Muḥammad ibn Mūsā al-Ḵwārizmī
The Islamic Caliphate (Islamic Empire) established across the Middle East, 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, they were not all 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. Some of the most important Islamic mathematicians were Persian.
Muḥammad ibn Mūsā al-Ḵwārizmī, a 9th century Persian mathematician and astronomer to the Caliph of Baghdad, wrote several important books on the Hindu-Arabic numerals and on methods for solving equations. His book On the Calculation with Hindu Numerals, written about 825, along with the work of the Arab mathematician Al-Kindi, were instrumental in spreading Indian mathematics and Indian numerals to the West. The word algorithm is derived from the Latinization of his name, Algoritmi, and the word algebra from the title of one of his works, Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing). Al-Khwarizmi is often called the "father of algebra", for his preservation of ancient algebraic methods and for his original contributions to the field. [12] [13]
Further developments in algebra were made by Abu Bakr al-Karaji (953—1029) in his treatise al-Fakhri, where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. In the 10th century, Abul Wafa translated the works of Diophantus into Arabic and developed the tangent function.
Omar Khayyam, the 12th century poet, was also a mathematician, and wrote Discussions of the Difficulties in Euclid, a book about flaws in Euclid's Elements. He gave a geometric solution to cubic equations, one of the most original developments in Islamic mathematics. He was also very influential in calendar reform. The Persian mathematician Nasir al-Din Tusi (Nasireddin) in the 13th century made advances in spherical trigonometry. He also wrote influential work on Euclid's parallel postulate.
In the 15th century, Ghiyath 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 notable Islamic mathematicians are al-Samawal, Abu'l-Hasan al-Uqlidisi, Jamshid al-Kashi, Thabit ibn Qurra, Abu Kamil and Abu Sahl al-Kuhi.
During the time of the Ottoman Empire (from the 15th century) the development of Islamic mathematics became stagnant. This parallels the stagnation of mathematics when the Romans conquered the Hellenistic world.
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 sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians four centuries earlier. In many respects, the mathematics studied today is far closer in style to that of Islamic mathematics than to that of Hellenistic mathematics.
Monday, February 26, 2007
Islamic mathematics (c. 700—1600)
Classical Indian mathematics (c. 400—1600)
Aryabhata
The Surya Siddhanta (c. 400) introduced the trigonometric functions of sine, cosine, and inverse sine, and laid down rules to determine the true motions of the luminaries, which conforms to their actual positions in the sky. The cosmological time cycles explained in the text, which was copied from an earlier work, corresponds to an average sidereal year of 365.2563627 days, which is only 1.4 seconds longer than the modern value of 365.25636305 days. This work was translated to Arabic and Latin during the Middle Ages.
Aryabhata in 499 introduced the versine function, produced the first trigonometric tables of sine, developed techniques and algorithms of algebra, infinitesimals, differential equations, and obtained whole number solutions to linear equations by a method equivalent to the modern method, along with accurate astronomical calculations based on a heliocentric system of gravitation. An Arabic translation of his Aryabhatiya was available from the 8th century, followed by a Latin translation in the 13th century. He also computed the value of π to the fourth decimal place as 3.1416. Madhava later in the 14th century computed the value of π to the eleventh decimal place as 3.14159265359.
In the 7th century, Brahmagupta identified the Brahmagupta theorem, Brahmagupta's identity and Brahmagupta's formula, and for the first time, in Brahma-sphuta-siddhanta, he lucidly explained the use of zero as both a placeholder and decimal digit and explained the Hindu-Arabic numeral system. It was from a translation of this Indian text on mathematics (around 770) that Islamic mathematicians were introduced to this numeral system, which they adapted as Arabic numerals. Islamic scholars carried knowledge of this number system to Europe by the 12th century, and it has now displaced all older number systems throughout the world. In the 10th century, Halayudha's commentary on Pingala's work contains a study of the Fibonacci sequence and Pascal's triangle, and describes the formation of a matrix.
In the 12th century, Bhaskara first conceived differential calculus, along with the concepts of the derivative, differential coefficient and differentiation. He also proved Rolle's theorem (a special case of the mean value theorem), studied Pell's equation, and investigated the derivative of the sine function. From the 14th century, Madhava and other Kerala School mathematicians, further developed his ideas. They developed the concepts of mathematical analysis and floating point numbers, and concepts fundamental to the overall development of calculus, including the mean value theorem, term by term integration, the relationship of an area under a curve and its antiderivative or integral, tests of convergence, iterative methods for solutions of non-linear equations, and a number of infinite series, power series, Taylor series and trigonometric series. In the 16th century, Jyeshtadeva consolidated many of the Kerala School's developments and theorems in the Yuktibhasa, the world's first differential calculus text, which also introduced concepts of integral calculus. Mathematical progress in India became stagnant from the late 16th century onwards due to subsequent political turmoil.
Classical Chinese mathematics (c. 400—1300)
Zu Chongzhi
Main article: Chinese mathematics
Zu Chongzhi (5th century) of the Southern and Northern Dynasties computed the value of π to seven decimal places, which remained the most accurate value of π for almost 1000 years.
In the thousand years following the Han dynasty, starting in the Tang dynasty and ending in the Song dynasty, Chinese mathematics thrived at a time when European mathematics did not exist. Developments first made in China, and only much later known in the West, include negative numbers, the binomial theorem, matrix methods for solving systems of linear equations and the Chinese remainder theorem. The Chinese also developed Pascal's triangle and the rule of three long before it was known in Europe.
Even after European mathematics began to flourish during the Renaissance, European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline, until the Jesuit missionaries carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries.
Wednesday, February 21, 2007
Greek and Hellenistic mathematics (c. 550 BC—AD 300)
Greek mathematics studied before the Hellenistic period refers only to the mathematics of Greece. 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. Greek mathematics from this point merged with Egyptian and Babylonian mathematics to give rise to Hellenistic mathematics. Most mathematical texts written in Greek were found in Greece, Egypt, Mesopotamia, Asia Minor, Sicily and Southern Italy.
Thales of Miletus
Although the earliest found Greek texts on mathematics were written after the Hellenistic period, many of these are considered to be copies of works written during and before the Hellenistic period. Nevertheless, the dates of Greek mathematics are more certain than the dates of earlier mathematical writing, since a large number of chronologies exist that, overlapping, record events year by year up to the present day. Even so, many dates are uncertain; but the doubt is a matter of decades rather than centuries.
Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures such as the Egyptians and Babylonians because those previous cultures used inductive reasoning which is using repeated observations to establish rules of mathematics, in other words, using general principles and applying them to specific examples. Inductive reasoning is abstract. The ancient Greek mathematicians, by contrast, used deductive reasoning, which is using specific examples and applying them to general principles. Deductive reasoning is concrete.
Greek mathematics is thought to have begun with Thales (c. 624—c.546 BC) and Pythagoras (c. 582—c. 507 BC). Although the extent of the influence is disputed, they were probably influenced by the ideas of Egypt, Mesopotamia and India. Pythagoras actually had travelled to Egypt for a while to learn about mathematics, geometry, and astronomy under the priests there. He received great mathematical knowledge while he was there. Pythagoras also is given credit for discovering the Pythagorean theorem, a theorem in trigonometry about how to find the square of the hypotenuse, or largest angle, of a right triangle, which is a triangle with one right angle,i.e., a 90 degree angle. In the Pythagorean theorem the squares of the legs, i.e., the shorter angles of the right triang are added to find the square of the hypotenuse. This mathematical equation is summed up as a squared+b squared=c squared. Pythagoras also made an algebraic version of the Pythagorean called Pythagorean triples, in which one number was squared and then another number was squared and the two numbers were added to each other and if they equalled a certain number squared they were considered Pythagorean triples, for example, 3 squared+4 squared=5 squared, that is a Pythagorean, because 3 squared=9 and 4 squared=16 and 9+16=25 which is 5 squared. Pythagoras also invented a means of translating musical notes into mathematical equations and mathematical equations into musical notes called Pythagorean tuning which used the ratio 3:2 to do this. He was on the first people to recognize that Venus as the morning star and Venus as the evening star were the same planet. Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. According to Proclus' commentary on Euclid, Pythagoras stated the Pythagorean theorem and constructed Pythagorean triples algebraically. It is generally conceded that Greek mathematics differed from that of its neighbors in its insistence on axiomatic proofs.[9]
Plato when he started his Academy taught mathematics there and had an inscription upon his academy that read "let none unversed in geometry enter here".
Greek and Hellenistic mathematicians were the first to give a proof for irrational numbers (due to the Pythagoreans). It is ironic that it was a Pythagorean who discovered the existence of irrational numbers because Pythagoras and his followers denied the existence of irrational numbers because it contradicted their philosophy of mathematics. Once when a bunch of Pythagoreans were on a ship one of them discovered an irrational number when he was working on a division problem because the answer was an irrational number. The other Pythagoreans now realized that this was true but they were not willing to let other people outside their group know that they had been wrong so to prevent any chance of this happening they murdered their member who had discovered the existence of irrational numbers by throwing him overboard into the sea. And the first to develop Eudoxus's method of exhaustion, and the Sieve of Eratosthenes for uncovering prime numbers. They took the ad hoc methods of constructing a circle or an ellipse and developed a comprehensive theory of conics; they took many various formulas for areas and volumes and deduced methods to separate the correct from the incorrect and generate general formulas. The first recorded abstract proofs are in Greek, and all extant studies of logic proceed from the methods set down by Aristotle. Euclid, in the Elements, wrote a book that would be used as a mathematics textbook throughout Europe, the Near East and North Africa for almost two thousand years. In addition to the familiar theorems of geometry, such as the Pythagorean theorem, The Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.
Some say the greatest of Greek mathematicians, if not of all time, was Archimedes (287—212 BC) of Syracuse. According to Plutarch, at the age of 75, while drawing mathematical formulas in the dust, he was run through with a spear by a Roman soldier. Roman society has left little evidence of an interest in pure mathematics.
Ancient Babylonian mathematics (c. 1800—550 BC)
Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (present-day Iraq) from the days of the early Sumerians until the beginning of the Hellenistic period. It is named Babylonian mathematics due to the central role of Babylon as a place of study, which ceased to exist during the Hellenistic period. From this point, Babylonian mathematics merged with Greek and Egyptian mathematics to give rise to Hellenistic mathematics.
In contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework. The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics which include fractions, algebra, quadratic and cubic equations, and the calculation of Pythagorean triples (see Plimpton 322).[7] The tablets also include multiplication tables, trigonometry tables and methods for solving linear and quadratic equations. The Babylonian tablet YBC 7289 gives an approximation to √2 accurate to five decimal places.
Babylonian mathematics was written using a sexagesimal (base-60) numeral system. From this we derive the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle. Babylonian advances in mathematics were facilitated by the fact that 60 has many divisors. Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values, much as in the decimal system. They lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.
Ancient Egyptian mathematics (c. 1850—600 BC)
Egyptian mathematics refers to mathematics written in the Egyptian language. From the Hellenistic period, Greek replaced Egyptian as the written language of Egyptian scholars, and from this point Egyptian mathematics merged with Greek and Babylonian mathematics to give rise to Hellenistic mathematics. Mathematical study in Egypt later continued under the Islamic Caliphate as part of Islamic mathematics, when Arabic became the written language of Egyptian scholars.
The oldest mathematical text discovered so far is the Moscow papyrus, which is an Egyptian Middle Kingdom papyrus dated c. 2000—1800 BC.[citations needed] Like many ancient mathematical texts, it consists of what are today called "word problems" or "story problems", which were apparently intended as entertainment. One problem is considered to be of particular importance because it gives a method for finding the volume of a frustum: "If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top. You are to square this 4, result 16. You are to double 4, result 8. You are to square 2, result 4. You are to add the 16, the 8, and the 4, result 28. You are to take one third of 6, result 2. Your are to take 28 twice, result 56. See, it is 56. You will find it right."
The Rhind papyrus (c. 1650 BC [1]) is another major Egyptian mathematical text, an instruction manual in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge (see [2]), including composite and prime numbers; arithmetic, geometric and harmonic means; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory (namely, that of the number 6)[3]. It also shows how to solve first order linear equations [4] as well as arithmetic and geometric series [5].
Also, three geometric elements contained in the Rhind papyrus suggest the simplest of underpinnings to analytical geometry: (1) first and foremost, how to obtain an approximation of π accurate to within less than one percent; (2) second, an ancient attempt at squaring the circle; and (3) third, the earliest known use of a kind of cotangent.
mathematic now
Long before the earliest written records, there are drawings that indicate a knowledge of mathematics and of measurement of time based on the stars. For example, paleontologists have discovered ochre rocks in a cave in South Africa adorned with scratched geometric patterns dating back to c. 70,000 BC.[1] Also prehistoric artifacts discovered in Africa and France, dated between 35,000 BC and 20,000 BC,[citations needed] indicate early attempts to quantify time.[citations needed] Evidence exists that early counting involved women who kept records of their monthly biological cycles; twenty-eight, twenty-nine, or thirty scratches on bone or stone, followed by a distinctive scratching on the bone or stone, for example. Moreover, hunters had the concepts of one, two, and many, as well as the idea of none or zero, when considering herds of animals.[2][3]
The Ishango Bone, found in the area of the headwaters of the Nile River (northeastern Congo), dates as early as 20,000 BC. One common interpretation is that the bone is the earliest known demonstration[4] of sequences of prime numbers and Ancient Egyptian multiplication. Predynastic Egyptians of the 5th millennium BC pictorially represented geometric spatial designs. It has been claimed that Megalithic monuments from as early as the 5th millennium BC in Egypt,[citations needed] and then subsequently England and Scotland from the 3rd millennium BC,[5] incorporate geometric ideas such as circles, ellipses, and Pythagorean triples in their design,[citations needed] as well as a possible understanding of the measurement of time based on the movement of the stars. From circa 3100 BC, Egyptians introduced the earliest known decimal system,[citations needed] allowing indefinite counting by way of introducing new symbols. Circa 2600 BC, Egypt's massive construction techniques represent not only precision surveying but also suggest knowledge of the golden ratio.[citations needed]
The earliest known mathematics in ancient India dates back to circa 3000-2600 BC in the Indus Valley Civilization (Harappan civilization) of North India and Pakistan, which developed a system of uniform weights and measures that used the decimal system, a surprisingly advanced brick technology which utilised ratios, streets laid out in perfect right angles, and a number of geometrical shapes and designs, including cuboids, barrels, cones, cylinders, and drawings of concentric and intersecting circles and triangles. Mathematical instruments discovered include an accurate decimal ruler with small and precise subdivisions, a shell instrument that served as a compass to measure angles on plane surfaces or in horizon in multiples of 40–360 degrees, a shell instrument used to measure 8–12 whole sections of the horizon and sky, and an instrument for measuring the positions of stars for navigational purposes. The Indus script has not yet been deciphered; hence very little is known about the written forms of Harappan mathematics. Archeological evidence has led some historians to believe that this civilization used a base 8 numeral system and possessed knowledge of the ratio of the length of the circumference of the circle to its diameter, thus a value of π.[6]
mathematic history
See timeline of mathematics for a timeline of events in mathematics. See list of mathematicians for a list of biographies of mathematicians.
The Compendious Book on Calculation by Completion and Balancing
The word "mathematics" comes from the Greek μάθημα (máthema) which means "science, knowledge, or learning"; μαθηματικός (mathematikós) means "fond of learning". Today, the term refers to a specific body of knowledge -- the deductive study of quantity, structure, space and change.
Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments come to light only in a few locales. The most ancient mathematical texts available are from ancient India circa 1500BC-500 BC (Rigveda - Sulba Sutras) and ancient Egypt in the Middle Kingdom period circa 1300-1200 BC (Berlin 6619), Mesopotamia circa 1800 BC (Plimpton 322). All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. The Han Dynasty in ancient China contributed the Sea Island Manual and The Nine Chapters on the Mathematical Art from the 2nd century BC to the 2nd century AD. Ancient Greece and the Hellenistic cultures of Egypt, Mesopotamia and the city of Syracuse increased mathematical knowledge. Jain mathematicians contributed from the 4th century BC to the 2nd century AD, while Hindu mathematicians from the 5th century and Islamic mathematicians from the 9th century made major contributions to mathematics.
One striking feature of the history of ancient and medieval mathematics is that bursts of mathematical development were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an ever increasing pace, and this continues to the present day.
