The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.
Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322 (Babylonian mathematics c. 1900 BC),^{[1]} the Rhind Mathematical Papyrus (Egyptian mathematics c. 20001800 BC)^{[2]} and the Moscow Mathematical Papyrus (Egyptian mathematics c. 1890 BC). All of these texts concern the socalled Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
The Greek and Hellenistic contribution greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics.^{[3]} The word "mathematics" itself derives from the ancient Greek μάθημα (mathema), meaning "subject of instruction".^{[4]} Chinese mathematics made early contributions, including a place value system. The HinduArabic numeral system and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in India and was transmitted to the west via Islamic mathematics.^{[5]}^{[6]} Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations.^{[7]} Many Greek and Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe.
From ancient times through the Middle Ages, bursts of mathematical creativity 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 increasing pace that continues through the present day.
The origins of mathematical thought lie in the concepts of number, magnitude, and form.^{[8]} Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in huntergatherer societies. That the concept of number evolved gradually over time is evident in that some languages today preserve the distinction between "one", "two", and "many", but not of numbers larger than two.^{[8]}
The oldest known mathematical object is the Lebombo bone, discovered in the Lebombo mountains of Swaziland and dated to approximately 35,000 BC.^{[9]} It consists of 29 distinct notches deliberately cut into a baboon's fibula.^{[10]} There is evidence that women used counting to keep track of their menstrual cycles; 28 to 30 scratches on bone or stone, followed by a distinctive marker.^{[11]} Also prehistoric artifacts discovered in Africa and France, dated between 35,000 and 20,000 years old,^{[12]} suggest early attempts to quantify time.^{[13]}
The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be as much as 20,000 years old and consists of a series of tally marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers^{[10]} or a six month lunar calendar.^{[14]} Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. It has been claimed that megalithic monuments in England and Scotland, dating from the 3rd millennium BC, incorporate geometric ideas such as circles, ellipses, and Pythagorean triples in their design.^{[15]}
Babylonian mathematics refers to any mathematics of the people of Mesopotamia (modern Iraq) from the days of the early Sumerians until the beginning of the Hellenistic period.^{[16]} It is named Babylonian mathematics due to the central role of Babylon as a place of study. During the Hellenistic period Babylonian mathematics merged with Greek and Egyptian mathematics to give rise to Hellenistic mathematics. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic 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.^{[17]} 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 earliest evidence of written mathematics dates back to the ancient Sumerians, who built the earliest civilization in Mesopotamia. They developed a complex system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.^{[18]}
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 regular reciprocal pairs.^{[19]} The tablets also include multiplication 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 were written using a sexagesimal (base60) numeral system. From this derives the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. 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 placevalue 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.
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 Arab Empire as part of Islamic mathematics, when Arabic became the written language of Egyptian scholars.
The most extensive Egyptian mathematical text is the Rhind papyrus (sometimes also called the Ahmes Papyrus after its author), dated to c. 1650 BC but likely a copy of an older document from the Middle Kingdom of about 20001800 BC.^{[20]} It is an instruction manual for students 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,^{[21]} 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).^{[22]} It also shows how to solve first order linear equations ^{[23]} as well as arithmetic and geometric series.^{[24]}
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.
Another significant Egyptian mathematical text is the Moscow papyrus, also from the Middle Kingdom period, dated to c. 1890 BC.^{[25]} 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. You are to take 28 twice, result 56. See, it is 56. You will find it right."
Finally, the Berlin papyrus (c. 1300 BC ^{[26]}) shows that ancient Egyptians could solve a secondorder algebraic equation.^{[27]}
Greek mathematics refers to mathematics written in the Greek language between about 600 BC and AD 300.^{[28]} Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics.
Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of preGreek mathematics show the use of inductive reasoning, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to derive conclusions from definitions and axioms, and used mathematical rigor to prove them.^{[29]}
Greek mathematics is thought to have begun with Thales of Miletus (c. 624–c.546 BC) and Pythagoras of Samos (c. 582–c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by Egyptian and Babylonian mathematics. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.
Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.^{[30]} Pythagoras established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number".^{[31]} It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The Pythagoreans are credited with the first proof of the Pythagorean theorem,^{[32]} though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers.^{[citation needed]}
Eudoxus (408–c.355 BC) developed the method of exhaustion, a precursor of modern integration. Aristotle (384—c.322 BC) first wrote down the laws of logic. Euclid (c. 300 BC) is the earliest example of the format still used in mathematics today, definition, axiom, theorem, and proof. He also studied conics. His book, Elements, was known to all educated people in the West until the middle of the 20th century.^{[33]} In addition to the familiar theorems of geometry, such as the Pythagorean theorem, Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers. The Sieve of Eratosthenes (c. 230 BC) was used to discover prime numbers.
Archimedes (c.287–212 BC) of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of Pi.^{[34]} He also studied the spiral bearing his name, formulas for the volumes of surfaces of revolution, and an ingenious system for expressing very large numbers.
Early Chinese mathematics is so different from that of other parts of the world that it is reasonable to assume independent development.^{[35]} The oldest extant mathematical text from China is the Chou Pei Suan Ching, variously dated to between 1200 BC and 100 BC, though a date of about 300 BC appears reasonable.^{[36]}
Of particular note is the use in Chinese mathematics of a decimal positional notation system, the socalled "rod numerals" in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten.^{[37]} Thus, the number 123 would be written using the symbol for "1", followed by the symbol for "100", then the symbol for "2" followed by the symbol for "10", followed by the symbol for "3". This was the most advanced number system in the world at the time, apparently in use several centuries before the common era and well before the development of the Indian numeral system.^{[38]} Rod numerals allowed the representation of numbers as large as desired 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.
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.
In 212 BC, the Emperor Qin Shi Huang (Shi Huangti) commanded all books in the Qin Empire other than officially sanctioned ones be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date. After the book burning of 212 BC, the Han dynasty (202 BC–220 AD) produced works of mathematics which presumably expanded 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.
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.^{[39]}
The oldest extant mathematical records from India are the Shatapatha Brahmana (c. 9th century BC), which approximates the value of π,^{[40]} and the Sulba Sutras (c. 800–500 BC), 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,^{[41]} solved linear and quadratic equations; developed Pythagorean triples algebraically, and gave a statement and numerical proof of the Pythagorean theorem.^{[citation needed]}
Pāṇini (c. 5th century BC) formulated the rules for Sanskrit grammar.^{[42]} His notation was similar to modern mathematical notation, and used metarules, transformations, and recursion. Pingala (roughly 3rd1st 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).^{[43]}
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.^{[44]} 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.
In the 5th century AD, Aryabhatta wrote the Aryabhatiya, a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology.^{[45]} Though about half of the entries are wrong, it is in the Aryabhatiya that the decimal placevalue system first appears. Several centuries later, the Muslim mathematician Abu Rayhan Biruni described the Aryabhatiya as a "mix of common pebbles and costly crystals".^{[46]}
In the 7th century, Brahmagupta identified the Brahmagupta theorem, Brahmagupta's identity and Brahmagupta's formula, and for the first time, in Brahmasphutasiddhanta, he lucidly explained the use of zero as both a placeholder and decimal digit, and explained the HinduArabic numeral system.^{[47]} 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.^{[citation needed]}
In the 12th century, Bhaskara^{[48]} 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. In the 14th century, Madhava of Sangamagrama found the Madhava–Leibniz series, and, using 21 terms, computed the value of π as 3.14159265359. Madhava and other Kerala School mathematicians 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 nonlinear equations, and a number of infinite series, power series, Taylor series, and trigonometric series.^{[citation needed]} 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.
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 nonArab scholars throughout the Islamic world at the time. Persians contributed to the world of Mathematics alongside Arabs.
In the 9th century, Muḥammad ibn Mūsā alḴwārizmī wrote several important books on the HinduArabic numerals and on methods for solving equations. His book On the Calculation with Hindu Numerals, written about 825, along with the work of AlKindi, 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, AlKitāb almukhtaṣar fī hīsāb alğabr wa’lmuqābala (The Compendious Book on Calculation by Completion and Balancing). AlKhwarizmi is often called the "father of algebra", for his fundamental contributions to the field.^{[49]} He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,^{[50]} and he was the first to teach algebra in an elementary form and for its own sake.^{[51]} 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 AlKhwarizmi originally described as aljabr.^{[52]} 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."^{[53]}
Further developments in algebra were made by AlKaraji in his treatise alFakhri, 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 AlKaraji around 1000 AD, who used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes.^{[54]} The historian of mathematics, F. Woepcke,^{[55]} praised AlKaraji 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 alHaytham 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.^{[56]}
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 nonEuclidean 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 alDīn alTūsī introduced the concept of a function,^{[57]} and he was the first to discover the derivative of cubic polynomials.^{[58]} 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.^{[59]}
In the 13th century, Nasir alDin Tusi (Nasireddin) made advances in spherical trigonometry. He also wrote influential work on Euclid's parallel postulate. In the 15th century, Ghiyath alKashi 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 alSamawal, Abu'lHasan alUqlidisi, Jamshid alKashi, Thabit ibn Qurra, Abu Kamil and Abu Sahl alKuhi.
Other achievements of Muslim mathematicians during this period include the development of algebra and algorithms, the development of spherical trigonometry,^{[60]} the addition of the decimal point notation to the Arabic numerals, the discovery of all the modern trigonometric functions besides the sine, alKindi's introduction of cryptanalysis and frequency analysis, the development of analytic geometry by Ibn alHaytham, the beginning of algebraic geometry by Omar Khayyam, the first refutations of Euclidean geometry and the parallel postulate by Nasīr alDīn alTūsī, the first attempt at a nonEuclidean geometry by Sadr alDin, the development of an algebraic notation by alQalasādī,^{[61]} 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.
Medieval European interest in mathematics was driven by concerns quite different from those of modern mathematicians. One driving element was the belief that mathematics provided the key to understanding the created order of nature, frequently justified by Plato's Timaeus and the apocryphal biblical passage (in the Book of Wisdom) that God had ordered all things in measure, and number, and weight^{[62]}.
Boethius provided a place for mathematics in the curriculum when he coined the term quadrivium to describe the study of arithmetic, geometry, astronomy, and music. He wrote De institutione arithmetica, a free translation from the Greek of Nicomachus's Introduction to Arithmetic; De institutione musica, also derived from Greek sources; and a series of excerpts from Euclid's Elements. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.^{[63]}^{[64]}
In the 12th century, European scholars traveled to Spain and Sicily seeking scientific Arabic texts, including alKhwarizmi's The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester, and the complete text of Euclid's Elements, translated in various versions by Adelard of Bath, Herman of Carinthia, and Gerard of Cremona.^{[65]}^{[66]}
These new sources sparked a renewal of mathematics. Fibonacci, writing in the Liber Abaci, in 1202 and updated in 1254, produced the first significant mathematics in Europe since the time of Eratosthenes, a gap of more than a thousand years. The work introduced HinduArabic numerals to Europe, and discussed many other mathematical problems.
The fourteenth century saw the development of new mathematical concepts to investigate a wide range of problems.^{[67]} One important contribution was development of mathematics of local motion.
Thomas Bradwardine proposed that speed (V) increases in arithmetic proportion as the ratio of force (F) to resistance (R) increases in geometric proportion. Bradwardine expressed this by a series of specific examples, but although the logarithm had not yet been conceived, we can express his conclusion anachronistically by writing: V = log (F/R).^{[68]} Bradwardine's analysis is an example of transferring a mathematical technique used by alKindi and Arnald of Villanova to quantify the nature of compound medicines to a different physical problem.^{[69]}
One of the 14thcentury Oxford Calculators, William Heytesbury, lacking differential calculus and the concept of limits, proposed to measure instantaneous speed "by the path that would be described by [a body] if... it were moved uniformly at the same degree of speed with which it is moved in that given instant".^{[70]}
Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (today solved by integration), stating that "a moving body uniformly acquiring or losing that increment [of speed] will traverse in some given time a [distance] completely equal to that which it would traverse if it were moving continuously through the same time with the mean degree [of speed]".^{[71]}
Nicole Oresme at the University of Paris and the Italian Giovanni di Casali independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance traveled.^{[72]} In a later mathematical commentary on Euclid's Elements, Oresme made a more detailed general analysis in which he demonstrated that a body will acquire in each successive increment of time an increment of any quality that increases as the odd numbers. Since Euclid had demonstrated the sum of the odd numbers are the square numbers, the total quality acquired by the body increases as the square of the time.^{[73]}
The development of mathematics and accounting was intertwined during the Renaissance.^{[75]} While there is no direct relationship between algebra and accounting, the teaching of the subjects and the books published often intended for the children of merchants who were sent to reckoning schools (in Flanders and Germany) or abacus schools (known as abbaco in Italy), where they learned the skills useful for trade and commerce. There is probably no need for algebra in performing bookkeeping operations, but for complex bartering operations or the calculation of compound interest, a basic knowledge of arithmetic was mandatory and knowledge of algebra was very useful.^{[76]}
Luca Pacioli's "Summa de Arithmetica, Geometria, Proportioni et Proportionalità" (Italian: "Review of Arithmetic, Geometry, Ratio and Proportion") was first printed and published in Venice in 1494. It included a 27page treatise on bookkeeping, "Particularis de Computis et Scripturis" (Italian: "Details of Calculation and Recording"). It was written primarily for, and sold mainly to, merchants who used the book as a reference text, as a source of pleasure from the mathematical puzzles it contained, and to aid the education of their sons. In Summa Arithmetica, Pacioli introduced symbols for plus and minus for the first time in a printed book, symbols that became standard notation in Italian Renaissance mathematics. Summa Arithmetica was also the first known book printed in Italy to contain algebra.^{[77]}
Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. Regiomontanus's table of sines and cosines was published in 1533.^{[78]}
The 17th century saw an unprecedented explosion of mathematical and scientific ideas across Europe. Galileo, an Italian, observed the moons of Jupiter in orbit about that planet, using a telescope based on a toy imported from Holland. Tycho Brahe, a Dane, had gathered an enormous quantity of mathematical data describing the positions of the planets in the sky. His student, Johannes Kepler, a German, began to work with this data. In part because he wanted to help Kepler in his calculations, John Napier, in Scotland, was the first to investigate natural logarithms. Kepler succeeded in formulating mathematical laws of planetary motion. The analytic geometry developed by René Descartes (1596–1650), a French mathematician and philosopher, allowed those orbits to be plotted on a graph, in Cartesian coordinates. Simon Stevin (1585) created the basis for modern decimal notation capable of describing all numbers, whether rational or irrational.
Building on earlier work by many predecessors, Isaac Newton, an Englishman, discovered the laws of physics explaining Kepler's Laws, and brought together the concepts now known as infinitesimal calculus. Independently, Gottfried Wilhelm Leibniz, in Germany, developed calculus and much of the calculus notation still in use today. Science and mathematics had become an international endeavor, which would soon spread over the entire world.^{[79]}
In addition to the application of mathematics to the studies of the heavens, applied mathematics began to expand into new areas, with the correspondence of Pierre de Fermat and Blaise Pascal. Pascal and Fermat set the groundwork for the investigations of probability theory and the corresponding rules of combinatorics in their discussions over a game of gambling. Pascal, with his wager, attempted to use the newly developing probability theory to argue for a life devoted to religion, on the grounds that even if the probability of success was small, the rewards were infinite. In some sense, this foreshadowed the development of utility theory in the 18th–19th century.
The most influential mathematician of the 1700s was arguably Leonhard Euler. His contributions range from founding the study of graph theory with the Seven Bridges of Königsberg problem to standardizing many modern mathematical terms and notations. For example, he named the square root of minus 1 with the symbol i, and he popularized the use of the Greek letter π to stand for the ratio of a circle's circumference to its diameter. He made numerous contributions to the study of topology, graph theory, calculus, combinatorics, and complex analysis, as evidenced by the multitude of theorems and notations named for him.
Other important European mathematicians of the 18th century included Joseph Louis Lagrange, who did pioneering work in number theory, algebra, differential calculus, and the calculus of variations, and Laplace who, in the age of Napoleon did important work on the foundations of celestial mechanics and on statistics.
Throughout the 19th century mathematics became increasingly abstract. In the 19th century lived Carl Friedrich Gauss (1777–1855). Leaving aside his many contributions to science, in pure mathematics he did revolutionary work on functions of complex variables, in geometry, and on the convergence of series. He gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law.
This century saw the development of the two forms of nonEuclidean geometry, where the parallel postulate of Euclidean geometry no longer holds. The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival, the Hungarian mathematician Janos Bolyai, independently defined and studied hyperbolic geometry, where uniqueness of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. Elliptic geometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed Riemannian geometry, which unifies and vastly generalizes the three types of geometry, and he defined the concept of a manifold, which generalize the ideas of curves and surfaces.
The 19th century saw the beginning of a great deal of abstract algebra. Hermann Grassmann in Germany gave a first version of vector spaces, William Rowan Hamilton in Ireland developed noncommutative algebra. The British mathematician George Boole devised an algebra that soon evolved into what is now called Boolean algebra, in which the only numbers were 0 and 1 and in which, 1 + 1 = 1. Boolean algebra is the starting point of mathematical logic and has important applications in computer science.
AugustinLouis Cauchy, Bernhard Riemann, and Karl Weierstrass reformulated the calculus in a more rigorous fashion.
Also, for the first time, the limits of mathematics were explored. Niels Henrik Abel, a Norwegian, and Évariste Galois, a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four. Other 19th century mathematicians utilized this in their proofs that straightedge and compass alone are not sufficient to trisect an arbitrary angle, to construct the side of a cube twice the volume of a given cube, nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks.
Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of group theory, and the associated fields of abstract algebra. In the 20th century physicists and other scientists have seen group theory as the ideal way to study symmetry.
In the later 19th century, Georg Cantor established the first foundations of set theory, which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics. Cantor's set theory, and the rise of mathematical logic in the hands of Peano, L. E. J. Brouwer, David Hilbert, Bertrand Russell, and A.N. Whitehead, initiated a long running debate on the foundations of mathematics.
The 19th century saw the founding of a number of national mathematical societies: the London Mathematical Society in 1865, the Société Mathématique de France in 1872, the Circolo Mathematico di Palermo in 1884, the Edinburgh Mathematical Society in 1883, and the American Mathematical Society in 1888.
The 20th century saw mathematics become a major profession. Every year, thousands of new Ph.D.s in mathematics are awarded, and jobs are available in both teaching and industry. In earlier centuries, there were few creative mathematicians in the world at any one time.
As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: by the end of the century there were hundreds of specialized areas in mathematics and the Mathematics Subject Classification was dozens of pages long^{[80]}. More and more mathematical journals were published and, by the end of the century, the development of the world wide web led to online publishing.
In a 1900 speech to the International Congress of Mathematicians, David Hilbert set out a list of 23 unsolved problems in mathematics. These problems, spanning many areas of mathematics, formed a central focus for much of 20th century mathematics. Today, 10 have been solved, 7 are partially solved, and 2 are still open. The remaining 4 are too loosely formulated to be stated as solved or not.
Notable historical conjectures were finally proved. In 1976, Wolfgang Haken and Kenneth Appel used a computer to prove the four color theorem. Andrew Wiles, building on the work of others, proved Fermat's Last Theorem in 1995. Paul Cohen and Kurt Gödel proved that the continuum hypothesis is independent of (could neither be proved nor disproved from) the standard axioms of set theory. In 1998 Thomas Callister Hales proved the Kepler conjecture.
Mathematical collaborations of unprecedented size and scope took place. An example is the classification of finite simple groups (also called the "enormous theorem"), whose proof between 1955 and 1983 required 500odd journal articles by about 100 authors, and filling tens of thousands of pages. A group of French mathematicians, including Jean Dieudonné and André Weil, publishing under the pseudonym "Nicolas Bourbaki," attempted to exposit all of known mathematics as a coherent rigorous whole. The resulting several dozen volumes has had a controversial influence on mathematical education.^{[81]}
Differential geometry came into its own when Einstein used it in general relativity. Entire new areas of mathematics such as mathematical logic, topology, and John von Neumann's game theory changed the kinds of questions that could be answered by mathematical methods. All kinds of structures were abstracted using axioms and given names like metric spaces, topological spaces etc. As mathematicians do, the concept of an abstract structure was itself abstracted and led to category theory. Grothendieck and Serre recast algebraic geometry using sheaf theory. Large advances were made in the qualitative study of dynamical systems that Poincaré had began in the 1890s. Measure theory was developed in the late 19th and early 20th century. Applications of measures include the Lebesgue integral, Kolmogorov's axiomatisation of probability theory, and ergodic theory. Knot theory greatly expanded. Quantum mechanics led to the deveopment of functional analysis. Other new areas include , Laurent Schwarz's distribution theory, fixed point theory, singularity theory and René Thom's catastrophe theory, model theory, and Mandelbrot's fractals.
The development and continual improvement of computers, at first mechanical analog machines and then digital electronic machines, allowed industry to deal with larger and larger amounts of data to facilitate mass production and distribution and communication, and new areas of mathematics were developed to deal with this: Alan Turing's computability theory; complexity theory; Claude Shannon's information theory; signal processing; data analysis; optimization and other areas of operations research. In the preceding centuries much mathematical focus was on calculus and continuous functions, but the rise of computing and communication networks led to an increasing importance of discrete concepts and the expansion of combinatorics including graph theory. The speed and data processing abilities of computers also enabled the handling of mathematical problems that were too timeconsuming to deal with by pencil and paper calculations, leading to areas such as numerical analysis and symbolic computation.
At the same time, deep insights were made about the limitations to mathematics. In 1929 and 1930, it was proved the truth or falsity of all statements formulated about the natural numbers plus one of addition and multiplication, was decidable, i.e. could be determined by algorithm. In 1931, Kurt Gödel found that this was not the case for the natural numbers plus both addition and multiplication; this system, known as Peano arithmetic, was in fact incompletable. (Peano arithmetic is adequate for a good deal of number theory, including the notion of prime number.) A consequence of Gödel's two incompleteness theorems is that in any mathematical system that includes Peano arithmetic (including all of analysis and geometry), truth necessarily outruns proof, i.e. there are true statements that cannot be proved within the system. Hence mathematics cannot be reduced to mathematical logic, and David Hilbert's dream of making all of mathematics complete and consistent died.
One of the more colorful figures in 20th century mathematics was Srinivasa Aiyangar Ramanujan (1887–1920) who, despite being largely selfeducated, conjectured or proved over 3000 theorems, including properties of highly composite numbers, the partition function and its asymptotics, and mock theta functions. He also made major investigations in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory.
Paul Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. Mathematicians have a game equivalent to the Kevin Bacon Game, which leads to the Erdős number of a mathematician. This describes the "collaborative distance" between a person and Paul Erdős, as measured by joint authorship of mathematical papers.
In 2000, the Clay Mathematics Institute announced the Millennium Prize Problems, and in 2003 the Poincaré conjecture was solved by Grigori Perelman.
Most mathematical journals now have online versions as well as print versions, and many onlineonly journals are launched. There is an increasing drive towards open access publishing, first popularized by the arXiv.
There are many observable trends in mathematics, the most notable being that the subject is growing ever larger, computers are ever more important and powerful, the application of mathematics to bioinformatics is rapidly expanding, the volume of data to be analyzed being produced by science and industry, facilitated by computers, is explosively expanding...
by Muhammad ibn Mūsā alKhwārizmī (c. AD 820)]]
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.
Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322 (Babylonian mathematics c. 1900 BC),^{[1]} the Rhind Mathematical Papyrus (Egyptian mathematics c. 20001800 BC)^{[2]} and the Moscow Mathematical Papyrus (Egyptian mathematics c. 1890 BC). All of these texts concern the socalled Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
The Greek and Hellenistic contribution greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics.^{[3]} The study of mathematics as a subject in its own right begins in the 6th century BCE with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction".^{[4]} Chinese mathematics made early contributions, including a place value system.^{[5]}^{[6]} The HinduArabic numeral system and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in India and was transmitted to the west via Islamic mathematics.^{[7]}^{[8]} Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations.^{[9]} Many Greek and Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe.
From ancient times through the Middle Ages, bursts of mathematical creativity 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 increasing pace that continues through the present day.
, dating to perhaps 18000 to 20000 B.C.]]
The origins of mathematical thought lie in the concepts of number, magnitude, and form.^{[10]} Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in huntergatherer societies. That the concept of number evolved gradually over time is evident in that some languages today preserve the distinction between "one", "two", and "many", but not of numbers larger than two.^{[10]}
The oldest known mathematical object is the Lebombo bone, discovered in the Lebombo mountains of Swaziland and dated to approximately 35,000 BC.^{[11]} It consists of 29 distinct notches deliberately cut into a baboon's fibula.^{[12]} There is evidence that women used counting to keep track of their menstrual cycles; 28 to 30 scratches on bone or stone, followed by a distinctive marker.^{[13]} Also prehistoric artifacts discovered in Africa and France, dated between 35,000 and 20,000 years old,^{[14]} suggest early attempts to quantify time.^{[15]}
The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be as much as 20,000 years old and consists of a series of tally marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers^{[12]} or a six month lunar calendar.^{[16]} Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. It has been claimed that megalithic monuments in England and Scotland, dating from the 3rd millennium BC, incorporate geometric ideas such as circles, ellipses, and Pythagorean triples in their design.^{[17]}
Babylonian mathematics refers to any mathematics of the people of Mesopotamia (modern Iraq) from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity.^{[18]} It is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic 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.^{[19]} 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 earliest evidence of written mathematics dates back to the ancient Sumerians, who built the earliest civilization in Mesopotamia. They developed a complex system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.^{[20]}
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 regular reciprocal pairs.^{[21]} The tablets also include multiplication 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 were written using a sexagesimal (base60) numeral system. From this derives the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. 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 placevalue 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.
Egyptian mathematics refers to mathematics written in the Egyptian language. From the Hellenistic period, Greek replaced Egyptian as the written language of Egyptian scholars. Mathematical study in Egypt later continued under the Arab Empire as part of Islamic mathematics, when Arabic became the written language of Egyptian scholars.
The most extensive Egyptian mathematical text is the Rhind papyrus (sometimes also called the Ahmes Papyrus after its author), dated to c. 1650 BC but likely a copy of an older document from the Middle Kingdom of about 20001800 BC.^{[22]} It is an instruction manual for students 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,^{[23]} 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).^{[24]} It also shows how to solve first order linear equations ^{[25]} as well as arithmetic and geometric series.^{[26]}
Another significant Egyptian mathematical text is the Moscow papyrus, also from the Middle Kingdom period, dated to c. 1890 BC.^{[27]} 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. You are to take 28 twice, result 56. See, it is 56. You will find it right."
Finally, the Berlin papyrus (c. 1300 BC ^{[28]}) shows that ancient Egyptians could solve a secondorder algebraic equation.^{[29]}
Greek mathematics refers to the mathematics written in the Greek language from the time of Thales of Miletus (~600 BC) to the closure of the Academy of Athens in 529 AD.^{[30]} Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics.
Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of preGreek mathematics show the use of inductive reasoning, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to derive conclusions from definitions and axioms, and used mathematical rigor to prove them.^{[31]}
Greek mathematics is thought to have begun with Thales of Miletus (c. 624–c.546 BC) and Pythagoras of Samos (c. 582–c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by Egyptian and Babylonian mathematics. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.
Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.^{[32]} Pythagoras established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number".^{[33]} It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The Pythagoreans are credited with the first proof of the Pythagorean theorem,^{[34]} though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers.^{[35]}^{[36]}
Eudoxus (408–c.355 BC) developed the method of exhaustion, a precursor of modern integration. Aristotle (384—c.322 BC) first wrote down the laws of logic. Euclid (c. 300 BC) is the earliest example of the format still used in mathematics today, definition, axiom, theorem, and proof. He also studied conics. His book, Elements, was known to all educated people in the West until the middle of the 20th century.^{[37]} In addition to the familiar theorems of geometry, such as the Pythagorean theorem, Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers. The Sieve of Eratosthenes (c. 230 BC) was used to discover prime numbers.
Archimedes (c.287–212 BC) of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of Pi.^{[38]} He also studied the spiral bearing his name, formulas for the volumes of surfaces of revolution, and an ingenious system for expressing very large numbers.
Early Chinese mathematics is so different from that of other parts of the world that it is reasonable to assume independent development.^{[39]} The oldest extant mathematical text from China is the Chou Pei Suan Ching, variously dated to between 1200 BC and 100 BC, though a date of about 300 BC appears reasonable.^{[40]}
Of particular note is the use in Chinese mathematics of a decimal positional notation system, the socalled "rod numerals" in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten.^{[41]} Thus, the number 123 would be written using the symbol for "1", followed by the symbol for "100", then the symbol for "2" followed by the symbol for "10", followed by the symbol for "3". This was the most advanced number system in the world at the time, apparently in use several centuries before the common era and well before the development of the Indian numeral system.^{[42]} Rod numerals allowed the representation of numbers as large as desired 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.
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 number of geometrical theorems as well.^{[43]}
In 212 BC, the Emperor Qin Shi Huang (Shi Huangti) commanded all books in the Qin Empire other than officially sanctioned ones be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date. After the book burning of 212 BC, the Han dynasty (202 BC–220 AD) produced works of mathematics which presumably expanded 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 values of π.^{[40]} It also made use of Cavalieri's principle on volume more than a thousand years before Cavalieri would propose it in the West.^{[citation needed]} It created mathematical proof for the Pythagorean theorem, and a mathematical formula for Gaussian elimination.^{[citation needed]} Liu Hui commented on the work by the 3rd century AD, and gave a value of π accurate to 5 decimal places. Though more of a matter of computational stamina than theoretical insight, in the 5th century AD Zu Chongzhi computed the value of π to seven decimal places, which remained the most accurate value of π for almost the next 1000 years.^{[44]}
The high water mark of Chinese mathematics occurs in the 13th century, with the development of Chinese algebra. The most important text form that period is the Precious Mirror of the Four Elements by Chu Shihchieh (fl. 12801303), dealing with the solution of simultaneous higher order algebraic equations using a method similar to Horner's method.^{[44]} The Precious Mirror also contains a diagram of Pascal's triangle with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100.^{[45]} 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 1238–1298).
Even after European mathematics began to flourish during the Renaissance, European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline from the 13th century onwards. Jesuit missionaries such as Matteo Ricci carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries, though at this point far more mathematical ideas were entering China than leaving.^{[45]}
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.^{[46]}
The oldest extant mathematical records from India are the Shatapatha Brahmana (c. 9th century BCE but estimates of the date vary widely), which gives a rough approximation of the value of π,^{[47]} and the Sulba Sutras (c. 800 BC–200 AD),^{[48]} appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others.^{[49]} In addition, the Sulba Sutras give a method for constructing a circle with approximately the same area as a given square,^{[50]} computing the square root of 2 to several decimal places, list Pythagorean triples, and give a statement of the Pythagorean theorem.^{[51]} Mesopotamian influence at this stage is considered likely.^{[48]}
Pāṇini (c. 5th century BC) formulated the rules for Sanskrit grammar.^{[52]} His notation was similar to modern mathematical notation, and used metarules, transformations, and recursion. Pingala (roughly 3rd1st 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).^{[53]}
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.^{[54]} This work was translated into to Arabic and Latin during the Middle Ages.
In the 5th century AD, Aryabhata wrote the Aryabhatiya, a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology.^{[55]} Though about half of the entries are wrong, it is in the Aryabhatiya that the decimal placevalue system first appears. Several centuries later, the Muslim mathematician Abu Rayhan Biruni described the Aryabhatiya as a "mix of common pebbles and costly crystals".^{[56]}
In the 7th century, Brahmagupta identified the Brahmagupta theorem, Brahmagupta's identity and Brahmagupta's formula, and for the first time, in Brahmasphutasiddhanta, he lucidly explained the use of zero as both a placeholder and decimal digit, and explained the HinduArabic numeral system.^{[57]} 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.^{[citation needed]}
In the 12th century, Bhāskara II^{[58]} lived in southern India and wrote extensively on all then known branches of mathematic. His work contains mathematical objects equivalent or approximately equivalent to infinitesimals, derivatives, the mean value theorem and the derivative of the sine function. To what extent he anticipated the invention of calculus is a controversial subject among historians of mathematics^{[59]}
In the 14th century, Madhava of Sangamagrama, the founder of the socalled Kerala School of Mathematics, found the Madhava–Leibniz series, and, using 21 terms, computed the value of π as 3.14159265359. Madhava also found the MadhavaGregory series to determine the arctangent, the MadhavaNewton power series to determine sine and cosine and the Taylor approximation for sine and cosine functions .^{[60]} In the 16th century, Jyesthadeva consolidated many of the Kerala School's developments and theorems in the Yuktibhasa.^{[61]} However, the Kerala School did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala.^{[62]}^{[63]}^{[64]}^{[65]} Mathematical progress in India stagnated from the late 16th century to onwards due to political turmoil.
]]
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 nonArab scholars throughout the Islamic world at the time. Persians contributed to the world of Mathematics alongside Arabs.
In the 9th century, the Persian mathematician Muḥammad ibn Mūsā Khwārizmī wrote several important books on the HinduArabic numerals and on methods for solving equations. His book On the Calculation with Hindu Numerals, written about 825, along with the work of AlKindi, 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, AlKitāb almukhtaṣar fī hīsāb alğabr wa’lmuqābala (The Compendious Book on Calculation by Completion and Balancing). Khwarizmi is often called the "father of algebra", for his fundamental contributions to the field.^{[66]} He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,^{[67]} and he was the first to teach algebra in an elementary form and for its own sake.^{[68]} 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 Khwarizmi originally described as aljabr.^{[69]} 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."^{[70]}
Further developments in algebra were made by AlKaraji in his treatise alFakhri, 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 AlKaraji around 1000 AD, who used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes.^{[71]} The historian of mathematics, F. Woepcke,^{[72]} praised AlKaraji 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 alHaytham 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.^{[73]}
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 nonEuclidean 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 alDīn alTūsī introduced the concept of a function,^{[74]} and he was the first to discover the derivative of cubic polynomials.^{[75]} 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.^{[76]}
In the 13th century, Nasir alDin Tusi (Nasireddin) made advances in spherical trigonometry. He also wrote influential work on Euclid's parallel postulate. In the 15th century, Ghiyath alKashi 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, alKindi's introduction of cryptanalysis and frequency analysis, the development of analytic geometry by Ibn alHaytham, the beginning of algebraic geometry by Omar Khayyam, the first attempt at a nonEuclidean geometry by Sadr alDin, and the development of an algebraic notation by alQalasādī.^{[77]}
During the time of the Ottoman Empire from the 15th century, the development of Islamic mathematics became stagnant.
Medieval European interest in mathematics was driven by concerns quite different from those of modern mathematicians. One driving element was the belief that mathematics provided the key to understanding the created order of nature, frequently justified by Plato's Timaeus and the biblical passage (in the Book of Wisdom) that God had ordered all things in measure, and number, and weight.^{[78]}
Boethius provided a place for mathematics in the curriculum when he coined the term quadrivium to describe the study of arithmetic, geometry, astronomy, and music. He wrote De institutione arithmetica, a free translation from the Greek of Nicomachus's Introduction to Arithmetic; De institutione musica, also derived from Greek sources; and a series of excerpts from Euclid's Elements. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.^{[79]}^{[80]}
In the 12th century, European scholars traveled to Spain and Sicily seeking scientific Arabic texts, including Khwarizmi's The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester, and the complete text of Euclid's Elements, translated in various versions by Adelard of Bath, Herman of Carinthia, and Gerard of Cremona.^{[81]}^{[82]}
These new sources sparked a renewal of mathematics. Fibonacci, writing in the Liber Abaci, in 1202 and updated in 1254, produced the first significant mathematics in Europe since the time of Eratosthenes, a gap of more than a thousand years. The work introduced HinduArabic numerals to Europe, and discussed many other mathematical problems.
The 14th century saw the development of new mathematical concepts to investigate a wide range of problems.^{[83]} One important contribution was development of mathematics of local motion.
Thomas Bradwardine proposed that speed (V) increases in arithmetic proportion as the ratio of force (F) to resistance (R) increases in geometric proportion. Bradwardine expressed this by a series of specific examples, but although the logarithm had not yet been conceived, we can express his conclusion anachronistically by writing: V = log (F/R).^{[84]} Bradwardine's analysis is an example of transferring a mathematical technique used by alKindi and Arnald of Villanova to quantify the nature of compound medicines to a different physical problem.^{[85]}
One of the 14thcentury Oxford Calculators, William Heytesbury, lacking differential calculus and the concept of limits, proposed to measure instantaneous speed "by the path that would be described by [a body] if... it were moved uniformly at the same degree of speed with which it is moved in that given instant".^{[86]}
Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (today solved by integration), stating that "a moving body uniformly acquiring or losing that increment [of speed] will traverse in some given time a [distance] completely equal to that which it would traverse if it were moving continuously through the same time with the mean degree [of speed]".^{[87]}
Nicole Oresme at the University of Paris and the Italian Giovanni di Casali independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance traveled.^{[88]} In a later mathematical commentary on Euclid's Elements, Oresme made a more detailed general analysis in which he demonstrated that a body will acquire in each successive increment of time an increment of any quality that increases as the odd numbers. Since Euclid had demonstrated the sum of the odd numbers are the square numbers, the total quality acquired by the body increases as the square of the time.^{[89]}
, 1495, (Museo di Capodimonte).The open book to which he is pointing may be his Summa de Arithmetica, Geometria, Proportioni et Proportionalità.^{[90]}]]
During the Renaissance, mathematics and accounting were closely linked.^{[91]} It is important to note that Pacioli himself had borrowed much of the work of Piero Della Francesca whom he plagiarized.
Simon Stevin's book De Thiende ('the art of tenths'), first published in Dutch in 1585, contained the first systematic treatment of decimal notation, which influenced all later work on the real number system.
Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. Regiomontanus's table of sines and cosines was published in 1533.^{[92]}
The 17th century saw an unprecedented explosion of mathematical and scientific ideas across Europe. Galileo, an Italian, observed the moons of Jupiter in orbit about that planet, using a telescope based on a toy imported from Holland. Tycho Brahe, a Dane, had gathered an enormous quantity of mathematical data describing the positions of the planets in the sky. His student, Johannes Kepler, a German, began to work with this data. In part because he wanted to help Kepler in his calculations, John Napier, in Scotland, was the first to investigate natural logarithms. Kepler succeeded in formulating mathematical laws of planetary motion. The analytic geometry developed by René Descartes (1596–1650), a French mathematician and philosopher, allowed those orbits to be plotted on a graph, in Cartesian coordinates. Simon Stevin (1585) created the basis for modern decimal notation capable of describing all numbers, whether rational or irrational.
Building on earlier work by many predecessors, Isaac Newton, an Englishman, discovered the laws of physics explaining Kepler's Laws, and brought together the concepts now known as infinitesimal calculus. Independently, Gottfried Wilhelm Leibniz, in Germany, developed calculus and much of the calculus notation still in use today. Science and mathematics had become an international endeavor, which would soon spread over the entire world.^{[93]}
In addition to the application of mathematics to the studies of the heavens, applied mathematics began to expand into new areas, with the correspondence of Pierre de Fermat and Blaise Pascal. Pascal and Fermat set the groundwork for the investigations of probability theory and the corresponding rules of combinatorics in their discussions over a game of gambling. Pascal, with his wager, attempted to use the newly developing probability theory to argue for a life devoted to religion, on the grounds that even if the probability of success was small, the rewards were infinite. In some sense, this foreshadowed the development of utility theory in the 18th–19th century.
by Emanuel Handmann.]]
The most influential mathematician of the 18th century was arguably Leonhard Euler. His contributions range from founding the study of graph theory with the Seven Bridges of Königsberg problem to standardizing many modern mathematical terms and notations. For example, he named the square root of minus 1 with the symbol i, and he popularized the use of the Greek letter $\backslash pi$ to stand for the ratio of a circle's circumference to its diameter. He made numerous contributions to the study of topology, graph theory, calculus, combinatorics, and complex analysis, as evidenced by the multitude of theorems and notations named for him.
Other important European mathematicians of the 18th century included Joseph Louis Lagrange, who did pioneering work in number theory, algebra, differential calculus, and the calculus of variations, and Laplace who, in the age of Napoleon did important work on the foundations of celestial mechanics and on statistics.
Throughout the 19th century mathematics became increasingly abstract. In the 19th century lived Carl Friedrich Gauss (1777–1855). Leaving aside his many contributions to science, in pure mathematics he did revolutionary work on functions of complex variables, in geometry, and on the convergence of series. He gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law.
This century saw the development of the two forms of nonEuclidean geometry, where the parallel postulate of Euclidean geometry no longer holds. The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival, the Hungarian mathematician Janos Bolyai, independently defined and studied hyperbolic geometry, where uniqueness of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. Elliptic geometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed Riemannian geometry, which unifies and vastly generalizes the three types of geometry, and he defined the concept of a manifold, which generalize the ideas of curves and surfaces.
The 19th century saw the beginning of a great deal of abstract algebra. Hermann Grassmann in Germany gave a first version of vector spaces, William Rowan Hamilton in Ireland developed noncommutative algebra. The British mathematician George Boole devised an algebra that soon evolved into what is now called Boolean algebra, in which the only numbers were 0 and 1 and in which, 1 + 1 = 1. Boolean algebra is the starting point of mathematical logic and has important applications in computer science.
AugustinLouis Cauchy, Bernhard Riemann, and Karl Weierstrass reformulated the calculus in a more rigorous fashion.
Also, for the first time, the limits of mathematics were explored. Niels Henrik Abel, a Norwegian, and Évariste Galois, a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four. Other 19th century mathematicians utilized this in their proofs that straightedge and compass alone are not sufficient to trisect an arbitrary angle, to construct the side of a cube twice the volume of a given cube, nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks.
Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of group theory, and the associated fields of abstract algebra. In the 20th century physicists and other scientists have seen group theory as the ideal way to study symmetry.
In the later 19th century, Georg Cantor established the first foundations of set theory, which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics. Cantor's set theory, and the rise of mathematical logic in the hands of Peano, L. E. J. Brouwer, David Hilbert, Bertrand Russell, and A.N. Whitehead, initiated a long running debate on the foundations of mathematics.
The 19th century saw the founding of a number of national mathematical societies: the London Mathematical Society in 1865, the Société Mathématique de France in 1872, the Circolo Mathematico di Palermo in 1884, the Edinburgh Mathematical Society in 1883, and the American Mathematical Society in 1888. The first international, specialinterest society, the Quaternion Society, was formed in 1899, in the context of a vector controversy.
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The 20th century saw mathematics become a major profession. Every year, thousands of new Ph.D.s in mathematics are awarded, and jobs are available in both teaching and industry. In earlier centuries, there were few creative mathematicians in the world at any one time.
In a 1900 speech to the International Congress of Mathematicians, David Hilbert set out a list of 23 unsolved problems in mathematics. These problems, spanning many areas of mathematics, formed a central focus for much of 20th century mathematics. Today, 10 have been solved, 7 are partially solved, and 2 are still open. The remaining 4 are too loosely formulated to be stated as solved or not.
Notable historical conjectures were finally proved. In 1976, Wolfgang Haken and Kenneth Appel used a computer to prove the four color theorem. Andrew Wiles, building on the work of others, proved Fermat's Last Theorem in 1995. Paul Cohen and Kurt Gödel proved that the continuum hypothesis is independent of (could neither be proved nor disproved from) the standard axioms of set theory. In 1998 Thomas Callister Hales proved the Kepler conjecture.
Mathematical collaborations of unprecedented size and scope took place. An example is the classification of finite simple groups (also called the "enormous theorem"), whose proof between 1955 and 1983 required 500odd journal articles by about 100 authors, and filling tens of thousands of pages. A group of French mathematicians, including Jean Dieudonné and André Weil, publishing under the pseudonym "Nicolas Bourbaki", attempted to exposit all of known mathematics as a coherent rigorous whole. The resulting several dozen volumes has had a controversial influence on mathematical education.^{[94]}
Differential geometry came into its own when Einstein used it in general relativity. Entire new areas of mathematics such as mathematical logic, topology, and John von Neumann's game theory changed the kinds of questions that could be answered by mathematical methods. All kinds of structures were abstracted using axioms and given names like metric spaces, topological spaces etc. As mathematicians do, the concept of an abstract structure was itself abstracted and led to category theory. Grothendieck and Serre recast algebraic geometry using sheaf theory. Large advances were made in the qualitative study of dynamical systems that Poincaré had begun in the 1890s. Measure theory was developed in the late 19th and early 20th century. Applications of measures include the Lebesgue integral, Kolmogorov's axiomatisation of probability theory, and ergodic theory. Knot theory greatly expanded. Quantum mechanics led to the development of functional analysis. Other new areas include , Laurent Schwarz's distribution theory, fixed point theory, singularity theory and René Thom's catastrophe theory, model theory, and Mandelbrot's fractals. Lie theory with its Lie groups and Lie algebras became one of the major areas of study.
The development and continual improvement of computers, at first mechanical analog machines and then digital electronic machines, allowed industry to deal with larger and larger amounts of data to facilitate mass production and distribution and communication, and new areas of mathematics were developed to deal with this: Alan Turing's computability theory; complexity theory; Claude Shannon's information theory; signal processing; data analysis; optimization and other areas of operations research. In the preceding centuries much mathematical focus was on calculus and continuous functions, but the rise of computing and communication networks led to an increasing importance of discrete concepts and the expansion of combinatorics including graph theory. The speed and data processing abilities of computers also enabled the handling of mathematical problems that were too timeconsuming to deal with by pencil and paper calculations, leading to areas such as numerical analysis and symbolic computation. Some of the most important methods and algorithms discovered in the 20th century are: the simplex algorithm, the Fast Fourier Transform and the Kalman filter.
At the same time, deep insights were made about the limitations to mathematics. In 1929 and 1930, it was proved the truth or falsity of all statements formulated about the natural numbers plus one of addition and multiplication, was decidable, i.e. could be determined by some algorithm. In 1931, Kurt Gödel found that this was not the case for the natural numbers plus both addition and multiplication; this system, known as Peano arithmetic, was in fact incompletable. (Peano arithmetic is adequate for a good deal of number theory, including the notion of prime number.) A consequence of Gödel's two incompleteness theorems is that in any mathematical system that includes Peano arithmetic (including all of analysis and geometry), truth necessarily outruns proof, i.e. there are true statements that cannot be proved within the system. Hence mathematics cannot be reduced to mathematical logic, and David Hilbert's dream of making all of mathematics complete and consistent needed to be reformulated.
One of the more colorful figures in 20th century mathematics was Srinivasa Aiyangar Ramanujan (1887–1920), an Indian autodidact who conjectured or proved over 3000 theorems, including properties of highly composite numbers, the partition function and its asymptotics, and mock theta functions. He also made major investigations in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory.
Paul Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. Mathematicians have a game equivalent to the Kevin Bacon Game, which leads to the Erdős number of a mathematician. This describes the "collaborative distance" between a person and Paul Erdős, as measured by joint authorship of mathematical papers.
As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: by the end of the century there were hundreds of specialized areas in mathematics and the Mathematics Subject Classification was dozens of pages long.^{[95]} More and more mathematical journals were published and, by the end of the century, the development of the world wide web led to online publishing.
In 2000, the Clay Mathematics Institute announced the Millennium Prize Problems, and in 2003 the Poincaré conjecture was solved by Grigori Perelman (who declined to accept any awards).
Most mathematical journals now have online versions as well as print versions, and many onlineonly journals are launched. There is an increasing drive towards open access publishing, first popularized by the arXiv.
There are many observable trends in mathematics, the most notable being that the subject is growing ever larger, computers are ever more important and powerful, the application of mathematics to bioinformatics is rapidly expanding, the volume of data to be analyzed being produced by science and industry, facilitated by computers, is explosively expanding...
