The following is a timeline of key developments of algebra:
|c. 1800 BC||The Old Babylonian Strassburg tablet seeks the solution of a quadratic elliptic equation.[ citation needed ]|
|c. 1800 BC||The Plimpton 322 tablet gives a table of Pythagorean triples in Babylonian Cuneiform script.|
|1800 BC||Berlin Papyrus 6619 (19th dynasty) contains a quadratic equation and its solution.|
|800 BC||Baudhayana, author of the Baudhayana Sulba Sutra, a Vedic Sanskrit geometric text, contains quadratic equations, and calculates the square root of 2 correct to five decimal places|
|c. 300 BC||Euclid's Elements gives a geometric construction with Euclidean tools for the solution of the quadratic equation for positive real roots. The construction is due to the Pythagorean School of geometry.[ citation needed ]|
|c. 300 BC||A geometric construction for the solution of the cubic is sought (doubling the cube problem). It is now well known that the general cubic has no such solution using Euclidean tools.|
|150 BC||Jain mathematicians in India write the “Sthananga Sutra”, which contains work on the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.|
|250 BC||Algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu (The Nine Chapters on the Mathematical Art), which contains solutions of linear equations solved using the rule of double false position, geometric solutions of quadratic equations, and the solutions of matrices equivalent to the modern method, to solve systems of simultaneous linear equations.|
|1st century AD||Hero of Alexandria gives the earliest fleeting reference to square roots of negative numbers.|
|c. 150||Greek mathematician Hero of Alexandria, treats algebraic equations in three volumes of mathematics.|
|c. 200||Hellenistic mathematician Diophantus, who lived in Alexandria and is often considered to be the "father of algebra", writes his famous Arithmetica , a work featuring solutions of algebraic equations and on the theory of numbers.|
|499||Indian mathematician Aryabhata, in his treatise Aryabhatiya, obtains whole-number solutions to linear equations by a method equivalent to the modern one, describes the general integral solution of the indeterminate linear equation, gives integral solutions of simultaneous indeterminate linear equations, and describes a differential equation.[ citation needed ]|
|c. 625||Chinese mathematician Wang Xiaotong finds numerical solutions to certain cubic equations.|
|c. 7th century|
Dates vary from the 3rd to the 12th centuries.
|The Bakhshali Manuscript written in ancient India uses a form of algebraic notation using letters of the alphabet and other signs, and contains cubic and quartic equations, algebraic solutions of linear equations with up to five unknowns, the general algebraic formula for the quadratic equation, and solutions of indeterminate quadratic equations and simultaneous equations.[ citation needed ]|
|7th century||Brahmagupta invents the method of solving indeterminate equations of the second degree and is the first to use algebra to solve astronomical problems. He also develops methods for calculations of the motions and places of various planets, their rising and setting, conjunctions, and the calculation of eclipses of the sun and the moon|
|628||Brahmagupta writes the Brahmasphuta-siddhanta , where zero is clearly explained, and where the modern place-value Indian numeral system is fully developed. It also gives rules for manipulating both negative and positive numbers, methods for computing square roots, methods of solving linear and quadratic equations, and rules for summing series, Brahmagupta's identity, and the Brahmagupta theorem|
|8th century||Virasena gives explicit rules for the Fibonacci sequence, gives the derivation of the volume of a frustum using an infinite procedure, and also deals with the logarithm to base 2 and knows its laws|
|c. 800||The Abbasid patrons of learning, al-Mansur, Haroun al-Raschid, and al-Mamun, has Greek, Babylonian, and Indian mathematical and scientific works translated into Arabic and begins a cultural, scientific and mathematical awakening after a century devoid of mathematical achievements.|
|820||The word algebra is derived from operations described in the treatise written by the Persian mathematician, Muḥammad ibn Mūsā al-Ḵhwārizmī, titled Al-Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing") on the systematic solution of linear and quadratic equations. Al-Khwarizmi is often considered the "father of algebra", for founding algebra as an independent discipline and for introducing the methods of "reduction" and "balancing" (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) which was what he originally used the term al-jabr to refer to. 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."|
|c. 850||Persian mathematician al-Mahani conceives the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.[ citation needed ]|
|c. 990||Persian mathematician Al-Karaji (also known as al-Karkhi), in his treatise Al-Fakhri, further develops algebra by extending Al-Khwarizmi's methodology to incorporate integral powers and integral roots of unknown quantities. He replaces geometrical operations of algebra with modern arithmetical operations, and defines the monomials x, x2, x3, .. and 1/x, 1/x2, 1/x3, .. and gives rules for the products of any two of these. He also discovers the first numerical solution to equations of the form ax2n + bxn = c. Al-Karaji is also regarded as the first person to free algebra from geometrical operations and replace them with the type of arithmetic operations which are at the core of algebra today. His work on algebra and polynomials, gave the rules for arithmetic operations to manipulate polynomials. The historian of mathematics F. Woepcke, in Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi (Paris, 1853), praised Al-Karaji for being "the first who introduced the theory of algebraic calculus". Stemming from this, Al-Karaji investigated binomial coefficients and Pascal's triangle.|
|895||Thabit ibn Qurra: the only surviving fragment of his original work contains a chapter on the solution and properties of cubic equations. He also generalized the Pythagorean theorem, and discovered the theorem by which pairs of amicable numbers can be found, (i.e., two numbers such that each is the sum of the proper divisors of the other).|
|953||Al-Karaji is the “first person to completely free algebra from geometrical operations and to replace them with the arithmetical type of operations which are at the core of algebra today. He [is] first to define the monomials , , , … and , , , … and to give rules for products of any two of these. He start[s] a school of algebra which flourished for several hundreds of years”. He also discovers the binomial theorem for integer exponents, which “was a major factor in the development of numerical analysis based on the decimal system.”|
|c. 1000||Abū Sahl al-Qūhī (Kuhi) solves equations higher than the second degree.|
|c. 1050||Chinese mathematician Jia Xian finds numerical solutions of polynomial equations of arbitrary degree.|
|1070||Omar Khayyám begins to write Treatise on Demonstration of Problems of Algebra and classifies cubic equations.|
|1072||Persian mathematician Omar Khayyam gives a complete classification of cubic equations with positive roots and gives general geometric solutions to these equations found by means of intersecting conic sections.|
|12th century||Bhaskara Acharya writes the “Bijaganita” (“Algebra”), which is the first text that recognizes that a positive number has two square roots|
|1130||Al-Samawal gives a definition of algebra: “[it is concerned] with operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.”|
|1135||Sharafeddin Tusi follows al-Khayyam's application of algebra to geometry, and writes a treatise on cubic equations which “represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry.”|
|c. 1200||Sharaf al-Dīn al-Tūsī (1135–1213) writes the Al-Mu'adalat (Treatise on Equations), which deals with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He uses what would later be known as the "Ruffini-Horner method" to numerically approximate the root of a cubic equation. He also develops the concepts of the maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understands the importance of the discriminant of the cubic equation and uses an early version of Cardano's formula to find algebraic solutions to certain types of cubic equations. Some scholars, such as Roshdi Rashed, argue that Sharaf al-Din discovered the derivative of cubic polynomials and realized its significance, while other scholars connect his solution to the ideas of Euclid and Archimedes.|
|1202||Leonardo Fibonacci of Pisa publishes his Liber Abaci , a work on algebra that introduces Arabic numerals to Europe.|
|c. 1300||Chinese mathematician Zhu Shijie deals with polynomial algebra, solves quadratic equations, simultaneous equations and equations with up to four unknowns, and numerically solves some quartic, quintic and higher-order polynomial equations.|
|c. 1400||Jamshīd al-Kāshī develops an early form of Newton's method to numerically solve the equation to find roots of N.|
|c. 1400||Indian mathematician Madhava of Sangamagrama finds the solution of transcendental equations by iteration, iterative methods for the solution of non-linear equations, and solutions of differential equations.[ citation needed ]|
|15th century||Nilakantha Somayaji, a Kerala school mathematician, writes the “Aryabhatiya Bhasya”, which contains work on infinite-series expansions, problems of algebra, and spherical geometry|
|1412–1482||Arab mathematician Abū al-Hasan ibn Alī al-Qalasādī takes "the first steps toward the introduction of algebraic symbolism." He uses "short Arabic words, or just their initial letters, as mathematical symbols."|
|1535||Scipione del Ferro and Niccolò Fontana Tartaglia, in Italy, independently solve the general cubic equation.|
|1545||Girolamo Cardano publishes Ars magna -The great art which gives del Ferro's solution to the cubic equation and Lodovico Ferrari's solution to the quartic equation.|
|1572||Rafael Bombelli recognizes the complex roots of the cubic and improves current notation.|
|1591||Franciscus Vieta develops improved symbolic notation for various powers of an unknown and uses vowels for unknowns and consonants for constants in In artem analyticam isagoge.[ citation needed ]|
|1608||Christopher Clavius publishes his Algebra|
|1619||René Descartes discovers analytic geometry. (Pierre de Fermat claimed that he also discovered it independently.)|
|1631||Thomas Harriot in a posthumous publication is the first to use symbols < and > to indicate "less than" and "greater than", respectively.|
|1637||Pierre de Fermat claims to have proven Fermat's Last Theorem in his copy of Diophantus' Arithmetica,|
|1637||René Descartes introduces the use of the letters z, y, and x for unknown quantities.|
|1637||The term imaginary number is first used by René Descartes; it is meant to be derogatory.|
|1682||Gottfried Wilhelm Leibniz develops his notion of symbolic manipulation with formal rules which he calls characteristica generalis.|
|1683||Japanese mathematician Kowa Seki, in his Method of solving the dissimulated problems, discovers the determinant, discriminant,[ citation needed ] and Bernoulli numbers.|
|1693||Leibniz solves systems of simultaneous linear equations using matrices and determinants.[ citation needed ]|
|1722||Abraham de Moivre states de Moivre's formula connecting trigonometric functions and complex numbers,|
|1750||Gabriel Cramer, in his treatise Introduction to the analysis of algebraic curves, states Cramer's rule and studies algebraic curves, matrices and determinants.|
|1797||Caspar Wessel associates vectors with complex numbers and studies complex number operations in geometrical terms,|
|1799||Carl Friedrich Gauss proves the fundamental theorem of algebra (every polynomial equation has a solution among the complex numbers),|
|1799||Paolo Ruffini partially proves the Abel–Ruffini theorem that quintic or higher equations cannot be solved by a general formula,|
|1806||Jean-Robert Argand publishes proof of the Fundamental theorem of algebra and the Argand diagram,|
|1824||Niels Henrik Abel proves that the general quintic equation is insoluble by radicals.|
|1832||Galois theory is developed by Évariste Galois in his work on abstract algebra.|
|1843||William Rowan Hamilton discovers quaternions.|
|1853||Arthur Cayley provides a modern definition of groups.|
|1847||George Boole formalizes symbolic logic in The Mathematical Analysis of Logic, defining what now is called Boolean algebra.|
|1873||Charles Hermite proves that e is transcendental.|
|1878||Charles Hermite solves the general quintic equation by means of elliptic and modular functions.|
|1926||Emmy Noether extends Hilbert's theorem on the finite basis problem to representations of a finite group over any field.|
|1929||Emmy Noether combines work on structure theory of associative algebras and the representation theory of groups into a single arithmetic theory of modules and ideals in rings satisfying ascending chain conditions, providing the foundation for modern algebra.|
|1981||Mikhail Gromov develops the theory of hyperbolic groups, revolutionizing both infinite group theory and global differential geometry,|
Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīsābūrī, commonly known as Omar Khayyam, was a polymath, known for his contributions to mathematics, astronomy, philosophy, and poetry. He was born in Nishapur, the initial capital of the Seljuk Empire. As a scholar, he was contemporary with the rule of the Seljuk dynasty around the time of the First Crusade.
Muḥammad ibn Mūsā al-Khwārizmī, or al-Khwarizmi, was a polymath from Khwarazm, who produced vastly influential works in mathematics, astronomy, and geography. Around 820 CE, he was appointed as the astronomer and head of the library of the House of Wisdom in Baghdad.
al-Jabr, also known as The Compendious Book on Calculation by Completion and Balancing, is an Arabic mathematical treatise on algebra written in Baghdad around 820 CE by the Persian polymath Muḥammad ibn Mūsā al-Khwārizmī. It was a landmark work in the history of mathematics, serving as the eponymous work and ultimate etymology of the word "algebra" itself, with its abbreviated title borrowed into Medieval Latin as algebrāica.
Abu Mahmud Hamid ibn al-Khidr al-Khujandi was a Muslim Transoxanian astronomer and mathematician born in Khujand who lived in the late 10th century and helped build an observatory, near the city of Ray, in Iran.
Kamal al-Din Hasan ibn Ali ibn Hasan al-Farisi or Abu Hasan Muhammad ibn Hasan ) was a Persian Muslim scientist. He made two major contributions to science, one on optics, the other on number theory. Farisi was a pupil of the astronomer and mathematician Qutb al-Din al-Shirazi, who in turn was a pupil of Nasir al-Din Tusi.
Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī known more often as Sharaf al-Dīn al-Ṭūsī or Sharaf ad-Dīn aṭ-Ṭūsī, was an Iranian mathematician and astronomer of the Islamic Golden Age.
Alī ibn Aḥmad al-Nasawī was a Persian mathematician from Khurasan, Iran. He flourished under the Buwayhid sultan Majd al-dowleh, who died in 1029-30AD, and under his successor. He wrote a book on arithmetic in Persian, and then Arabic, entitled the "Satisfying on Hindu Calculation". He also wrote on Archimedes's Book of Lemmas and Menelaus's theorem, where he made corrections to the Book of Lemmas as translated into Arabic by Thabit ibn Qurra and last revised by Nasir al-Din al-Tusi.
Abu-Abdullah Muhammad ibn Īsa Māhānī was a Persian mathematician and astronomer born in Mahan, and active in Baghdad, Abbasid Caliphate. His known mathematical works included his commentaries on Euclid's Elements, Archimedes' On the Sphere and Cylinder and Menelaus' Sphaerica, as well as two independent treatises. He unsuccessfully tried to solve a problem posed by Archimedes of cutting a sphere into two volumes of a given ratio, which was later solved by 10th century mathematician Abū Ja'far al-Khāzin. His only known surviving work on astronomy was on the calculation of azimuths. He was also known to make astronomical observations, and claimed his estimates of the start times of three consecutive lunar eclipses were accurate to within half an hour.
This timeline of science and engineering in the Muslim world covers the time period from the eighth century AD to the introduction of European science to the Muslim world in the nineteenth century. All year dates are given according to the Gregorian calendar except where noted.
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built on Greek mathematics and Indian mathematics. Important progress was made, such as full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra, and advances in geometry and trigonometry.
Shams al-Dīn Muḥammad ibn Ashraf al-Ḥusaynī al-Samarqandī was a 13th century Persian astronomer and mathematician from Samarkand, now in Uzbekistan.
Abū ʿAbd Allāh Muḥammad ibn Muʿādh al-Jayyānī was an Arab, mathematician, Islamic scholar, and Qadi from Al-Andalus. Al-Jayyānī wrote important commentaries on Euclid's Elements and he wrote the first known treatise on spherical trigonometry.
Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra.
Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī was a Persian astronomer and mathematician during the reign of Tamerlane.
Qāḍī Zāda al-Rūmī, whose actual name was Salah al-Din Musa Pasha, was a Turkish astronomer and mathematician who worked at the observatory in Samarkand. He computed sin 1° to an accuracy of 10−12.
Algebra is the study of variables and the rules for manipulating these variables in formulas; it is a unifying thread of almost all of mathematics.
Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.
A timeline of numerals and arithmetic.
Roshdi Rashed, born in Cairo in 1936, is a mathematician, philosopher and historian of science, whose work focuses largely on mathematics and physics of the medieval Arab world. His work explores and illuminates the unrecognized Arab scientific tradition, being one of the first historians to study in detail the ancient and medieval texts, their journey through the Eastern schools and courses, their immense contributions to Western science, particularly in regarding the development of algebra and the first formalization of physics.
Rashed has argued that Sharaf al-Din discovered the derivative of cubic polynomials and realized its significance for investigating conditions under which cubic equations were solvable; however, other scholars have suggested quite difference explanations of Sharaf al-Din's thinking, which connect it with mathematics found in Euclid or Archimedes.