The earliest extant Chinese mathematics dates from the Shang Dynasty (1600–1046 BC), and consists of numbers scratched on a tortoise shell [9] [10]. These numbers were represented by means of a decimal notation. For example, the number 123 is written (from top to bottom) as the symbol for 1 followed by the symbol for 100, then the symbol for 2 followed by the symbol for 10, then the symbol for 3. This was the most advanced number system in the world at the time, and allowed calculations to be carried out on the suan pan or (Chinese abacus). The date of the invention of the suan pan is not certain, but the earliest written mention dates from AD 190, in Xu Yue's Supplementary Notes on the Art of Figures.
In China, the Emperor Qin Shi Huang (Shi Huang-ti) commanded in 212 BC that all books in Qin Empire other than officially sanctioned ones should be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics.
From the Western Zhou Dynasty (from 1046 BC), the oldest mathematical work to survive the book burning is the I Ching, which uses the 8 binary 3-tuples (trigrams) and 64 binary 6-tuples (hexagrams) for philosophical, mathematical, and mystical purposes. The binary tuples are composed of broken and solid lines, called yin (female) and yang (male), respectively (see King Wen sequence).
The oldest existent work on geometry in China comes from the philosophical Mohist canon c. 330 BC, compiled by the followers of Mozi (470–390 BC). The Mo Jing described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well.
After the book burning, the Han dynasty (202 BC–220 AD) produced works of mathematics which presumably expand on works that are now lost. The most important of these is The Nine Chapters on the Mathematical Art, the full title of which appeared by AD 179, but existed in part under other titles beforehand. It consists of 246 word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for Chinese pagoda towers, engineering, surveying, and includes material on right triangles and π. It also made use of Cavalieri's principle on volume more than a thousand years before Cavalieri would propose it in the West. It created mathematical proof for the Pythagorean theorem, and a mathematical formula for Gaussian elimination. Liu Hui commented on the work by the 3rd century AD.
In addition, the mathematical works of the Han astronomer and inventor Zhang Heng (AD 78–139) had a formulation for pi as well, which differed from Liu Hui's calculation. Zhang Heng used his formula of pi to find spherical volume. There was also the written work of the mathematician and music theorist Jing Fang (78–37 BC); by using the Pythagorean comma, Jing observed that 53 just fifths approximates 31 octaves. This would later lead to the discovery of 53 equal temperament, and was not calculated precisely elsewhere until the German Nicholas Mercator did so in the 17th century.
The Chinese also made use of the complex combinatorial diagram known as the magic square and magic circles, described in ancient times and perfected by Yang Hui (AD 1238–1398).
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.
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 such as Matteo Ricci carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries.
[edit] Indian mathematics (c. 800 BC–1600 AD)
Statue of
Aryabhata. As there is no known information regarding his appearance, any image of Aryabhata originates from an artist's conception.
The earliest civilization on the Indian subcontinent is the Indus Valley Civilization that flourished between 2600 and 1900 BC in the Indus river basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization.[27]
Vedic mathematics began in India in the early Iron Age. The Shatapatha Brahmana (c. 9th century BC), which approximates the value of π,[28] and the Sulba Sutras (c. 800–500 BC) were geometry texts that used irrational numbers, prime numbers, the rule of three and cube roots; computed the square root of 2 to one part in one hundred thousand; gave the method for constructing a circle with approximately the same area as a given square,[29] solved linear and quadratic equations; developed Pythagorean triples algebraically, and gave a statement and numerical proof of the Pythagorean theorem.
Pāṇini (c. 5th century BC) formulated the rules for Sanskrit grammar. His notation was similar to modern mathematical notation, and used metarules, transformations, and recursion. Pingala (roughly 3rd-1st centuries BC) in his treatise of prosody uses a device corresponding to a binary numeral system. His discussion of the combinatorics of meters corresponds to an elementary version of the binomial theorem. Pingala's work also contains the basic ideas of Fibonacci numbers (called mātrāmeru).[30]
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, correspond 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 into to Arabic and Latin during the Middle Ages.
Aryabhata, in 499, introduced the versine function, produced the first Indian trigonometric tables of sine, developed techniques and algorithms of algebra, infinitesimals, and differential equations, and obtained whole number solutions to linear equations by a method equivalent to modern methods, 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 gave a value of π corresponding to 62832/20000 = 3.1416. In the 14th century, Madhava of Sangamagrama found the Madhava–Leibniz series, and, using 21 terms, computed the value of π 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 (c. 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 stated 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, the integral test for convergence, iterative methods for solutions to 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 stagnated from the late 16th century to the 20th century, due to political turmoil.
[edit] Islamic mathematics (c. 800–1500)
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 9th century, Muḥammad ibn Mūsā al-Ḵwārizmī 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 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 fundamental contributions to the field.[31] He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,[32] and he was the first to teach algebra in an elementary form and for its own sake.[33] He also introduced the fundamental method of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which Al-Khwarizmi originally described as al-jabr.[34] His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."[35]
Further developments in algebra were made by Al-Karaji in his treatise al-Fakhri, where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. The first known proof by mathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes.[36] The historian of mathematics, F. Woepcke,[37] praised Al-Karaji for being "the first who introduced the theory of algebraic calculus." Also in the 10th century, Abul Wafa translated the works of Diophantus into Arabic and developed the tangent function. Ibn al-Haytham was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a paraboloid, and was able to generalize his result for the integrals of polynomials up to the fourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.[38]
In the late 11th century, Omar Khayyam wrote Discussions of the Difficulties in Euclid, a book about flaws in Euclid's Elements, especially the parallel postulate, and laid the foundations for analytic geometry and non-Euclidean geometry.[citation needed] He was also the first to find the general geometric solution to cubic equations. He was also very influential in calendar reform.[citation needed]
In the late 12th century, Sharaf al-Dīn al-Tūsī introduced the concept of a function,[39] and he was the first to discover the derivative of cubic polynomials.[40] His Treatise on Equations developed concepts related to differential calculus, such as the derivative function and the maxima and minima of curves, in order to solve cubic equations which may not have positive solutions.[41]
In the 13th century, Nasir al-Din Tusi (Nasireddin) 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 Muslim mathematicians included al-Samawal, Abu'l-Hasan al-Uqlidisi, Jamshid al-Kashi, Thabit ibn Qurra, Abu Kamil and Abu Sahl al-Kuhi.
Other achievements of Muslim mathematicians during this period include the development of algebra and algorithms (see Muhammad ibn Mūsā al-Khwārizmī), the development of spherical trigonometry,[42] the addition of the decimal point notation to the Arabic numerals, the discovery of all the modern trigonometric functions besides the sine, al-Kindi's introduction of cryptanalysis and frequency analysis, the development of analytic geometry by Ibn al-Haytham, the beginning of algebraic geometry by Omar Khayyam, the first refutations of Euclidean geometry and the parallel postulate by Nasīr al-Dīn al-Tūsī, the first attempt at a non-Euclidean geometry by Sadr al-Din, the development of an algebraic notation by al-Qalasādī,[43] and many other advances in algebra, arithmetic, calculus, cryptography, geometry, number theory and trigonometry.
During the time of the Ottoman Empire from the 15th century, the development of Islamic mathematics became stagnant.
