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1650 BC - Rhind Mathematical Papyrus, copy of a lost scroll from around 1850 BC, the scribe Ahmes presents one of the first known approximate values of ? at 3.16, the first attempt at squaring the circle, earliest known use of a sort of cotangent, and knowledge of solving first order linear equations.
1st millennium BC
c. 1000 BC - Simple fractions used by the Egyptians. However, only unit fractions are used (i.e., those with 1 as the numerator) and interpolation tables are used to approximate the values of the other fractions.
first half of 1st millennium BC - Vedic India - Yajnavalkya, in his Shatapatha Brahmana, describes the motions of the sun and the moon, and advances a 95-year cycle to synchronize the motions of the sun and the moon.
c. 8th century BC - the Yajur Veda, one of the four HinduVedas, contains the earliest concept of infinity, and states "if you remove a part from infinity or add a part to infinity, still what remains is infinity."
1046 BC to 256 BC - China, Zhoubi Suanjing, arithmetic, geometric algorithms, and proofs.
5th century BC - Apastamba, author of the Apastamba Sulba Sutra, another Vedic Sanskrit geometric text, makes an attempt at squaring the circle and also calculates the square root of 2 correct to five decimal places.
c. 400 BC - Jaina mathematicians in India write the Surya Prajinapti, a mathematical text classifying all numbers into three sets: enumerable, innumerable and infinite. It also recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.
260 BC - Archimedes proved that the value of ? lies between 3 + 1/7 (approx. 3.1429) and 3 + 10/71 (approx. 3.1408), that the area of a circle was equal to ? multiplied by the square of the radius of the circle and that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He also gave a very accurate estimate of the value of the square root of 3.
c. 250 BC - late Olmecs had already begun to use a true zero (a shell glyph) several centuries before Ptolemy in the New World. See 0 (number).
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.
final centuries BC - Indian astronomer Lagadha writes the Vedanga Jyotisha, a Vedic text on astronomy that describes rules for tracking the motions of the sun and the moon, and uses geometry and trigonometry for astronomy.
500 - Aryabhata writes the Aryabhata-Siddhanta, which first introduces the trigonometric functions and methods of calculating their approximate numerical values. It defines the concepts of sine and cosine, and also contains the earliest tables of sine and cosine values (in 3.75-degree intervals from 0 to 90 degrees).
6th century - Aryabhata gives accurate calculations for astronomical constants, such as the solar eclipse and lunar eclipse, computes ? to four decimal places, and obtains whole number solutions to linear equations by a method equivalent to the modern method.
7th century - Bhaskara I gives a rational approximation of the sine function.
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.
8th century - Shridhara gives the rule for finding the volume of a sphere and also the formula for solving quadratic equations.
773 - Kanka brings Brahmagupta's Brahma-sphuta-siddhanta to Baghdad to explain the Indian system of arithmetic astronomy and the Indian numeral system.
773 - Al-Fazari translates the Brahma-sphuta-siddhanta into Arabic upon the request of King Khalif Abbasid Al Mansoor.
9th century - Govindsvamin discovers the Newton-Gauss interpolation formula, and gives the fractional parts of Aryabhata's tabular sines.
810 - The House of Wisdom is built in Baghdad for the translation of Greek and Sanskrit mathematical works into Arabic.
820 - Al-Khwarizmi - Persian mathematician, father of algebra, writes the Al-Jabr, later transliterated as Algebra, which introduces systematic algebraic techniques for solving linear and quadratic equations. Translations of his book on arithmetic will introduce the Hindu-Arabicdecimal number system to the Western world in the 12th century. The term algorithm is also named after him.
953 - The arithmetic of the Hindu-Arabic numeral system at first required the use of a dust board (a sort of handheld blackboard) because "the methods required moving the numbers around in the calculation and rubbing some out as the calculation proceeded." Al-Uqlidisi modified these methods for pen and paper use. Eventually the advances enabled by the decimal system led to its standard use throughout the region and the world.
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 was first to define the monomials, , , ... and , , , ... and to give rules for products of any two of these. He started a school of algebra which flourished for several hundreds of years". He also discovered the binomial theorem for integerexponents, which "was a major factor in the development of numerical analysis based on the decimal system".
975 - Al-Batani extended the Indian concepts of sine and cosine to other trigonometrical ratios, like tangent, secant and their inverse functions. Derived the formulae: and .
1030 - Ali Ahmad Nasawi writes a treatise on the decimal and sexagesimal number systems. His arithmetic explains the division of fractions and the extraction of square and cubic roots (square root of 57,342; cubic root of 3, 652, 296) in an almost modern manner.
1070 - Omar Khayyám begins to write Treatise on Demonstration of Problems of Algebra and classifies cubic equations.
12th century - the Hindu-Arabic numeral system reaches Europe through the Arabs.
12th century - Bhaskara Acharya writes the Lilavati, which covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations.
12th century - Bh?skara II (Bhaskara Acharya) writes the Bijaganita (Algebra), which is the first text to recognize that a positive number has two square roots.
1130 - Al-Samawal gave 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 followed al-Khayyam's application of algebra to geometry, and wrote a treatise on cubic equations that "represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry".
1248 - Li Ye writes Ceyuan haijing, a 12 volume mathematical treatise containing 170 formulas and 696 problems mostly solved by polynomial equations using the method tian yuan shu.
1260 - Al-Farisi gave a new proof of Thabit ibn Qurra's theorem, introducing important new ideas concerning factorization and combinatorial methods. He also gave the pair of amicable numbers 17296 and 18416 that have also been joint attributed to Fermat as well as Thabit ibn Qurra.
14th century - Madhava is considered the father of mathematical analysis, who also worked on the power series for π and for sine and cosine functions, and along with other Kerala school mathematicians, founded the important concepts of calculus.
1400 - Madhava discovers the series expansion for the inverse-tangent function, the infinite series for arctan and sin, and many methods for calculating the circumference of the circle, and uses them to compute ? correct to 11 decimal places.
c. 1400 - Ghiyath al-Kashi "contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real numbers such as ?. His contribution to decimal fractions is so major that for many years he was considered as their inventor. Although not the first to do so, al-Kashi gave an algorithm for calculating nth roots, which is a special case of the methods given many centuries later by [Paolo] Ruffini and [William George] Horner." He is also the first to use the decimal point notation in arithmetic and Arabic numerals. His works include The Key of arithmetics, Discoveries in mathematics, The Decimal point, and The benefits of the zero. The contents of the Benefits of the Zero are an introduction followed by five essays: "On whole number arithmetic", "On fractional arithmetic", "On astrology", "On areas", and "On finding the unknowns [unknown variables]". He also wrote the Thesis on the sine and the chord and Thesis on finding the first degree sine.
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.
1424 - Ghiyath al-Kashi computes ? to sixteen decimal places using inscribed and circumscribed polygons.
1427 - Al-Kashi completes The Key to Arithmetic containing work of great depth on decimal fractions. It applies arithmetical and algebraic methods to the solution of various problems, including several geometric ones.
1464 - Regiomontanus writes De Triangulis omnimodus which is one of the earliest texts to treat trigonometry as a separate branch of mathematics.
1825 - Augustin-Louis Cauchy presents the Cauchy integral theorem for general integration paths--he assumes the function being integrated has a continuous derivative, and he introduces the theory of residues in complex analysis.
1832 - Lejeune Dirichlet proves Fermat's Last Theorem for n = 14.
1835 - Lejeune Dirichlet proves Dirichlet's theorem about prime numbers in arithmetical progressions.
1837 - Pierre Wantzel proves that doubling the cube and trisecting the angle are impossible with only a compass and straightedge, as well as the full completion of the problem of constructability of regular polygons.
1981 - Richard Feynman gives an influential talk "Simulating Physics with Computers" (in 1980 Yuri Manin proposed the same idea about quantum computations in "Computable and Uncomputable" (in Russian)).
1983 - Gerd Faltings proves the Mordell conjecture and thereby shows that there are only finitely many whole number solutions for each exponent of Fermat's Last Theorem.