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## #1 2015-09-27 15:39:53

Jai Ganesh
Registered: 2005-06-28
Posts: 47,777

### Greatest Mathematicians from 1 CE ...

Greatest Mathematicians  from 1 CE (also called AD) :

10-70 AD    Heron (or Hero) of Alexandria    : Greek : Heron’s Formula for finding the area of a triangle from its side lengths, Heron’s Method for iteratively computing a square root

90-168 AD:  Ptolemy : Greek/Egyptian : Develop even more detailed trigonometry tables

200 AD : Sun Tzu : Chinese : First definitive statement of Chinese Remainder Theorem

200 AD :     Indian : Refined and perfected decimal place value number system

200-284 AD : Diophantus : Greek :    Diophantine Analysis of complex algebraic problems, to find rational solutions to equations with several unknowns

220-280 AD : Liu Hui : Chinese : Solved linear equations using a matrices (similar to Gaussian elimination), leaving roots unevaluated, calculated value of π correct to five decimal places, early forms of integral and differential calculus

400 AD :     Indian : “Surya Siddhanta” contains roots of modern trigonometry, including first real use of sines, cosines, inverse sines, tangents and secants

476-550 AD : Aryabhata     Indian : Definitions of trigonometric functions, complete and accurate sine and versine tables, solutions to simultaneous quadratic equations, accurate approximation for π (and recognition that π is an irrational number)

598-668 AD : Brahmagupta : Indian : Basic mathematical rules for dealing with zero (+, - and x), negative numbers, negative roots of quadratic equations, solution of quadratic equations with two unknowns

600-680 AD : Bhaskara I : Indian    First to write numbers in Hindu-Arabic decimal system with a circle for zero, remarkably accurate approximation of the sine function

780-850 AD : Muhammad Al-Khwarizmi :Persian : Advocacy of the Hindu numerals 1 - 9 and 0 in Islamic world, foundations of modern algebra, including algebraic methods of “reduction” and “balancing”, solution of polynomial equations up to second degree

908-946 AD : Ibrahim ibn Sinan : Arabic    Continued Archimedes' investigations of areas and volumes, tangents to a circle

953-1029 AD : Muhammad Al-Karaji : Persian    First use of proof by mathematical induction, including to p
prove the binomial theorem

966-1059 AD :    Ibn al-Haytham (Alhazen) : Persian/Arabic    Derived a formula for the sum of fourth powers using a readily generalizable method, “Alhazen's problem”, established beginnings of link between algebra and geometry

1048-1131 AD :  Omar Khayyam :    Persian    Generalized Indian methods for extracting square and cube roots to include fourth, fifth and higher roots, noted existence of different sorts of cubic equations

1114-1185 AD :Bhaskara II : Indian    Established that dividing by zero yields infinity, found solutions to quadratic, cubic and quartic equations (including negative and irrational solutions) and to second order Diophantine equations, introduced some preliminary concepts of calculus

1170-1250 AD :Leonardo of Pisa  (Fibonacci) : Italian :     Fibonacci Sequence of numbers, advocacy of the use of the Hindu-Arabic numeral system in Europe, Fibonacci's identity (product of two sums of two squares is itself a sum of two squares)

1201-1274 AD: Nasir al-Din al-Tusi : Persian :
Developed field of spherical trigonometry, formulated law of sines for plane triangles

1202-1261 AD : Qin Jiushao :     Chinese    : Solutions to quadratic, cubic and higher power equations using a method of repeated approximations

1238-1298 AD : Yang Hui :     Chinese    Culmination of Chinese “magic” squares, circles and triangles, Yang Hui’s Triangle (earlier version of Pascal’s Triangle of binomial co-efficients)

1267-1319 AD : Kamal al-Din al-Farisi    Persian    Applied theory of conic sections to solve optical problems, explored amicable numbers, factorization and combinatorial methods

1350-1425 AD : Madhava : Indian :    Use of infinite series of fractions to give an exact formula for π, sine formula and other trigonometric functions, important step towards development of calculus

1323-1382 AD : Nicole Oresme : French : System of rectangular coordinates, such as for a time-speed-distance graph, first to use fractional exponents, also worked on infinite series
1446-1517 AD : Luca Pacioli :    Italian : Influential book on arithmetic, geometry and book-keeping, also introduced standard symbols for plus and minus

1499-1557 AD : Niccolò Fontana Tartaglia : Italian    Formula for solving all types of cubic equations, involving first real use of complex numbers (combinations of real and imaginary numbers), Tartaglia’s Triangle (earlier version of Pascal’s Triangle)

1501-1576 AD : Gerolamo Cardano : Italian :    Published solution of cubic and quartic equations (by Tartaglia and Ferrari), acknowledged existence of imaginary numbers (based on √-1)

1522-1565 AD : Lodovico Ferrari :    Italian : Devised formula for solution of quartic equations

1550-1617 AD : John Napier :    British : Invention of natural logarithms, popularized the use of the decimal point, Napier’s Bones tool for lattice multiplication

1588-1648 AD : Marin Mersenne :    French : Clearing house for mathematical thought during 17th Century, Mersenne primes (prime numbers that are one less than a power of 2)
1591-1661 AD : Girard Desargues : French :    Early development of projective geometry and “point at infinity”, perspective theorem

1596-1650 AD : René Descartes :    French :    Development of Cartesian coordinates and analytic geometry (synthesis of geometry and algebra), also credited with the first use of superscripts for powers or exponents

1598-1647 AD : Bonaventura Cavalieri :    Italian : “Method of indivisibles” paved way for the later development of infinitesimal calculus

1601-1665 AD : Pierre de Fermat    : French :    Discovered many new numbers patterns and theorems (including Little Theorem, Two-Square Theorem  and Last Theorem), greatly extending knowlege of number theory, also contributed to probability theory

1616-1703 AD : John Wallis :    British :    Contributed towards development of calculus, originated idea of number line, introduced symbol ∞ for infinity, developed standard notation for powers

1623-1662 AD : Blaise Pascal : French    Pioneer (with Fermat) of probability theory, Pascal’s Triangle of binomial coefficients

1643-1727 AD : Isaac Newton : British :    Development of infinitesimal calculus (differentiation and integration), laid ground work for almost all of classical mechanics, generalized binomial theorem, infinite power series

1646-1716 AD :Gottfried Leibniz :    German    Independently developed infinitesimal calculus (his calculus notation is still used), also practical calculating machine using binary system (forerunner of the computer), solved linear equations using a matrix

1654-1705 AD : Jacob Bernoulli    Swiss    Helped to consolidate infinitesimal calculus, developed a technique for solving separable differential equations, added a theory of permutations and combinations to probability theory, Bernoulli Numbers sequence, transcendental curves

1667-1748 AD : Johann Bernoulli :    Swiss : Further developed infinitesimal calculus, including the “calculus of variation”, functions for curve of fastest descent (brachistochrone) and catenary curve

1667-1754 AD : Abraham de Moivre : French    De Moivre's formula, development of analytic geometry, first statement of the formula for the normal distribution curve, probability theory

1690-1764 AD : Christian Goldbach : German : Goldbach Conjecture, Goldbach-Euler Theorem on perfect powers

1707-1783 AD : Leonhard Euler : Swiss : Made important contributions in almost all fields and found unexpected links between different fields, proved numerous theorems, pioneered new methods, standardized mathematical notation and wrote many influential textbooks

1728-1777 AD : Johann Lambert :     Swiss : Rigorous proof that π is irrational, introduced hyperbolic functions into trigonometry, made conjectures on non-Euclidean space and hyperbolic triangles

1736-1813 AD : Joseph Louis Lagrange : Italian/French : Comprehensive treatment of classical and celestial mechanics, calculus of variations, Lagrange’s theorem of finite groups, four-square theorem, mean value theorem

1746-1818 AD : Gaspard Monge :     French    Inventor of descriptive geometry, orthographic projection

1749-1827 AD : Pierre-Simon Laplace :    French : Celestial mechanics translated geometric study of classical mechanics to one based on calculus, Bayesian interpretation of probability, belief in scientific determinism

1752-1833 AD : Adrien-Marie Legendre : French    Abstract algebra, mathematical analysis, least squares method for curve-fitting and linear regression, quadratic reciprocity law, prime number theorem, elliptic functions

1768-1830 AD : Joseph Fourier :    French : Studied periodic functions and infinite sums in which the terms are trigonometric functions (Fourier series)

1777-1825 AD : Carl Friedrich Gauss :    German : Pattern in occurrence of prime numbers, construction of heptadecagon, Fundamental Theorem of Algebra, exposition of complex numbers, least squares approximation method, Gaussian distribution, Gaussian function, Gaussian error curve, non-Euclidean geometry, Gaussian curvature

1789-1857 AD : Augustin-Louis Cauchy    French    Early pioneer of mathematical analysis, reformulated and proved theorems of calculus in a rigorous manner, Cauchy's theorem (a fundamental theorem of group theory)

1790-1868 AD : August Ferdinand Möbius :    German    Möbius strip (a two-dimensional surface with only one side), Möbius configuration, Möbius transformations, Möbius transform (number theory), Möbius function, Möbius inversion formula

1791-1858 AD : George Peacock :    British : Inventor of symbolic algebra (early attempt to place algebra on a strictly logical basis)

1791-1871 AD : Charles Babbage :British : Designed a "difference engine" that could automatically perform computations based on instructions stored on cards or tape, forerunner of programmable computer.

1792-1856 AD : Nikolai Lobachevsky :    Russian :    Developed theory of hyperbolic geometry and curved spaces independendly of Bolyai

1802-1829 AD :  Niels Henrik Abel     :Norwegian :    Proved impossibility of solving quintic equations, group theory, abelian groups, abelian categories, abelian variety

1802-1860 AD : János Bolyai : Hungarian :    Explored hyperbolic geometry and curved spaces independently of Lobachevsky

1804-1851 AD : Carl Jacobi :    German :    Important contributions to analysis, theory of periodic and elliptic functions, determinants and matrices

1805-1865 AD : William Hamilton :    Irish : Theory of quaternions (first example of a non-commutative algebra)

1811-1832 AD : Évariste Galois    French    Proved that there is no general algebraic method for solving polynomial equations of degree greater than four, laid groundwork for abstract algebra, Galois theory, group theory, ring theory, etc

1815-1864  AD :  George Boole : British :  Devised Boolean algebra (using operators AND, OR and NOT), starting point of modern mathematical logic, led to the development of computer science

1815-1897 AD : Karl Weierstrass :    German :    Discovered a continuous function with no derivative, advancements in calculus of variations, reformulated calculus in a more rigorous fashion, pioneer in development of mathematical analysis

1821-1895 AD :    Arthur Cayley :    British : Pioneer of modern group theory, matrix algebra, theory of higher singularities, theory of invariants, higher dimensional geometry, extended Hamilton's quaternions to create octonions

1826-1866 AD :  Bernhard Riemann : German : Non-Euclidean elliptic geometry, Riemann surfaces, Riemannian geometry (differential geometry in multiple dimensions), complex manifold theory, zeta function, Riemann Hypothesis

1831-1916 AD : Richard Dedekind :  German :  Defined some important concepts of set theory such as similar sets and infinite sets, proposed Dedekind cut (now a standard definition of the real numbers)

1834-1923 AD : John Venn :    British : Introduced Venn diagrams into set theory (now a ubiquitous tool in probability, logic and statistics)

1842-1899 AD :  Marius Sophus Lie : Norwegian :    Applied algebra to geometric theory of differential equations, continuous symmetry, Lie groups of transformations

1845-1918 AD :  Georg Cantor : German : Creator of set theory, rigorous treatment of the notion of infinity and transfinite numbers, Cantor's theorem (which implies the existence of an “infinity of infinities”)

1848-1925 AD : Gottlob Frege : German : One of the founders of modern logic, first rigorous treatment of the ideas of functions and variables in logic, major contributor to study of the foundations of mathematics

1849-1925 AD : Felix Klein :     German    : Klein bottle (a one-sided closed surface in four-dimensional space), Erlangen Program to classify geometries by their underlying symmetry groups, work on group theory and function theory

1854-1912 AD :  Henri Poincaré :     French :     Partial solution to “three body problem”, foundations of modern chaos theory, extended theory of mathematical topology, Poincaré conjecture

1858-1932 AD :  Giuseppe Peano : Italian :    Peano axioms for natural numbers, developer of mathematical logic and set theory notation, contributed to modern method of mathematical induction

1861-1947 AD : Alfred North Whitehead :  British    : Co-wrote “Principia Mathematica” (attempt to ground mathematics on logic)

1862-1943 AD : David Hilbert : German : 23 “Hilbert problems”, finiteness theorem, “Entscheidungsproblem“ (decision problem), Hilbert space, developed modern axiomatic approach to mathematics, formalism

1864-1909 AD :  Hermann Minkowski :    German :    Geometry of numbers (geometrical method in multi-dimensional space for solving number theory problems), Minkowski space-time

1872-1970 AD : Bertrand Russell :British    Russell’s paradox, co-wrote “Principia Mathematica” (attempt to ground mathematics on logic), theory of types

1877-1947 AD :    G.H. Hardy :    British :    Progress toward solving Riemann hypothesis (proved infinitely many zeroes on the critical line), encouraged new tradition of pure mathematics in Britain, taxicab numbers

1878-1929 AD : Pierre Fatou :    French :    Pioneer in field of complex analytic dynamics, investigated iterative and recursive processes

1881-1966 AD :  L.E.J. Brouwer :    Dutch :    Proved several theorems marking breakthroughs in topology (including fixed point theorem and topological invariance of dimension)

1887-1920 AD : Srinivasa Ramanujan    : Indian :    Proved over 3,000 theorems, identities and equations, including on highly composite numbers, partition function and its asymptotics, and mock theta functions

1893-1978 AD :Gaston Julia :    French :    Developed complex dynamics, Julia set formula

1903-1957 AD : John von Neumann :    Hungarian/ American :    Pioneer of game theory, design model for modern computer architecture, work in quantum and nuclear physics

1906-1978 AD :Kurt Gödel :    Austria :    Incompleteness theorems (there can be solutions to mathematical problems which are true but which can never be proved), Gödel numbering, logic and set theory

1906-1998 AD : André Weil :    French :    Theorems allowed connections between algebraic geometry and number theory, Weil conjectures (partial proof of Riemann hypothesis for local zeta functions), founding member of influential Bourbaki group

1912-1954 AD :Alan Turing    :British :    Breaking of the German enigma code, Turing machine (logical forerunner of computer), Turing test of artificial intelligence

1913-1996 AD : Paul Erdös :    Hungarian : Set and solved many problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory and probability theory

1917-2008 AD :  Edward Lorenz :    American    Pioneer in modern chaos theory, Lorenz attractor, fractals, Lorenz oscillator, coined term “butterfly effect”

1919-1985 AD : Julia Robinson :    American : Work on decision problems and Hilbert's tenth problem, Robinson hypothesis

1924-2010 AD : Benoît Mandelbrot  :    French :    Mandelbrot set fractal, computer plottings of Mandelbrot and Julia sets

Born 1928 AD -     Alexander Grothendieck    : French :    Mathematical structuralist, revolutionary advances in algebraic geometry, theory of schemes, contributions to algebraic topology, number theory, category theory, etc.

1928 - 2015 AD : John Forbes Nash, Jr.   :  American : Mathematician with fundamental contributions in game theory, differential geometry, and partial differential equations. Has provided insight into the factors that govern chance and decision making inside complex systems found in daily life.

1934-2007 AD : Paul Cohen :    American :    Proved that continuum hypothesis could be both true and not true (i.e. independent from Zermelo-Fraenkel set theory)

Born 1937-    John Horton Conway :    British : Important contributions to game theory, group theory, number theory, geometry and (especially) recreational mathematics, notably with the invention of the cellular automaton called the "Game of Life"

Born 1947- Yuri Matiyasevich : Russian : Final proof that Hilbert’s tenth problem is impossible (there is no general method for determining whether Diophantine equations have a solution)

Born 1953 - Andrew Wiles :    British :    Finally proved Fermat’s Last Theorem for all numbers (by proving the Taniyama-Shimura conjecture for semistable elliptic curves)

Born 1966-    Grigori Perelman :    Russian :    Finally proved Poincaré Conjecture (by proving Thurston's geometrization conjecture), contributions to Riemannian geometry and geometric topology

Last edited by Jai Ganesh (2015-09-27 17:52:20)

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #2 2015-09-27 15:46:32

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Greatest Mathematicians from 1 CE ...

Hi;

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #3 2015-09-27 17:54:59

Jai Ganesh
Registered: 2005-06-28
Posts: 47,777

### Re: Greatest Mathematicians from 1 CE ...

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #4 2023-10-12 15:20:29

Jai Ganesh
Registered: 2005-06-28
Posts: 47,777

### Re: Greatest Mathematicians from 1 CE ...

(1) Heron (or Hero) of Alexandria

Hero of Alexandria was a Greek mathematician and engineer who was active in his native city of Alexandria in Egypt during the Roman era. He is often considered the greatest experimenter of antiquity and his work is representative of the Hellenistic scientific tradition.

Hero published a well-recognized description of a steam-powered device called an aeolipile (sometimes called a "Hero engine"). Among his most famous inventions was a windwheel, constituting the earliest instance of wind harnessing on land. He is said to have been a follower of the atomists. In his work Mechanics, he described pantographs. Some of his ideas were derived from the works of Ctesibius.

In mathematics he is mostly remembered for Heron's formula, a way to calculate the area of a triangle using only the lengths of its sides.

Much of Hero's original writings and designs have been lost, but some of his works were preserved including in manuscripts from the Eastern Roman Empire and to a lesser extent, in Latin or Arabic translations.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #5 2023-10-13 15:04:51

Jai Ganesh
Registered: 2005-06-28
Posts: 47,777

### Re: Greatest Mathematicians from 1 CE ...

(2) Ptolemy

Claudius Ptolemy (Ptolemaios; Latin: Claudius Ptolemaeus; c. 100 – c. 170 AD) was an Alexandrian mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importance to later Byzantine, Islamic, and Western European science. The first is the astronomical treatise now known as the Almagest, although it was originally entitled the Mathēmatikē Syntaxis or Mathematical Treatise, and later known as The Greatest Treatise. The second is the Geography, which is a thorough discussion on maps and the geographic knowledge of the Greco-Roman world. The third is the astrological treatise in which he attempted to adapt horoscopic astrology to the Aristotelian natural philosophy of his day. This is sometimes known as the Apotelesmatika (lit. "On the Effects") but more commonly known as the Tetrábiblos, from the Koine Greek meaning "Four Books", or by its Latin equivalent Quadripartite.

Unlike most Greek mathematicians, Ptolemy's writings (foremost the Almagest) never ceased to be copied or commented upon, both in Late Antiquity and in the Middle Ages. However, it is likely that only a few truly mastered the mathematics necessary to understand his works, as evidenced particularly by the many abridged and watered-down introductions to Ptolemy's astronomy that were popular among the Arabs and Byzantines.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #6 2023-10-22 00:47:27

Jai Ganesh
Registered: 2005-06-28
Posts: 47,777

### Re: Greatest Mathematicians from 1 CE ...

(3) Sun Tzu

Sun Tzu or Sun Zi was a Chinese mathematician of the third century CE.

His interests were in astronomy. He tried to develop a calendar and for this he investigated Diophantine equations. He is best known for authoring Sun Tzu Suan Ching (pinyin: Sun Zi Suan Jing; literally, "Sun Tzu's Calculation Classic"), which contains the Chinese remainder theorem.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #7 2023-11-06 16:22:36

Jai Ganesh
Registered: 2005-06-28
Posts: 47,777

### Re: Greatest Mathematicians from 1 CE ...

(4) Diophantus

Diophantus of Alexandria (born c. AD 200 – c. 214; died c. AD 284 – c. 298) was a Greek mathematician, who was the author of a series of books called Arithmetica, many of which deal with solving algebraic equations.

Diophantus is considered "the father of algebra" by many mathematicians because of his contributions to number theory, mathematical equations, and the earliest known use of algebraic notation and symbolism in his works. In modern use, Diophantine equations are algebraic equations with integer coefficients, for which integer solutions are sought.

Diophantine equations, Diophantine geometry, and Diophantine approximations are subareas of number theory that are named after him. Diophantus coined the term (parisotes) to refer to an approximate equality. This term was rendered as adaequalitas in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized positive rational numbers as numbers, by allowing fractions for coefficients and solutions.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #8 2023-11-12 15:50:19

Jai Ganesh
Registered: 2005-06-28
Posts: 47,777

### Re: Greatest Mathematicians from 1 CE ...

(5) Liu Hui

Liu Hui (fl.  3rd century CE) was a Chinese mathematician who published a commentary in 263 CE on Jiu Zhang Suan Shu (The Nine Chapters on the Mathematical Art). He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state of Cao Wei during the Three Kingdoms period (220–280 CE) of China.

His major contributions as recorded in his commentary on The Nine Chapters on the Mathematical Art include a proof of the Pythagorean theorem, theorems in solid geometry, an improvement on Archimedes's approximation of π, and a systematic method of solving linear equations in several unknowns. In his other work, Haidao Suanjing (The Sea Island Mathematical Manual), he wrote about geometrical problems and their application to surveying. He probably visited Luoyang, where he measured the sun's shadow.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #9 2023-11-19 17:07:13

Jai Ganesh
Registered: 2005-06-28
Posts: 47,777

### Re: Greatest Mathematicians from 1 CE ...

6) Aryabhata

Aryabhata (Āryabhaṭa) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the Āryabhaṭīya (which mentions that in 3600 Kali Yuga, 499 CE, he was 23 years old) and the Arya-siddhanta.

For his explicit mention of the relativity of motion, he also qualifies as a major early physicist.

Biography:

Name

While there is a tendency to misspell his name as "Aryabhatta" by analogy with other names having the "bhatta" suffix, his name is properly spelled Aryabhata: every astronomical text spells his name thus, including Brahmagupta's references to him "in more than a hundred places by name". Furthermore, in most instances "Aryabhatta" would not fit the metre either.

Time and place of birth

Aryabhata mentions in the Aryabhatiya that he was 23 years old 3,600 years into the Kali Yuga, but this is not to mean that the text was composed at that time. This mentioned year corresponds to 499 CE, and implies that he was born in 476. Aryabhata called himself a native of Kusumapura or Pataliputra (present day Patna, Bihar).

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one belonging to the Aśmaka country." During the Buddha's time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers in central India.

It has been claimed that the aśmaka (Sanskrit for "stone") where Aryabhata originated may be the present day Kodungallur which was the historical capital city of Thiruvanchikkulam of ancient Kerala. This is based on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, old records show that the city was actually Koṭum-kol-ūr ("city of strict governance"). Similarly, the fact that several commentaries on the Aryabhatiya have come from Kerala has been used to suggest that it was Aryabhata's main place of life and activity; however, many commentaries have come from outside Kerala, and the Aryasiddhanta was completely unknown in Kerala. K. Chandra Hari has argued for the Kerala hypothesis on the basis of astronomical evidence.

Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but his "Lanka" is an abstraction, standing for a point on the equator at the same longitude as his Ujjayini.

Education

It is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time. Both Hindu and Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna. A verse mentions that Aryabhata was the head of an institution (kulapa) at Kusumapura, and, because the university of Nalanda was in Pataliputra at the time and had an astronomical observatory, it is speculated that Aryabhata might have been the head of the Nalanda university as well. Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #10 2024-01-26 20:23:38

Jai Ganesh
Registered: 2005-06-28
Posts: 47,777

### Re: Greatest Mathematicians from 1 CE ...

7) Brahmagupta

Brahmagupta (c. 598 – c. 668 CE) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the Brāhmasphuṭasiddhānta (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical treatise, and the Khaṇḍakhādyaka ("edible bite", dated 665), a more practical text.

In 628 CE, Brahmagupta first described gravity as an attractive force, and used the term "gurutvākarṣaṇam" in Sanskrit to describe it.

Brahmagupta is credited with first clear description of the quadratic formula (the solution of the quadratic equation) in his main work, the Brāhma-sphuṭa-siddhānta.

Life and career

Brahmagupta, according to his own statement, was born in 598 CE. Born in Bhillamāla in Gurjaradesa (modern Bhinmal in Rajasthan, India) during the reign of the Chavda dynasty ruler Vyagrahamukha, his ancestors were probably from Sindh. He was the son of Jishnugupta and was a Hindu by religion, in particular, a Shaivite. He lived and worked there for a good part of his life. Prithudaka Svamin, a later commentator, called him Bhillamalacharya, the teacher from Bhillamala.

Bhillamala was the capital of the Gurjaradesa, the second-largest kingdom of Western India, comprising southern Rajasthan and northern Gujarat in modern-day India. It was also a centre of learning for mathematics and astronomy. He became an astronomer of the Brahmapaksha school, one of the four major schools of Indian astronomy during this period. He studied the five traditional Siddhantas on Indian astronomy as well as the work of other astronomers including Aryabhata I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin and Vishnuchandra.

In the year 628, at the age of 30, he composed the Brāhmasphuṭasiddhānta ("improved treatise of Brahma") which is believed to be a revised version of the received Siddhanta of the Brahmapaksha school of astronomy. Scholars state that he incorporated a great deal of originality into his revision, adding a considerable amount of new material. The book consists of 24 chapters with 1008 verses in the ārya metre. A good deal of it is astronomy, but it also contains key chapters on mathematics, including algebra, geometry, trigonometry and algorithmics, which are believed to contain new insights due to Brahmagupta himself.

Later, Brahmagupta moved to Ujjaini, Avanti, a major centre for astronomy in central India. At the age of 67, he composed his next well-known work Khanda-khādyaka, a practical manual of Indian astronomy in the karana category meant to be used by students.

Brahmagupta died in 668 CE, and he is presumed to have died in Ujjain.

Reception

Brahmagupta's mathematical advances were carried on further by Bhāskara II, a lineal descendant in Ujjain, who described Brahmagupta as the ganaka-chakra-chudamani (the gem of the circle of mathematicians). Prithudaka Svamin wrote commentaries on both of his works, rendering difficult verses into simpler language and adding illustrations. Lalla and Bhattotpala in the 8th and 9th centuries wrote commentaries on the Khanda-khadyaka. Further commentaries continued to be written into the 12th century.

A few decades after the death of Brahmagupta, Sindh came under the Arab Caliphate in 712 CE. Expeditions were sent into Gurjaradesa ("Al-Baylaman in Jurz", as per Arab historians). The kingdom of Bhillamala seems to have been annihilated but Ujjain repulsed the attacks. The court of Caliph Al-Mansur (754–775) received an embassy from Sindh, including an astrologer called Kanaka, who brought (possibly memorised) astronomical texts, including those of Brahmagupta. Brahmagupta's texts were translated into Arabic by Muhammad al-Fazari, an astronomer in Al-Mansur's court, under the names Sindhind and Arakhand. An immediate outcome was the spread of the decimal number system used in the texts. The mathematician Al-Khwarizmi (800–850 CE) wrote a text called al-Jam wal-tafriq bi hisal-al-Hind (Addition and Subtraction in Indian Arithmetic), which was translated into Latin in the 13th century as Algorithmi de numero indorum. Through these texts, the decimal number system and Brahmagupta's algorithms for arithmetic have spread throughout the world. Al-Khwarizmi also wrote his own version of Sindhind, drawing on Al-Fazari's version and incorporating Ptolemaic elements. Indian astronomic material circulated widely for centuries, even making its way into medieval Latin texts.

The historian of science George Sarton called Brahmagupta "one of the greatest scientists of his race and the greatest of his time."

Zero

Brahmagupta's Brahmasphuṭasiddhānta is the first book that provides rules for arithmetic manipulations that apply to zero and to negative numbers. The Brāhmasphuṭasiddhānta is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for lack of quantity as was done by Ptolemy and the Romans. In chapter eighteen of his Brāhmasphuṭasiddhānta, Brahmagupta describes operations on negative numbers. He first describes addition and subtraction.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #11 2024-04-28 16:37:11

Jai Ganesh
Registered: 2005-06-28
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### Re: Greatest Mathematicians from 1 CE ...

8) Bhāskara I

Bhāskara (c. 600 – c. 680) (commonly called Bhāskara I to avoid confusion with the 12th-century mathematician Bhāskara II) was a 7th-century Indian mathematician and astronomer who was the first to write numbers in the Hindu–Arabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work. This commentary, Āryabhaṭīyabhāṣya, written in 629 CE, is among the oldest known prose works in Sanskrit on mathematics and astronomy. He also wrote two astronomical works in the line of Aryabhata's school: the Mahābhāskarīya ("Great Book of Bhāskara") and the Laghubhāskarīya ("Small Book of Bhāskara").

On 7 June 1979, the Indian Space Research Organisation launched the Bhāskara I satellite, named in honour of the mathematician.

Biography

Little is known about Bhāskara's life, except for what can be deduced from his writings. He was born in India in the 7th century, and was probably an astronomer. Bhāskara I received his astronomical education from his father.

There are references to places in India in Bhāskara's writings, such as Vallabhi (the capital of the Maitraka dynasty in the 7th century) and Sivarajapura, both of which are in the Saurastra region of the present-day state of Gujarat in India. Also mentioned are Bharuch in southern Gujarat, and Thanesar in the eastern Punjab, which was ruled by Harsha. Therefore, a reasonable guess would be that Bhāskara was born in Saurastra and later moved to Aśmaka.

Bhāskara I is considered the most important scholar of Aryabhata's astronomical school. He and Brahmagupta are two of the most renowned Indian mathematicians; both made considerable contributions to the study of fractions.

Representation of numbers

The most important mathematical contribution of Bhāskara I concerns the representation of numbers in a positional numeral system. The first positional representations had been known to Indian astronomers approximately 500 years before Bhāskara's work. However, these numbers were written not in figures, but in words or allegories and were organized in verses. For instance, the number 1 was given as moon, since it exists only once; the number 2 was represented by wings, twins, or eyes since they always occur in pairs; the number 5 was given by the (5) senses. Similar to our current decimal system, these words were aligned such that each number assigns the factor of the power of ten corresponding to its position, only in reverse order: the higher powers were to the right of the lower ones.

Bhāskara's numeral system was truly positional, in contrast to word representations, where the same word could represent multiple values (such as 40 or 400). He often explained a number given in his numeral system by stating ankair api ("in figures this reads"), and then repeating it written with the first nine Brahmi numerals, using a small circle for the zero. Contrary to the word system, however, his numerals were written in descending values from left to right, exactly as we do it today. Therefore, since at least 629, the decimal system was definitely known to Indian scholars. Presumably, Bhāskara did not invent it, but he was the first to openly use the Brahmi numerals in a scientific contribution in Sanskrit.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #12 2024-06-09 16:30:41

Jai Ganesh
Registered: 2005-06-28
Posts: 47,777

### Re: Greatest Mathematicians from 1 CE ...

Muhammad ibn Musa al-Khwarizmi (c. 780 – c. 850), often referred to as simply al-Khwarizmi, was a Persian polymath who produced vastly influential Arabic-language works in mathematics, astronomy, and geography. Hailing from Khwarazm, he was appointed as the astronomer and head of the House of Wisdom in the city of Baghdad around 820 CE.

His popularizing treatise on algebra, compiled between 813–33 as Al-Jabr (The Compendious Book on Calculation by Completion and Balancing),  presented the first systematic solution of linear and quadratic equations. One of his achievements in algebra was his demonstration of how to solve quadratic equations by completing the square, for which he provided geometric justifications.  Because al-Khwarizmi was the first person to treat algebra as an independent discipline and introduced 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), he has been described as the father or founder of algebra. The English term algebra comes from the short-hand title of his aforementioned treatise (Al-Jabr, transl. "completion" or "rejoining"). His name gave rise to the English terms algorism and algorithm; the Spanish, Italian, and Portuguese terms algoritmo; and the Spanish term guarismo and Portuguese term algarismo, both meaning "digit".

In the 12th century, Latin-language translations of al-Khwarizmi's textbook on Indian arithmetic (Algorithmo de Numero Indorum), which codified the various Indian numerals, introduced the decimal-based positional number system to the Western world. Likewise, Al-Jabr, translated into Latin by the English scholar Robert of Chester in 1145, was used until the 16th century as the principal mathematical textbook of European universities.

Al-Khwarizmi revised Geography, the 2nd-century Greek-language treatise by the Roman polymath Claudius Ptolemy, listing the longitudes and latitudes of cities and localities.  He further produced a set of astronomical tables and wrote about calendric works, as well as the astrolabe and the sundial. Al-Khwarizmi made important contributions to trigonometry, producing accurate sine and cosine tables and the first table of tangents.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #13 2024-07-12 21:12:19

Jai Ganesh
Registered: 2005-06-28
Posts: 47,777

### Re: Greatest Mathematicians from 1 CE ...

10) Ibrahim ibn Sinan

Ibrahim ibn Sinan (born 295 – 296 AH/c. 908 in Baghdad, died: 334-335 AH/946 in Baghdad, aged 38) was a mathematician and astronomer who belonged to a family of scholars originally from Harran in northern Mesopotamia. He was the son of Sinan ibn Thabit (c. 880 – 943) and the grandson of Thābit ibn Qurra (c. 830 – 901). Like his grandfather, he belonged to a religious sect of star worshippers known as the Sabians of Harran.

Ibrahim ibn Sinan studied geometry, in particular tangents to circles. He made advances in the quadrature of the parabola and the theory of integration, generalizing the work of Archimedes, which was unavailable at the time. Ibrahim ibn Sinan is often considered to be one of the most important mathematicians of his time.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #14 2024-07-26 15:17:20

Jai Ganesh
Registered: 2005-06-28
Posts: 47,777

### Re: Greatest Mathematicians from 1 CE ...

Abū Bakr Muḥammad ibn al Ḥasan al-Karajī (c. 953 – c. 1029) was a 10th-century Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works are mathematical: Al-Badi' fi'l-hisab (Wonderful on calculation), Al-Fakhri fi'l-jabr wa'l-muqabala (Glorious on algebra), and Al-Kafi fi'l-hisab (Sufficient on calculation).

Work

Al-Karaji wrote on mathematics and engineering. Some consider him to be merely reworking the ideas of others (he was influenced by Diophantus) but most regard him as more original, in particular for the beginnings of freeing algebra from geometry. Among historians, his most widely studied work is his algebra book al-fakhri fi al-jabr wa al-muqabala, which survives from the medieval era in at least four copies.

In his book "Extraction of hidden waters" he has mentioned that earth is spherical in shape but considers it the centre of the universe long before Galileo Galilei, Johannes Kepler or Isaac Newton, but long after Aristotle and Ptolemy. He expounded the basic principles of hydrology and this book reveals his profound knowledge of this science and has been described as the oldest extant text in this field.

He systematically studied the algebra of exponents, and was the first to define the rules for monomials like x,x²,x³ and their reciprocals in the cases of multiplication and division. However, since for example the product of a square and a cube would be expressed, in words rather than in numbers, as a square-cube, the numerical property of adding exponents was not clear.

His work on algebra and polynomials gave the rules for arithmetic operations for adding, subtracting and multiplying polynomials; though he was restricted to dividing polynomials by monomials.

F. Woepcke was the first historian to realise the importance of al-Karaji's work and later historians mostly agree with his interpretation. He praised Al-Karaji for being the first who introduced the theory of algebraic calculus.

Al-Karaji gave the first formulation of the binomial coefficients and the first description of Pascal's triangle. He is also credited with the discovery of the binomial theorem.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #15 2024-09-03 15:28:19

Jai Ganesh
Registered: 2005-06-28
Posts: 47,777

### Re: Greatest Mathematicians from 1 CE ...

12) Ibn al-Haytham (Alhazen)

Ḥasan Ibn al-Haytham (c. 965 – c. 1040) was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq. Referred to as "the father of modern optics", he made significant contributions to the principles of optics and visual perception in particular. His most influential work is titled Kitāb al-Manāẓir "Book of Optics"), written during 1011–1021, which survived in a Latin edition. The works of Alhazen were frequently cited during the scientific revolution by Isaac Newton, Johannes Kepler, Christiaan Huygens, and Galileo Galilei.

Ibn al-Haytham was the first to correctly explain the theory of vision, and to argue that vision occurs in the brain, pointing to observations that it is subjective and affected by personal experience. He also stated the principle of least time for refraction which would later become the Fermat's principle. He made major contributions to catoptrics and dioptrics by studying reflection, refraction and nature of images formed by light rays. Ibn al-Haytham was an early proponent of the concept that a hypothesis must be supported by experiments based on confirmable procedures or mathematical reasoning—an early pioneer in the scientific method five centuries before Renaissance scientists, he is sometimes described as the world's "first true scientist". He was also a polymath, writing on philosophy, theology and medicine.

Born in Basra, he spent most of his productive period in the Fatimid capital of Cairo and earned his living authoring various treatises and tutoring members of the nobilities. Ibn al-Haytham is sometimes given the byname al-Baṣrī after his birthplace, or al-Miṣrī ("the Egyptian"). Al-Haytham was dubbed the "Second Ptolemy" by Abu'l-Hasan Bayhaqi and "The Physicist" by John Peckham. Ibn al-Haytham paved the way for the modern science of physical optics.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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