Timeline of Mathematics - 1st Millennium AD

1st Millennium AD

• 1st century — Heron of Alexandria, the earliest fleeting reference to square roots of negative numbers.
• c. 3rd century — Ptolemy of Alexandria wrote the Almagest
• 250 — Diophantus uses symbols for unknown numbers in terms of syncopated algebra, and writes Arithmetica, one of the earliest treatises on algebra
• 300 — the earliest known use of zero as a decimal digit is introduced by Indian mathematicians
• c. 340 — Pappus of Alexandria states his hexagon theorem and his centroid theorem
• c. 400 — the “Bakhshali manuscript” is written by Jaina mathematicians, which describes a theory of the infinite containing different levels of infinity, shows an understanding of indices, as well as logarithms to base 2, and computes square roots of numbers as large as a million correct to at least 11 decimal places
• 450 — Zu Chongzhi computes π to seven decimal places,
• 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 π to four decimal places, and obtains whole number solutions to linear equations by a method equivalent to the modern method
• 550 — Hindu mathematicians give zero a numeral representation in the positional notation Indian numeral system
• 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
• 628 — Brahmagupta writes the Brahma-sphuta-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
• 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 Fazaii 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-Arabic decimal number system to the Western world in the 12th century. The term algorithm is also named after him.
• 820 — Al-Mahani conceived the idea of reducing geometrical problems such as doubling the cube to problems in algebra.
• c. 850 — Al-Kindi pioneers cryptanalysis and frequency analysis in his book on cryptography.
• 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).
• c. 900 — Abu Kamil of Egypt had begun to understand what we would write in symbols as
• 940 — Abu'l-Wafa al-Buzjani extracts roots using the Indian numeral system.
• 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 integer exponents, 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 .