200 he returned to Pisa and used the knowledge he had gained on his travels to write Liber abaci in which he introduced the Latin-speaking world to the decimal number system.