# Timeline of mathematics

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This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the development of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.

## Syncopated stage

### 1st millennium BC

• 1st century – Greece, Heron of Alexandria, Hero, the earliest, fleeting reference to square roots of negative numbers.
• c 100 – Greece, Theon of Smyrna
• 60 – 120 – Greece, Nicomachus
• 70 – 140 – Greece, Menelaus of Alexandria Spherical trigonometry
• 78 – 139 – China, Zhang Heng
• c. 2nd century – Greece, Ptolemy of Alexandria wrote the Almagest .
• 132 – 192 – China, Cai Yong
• 240 – 300 – Greece, Sporus of Nicaea
• 250 – Greece, Diophantus uses symbols for unknown numbers in terms of syncopated algebra, and writes Arithmetica , one of the earliest treatises on algebra.
• 263 – China, Liu Hui computes π using Liu Hui's π algorithm.
• 300 – the earliest known use of zero as a decimal digit is introduced by Indian mathematicians.
• 234 – 305 – Greece, Porphyry (philosopher)
• 300 – 360 – Greece, Serenus of Antinoöpolis
• 335 – 405– Greece, Theon of Alexandria
• c. 340 – Greece, Pappus of Alexandria states his hexagon theorem and his centroid theorem.
• 350 – 415 – Byzantine Empire, Hypatia
• c. 400 – India, the Bakhshali manuscript , 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.
• 300 to 500 – the Chinese remainder theorem is developed by Sun Tzu.
• 300 to 500 – China, a description of rod calculus is written by Sun Tzu.
• 412 – 485 – Greece, Proclus
• 420 – 480 – Greece, Domninus of Larissa
• b 440 – Greece, Marinus of Neapolis "I wish everything was mathematics."
• 450 – China, Zu Chongzhi computes π to seven decimal places. This calculation remains the most accurate calculation for π for close to a thousand years.
• c. 474 – 558 – Greece, Anthemius of Tralles
• 500 – India, 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).
• 480 – 540 – Greece, Eutocius of Ascalon
• 490 – 560 – Greece, Simplicius of Cilicia
• 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.
• 505 – 587 – India, Varāhamihira
• 6th century – India, Yativṛṣabha
• 535 – 566 – China, Zhen Luan
• 550 Hindu mathematicians give zero a numeral representation in the positional notation Indian numeral system.
• 600 – China, Liu Zhuo uses quadratic interpolation.
• 602 – 670 – China, Li Chunfeng
• 625 China, Wang Xiaotong writes the Jigu Suanjing, where cubic and quartic equations are solved.
• 7th century – India, Bhāskara I gives a rational approximation of the sine function.
• 7th century – India, 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.
• 721 – China, Zhang Sui (Yi Xing) computes the first tangent table.
• 8th century – India, 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 – India, Sridhara gives the rule for finding the volume of a sphere and also the formula for solving quadratic equations.
• 773 – Iraq, Kanka brings Brahmagupta's Brahma-sphuta-siddhanta to Baghdad to explain the Indian system of arithmetic astronomy and the Indian numeral system.
• 773 Muḥammad ibn Ibrāhīm al-Fazārī translates the Brahma-sphuta-siddhanta into Arabic upon the request of King Khalif Abbasid Al Mansoor.
• 9th century – India, Govindasvāmi 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 – Iran, Al-Mahani conceived the idea of reducing geometrical problems such as doubling the cube to problems in algebra.
• c. 850 – Iraq, al-Kindi pioneers cryptanalysis and frequency analysis in his book on cryptography.
• c. 850 – India, Mahāvīra writes the Gaṇitasārasan̄graha otherwise known as the Ganita Sara Samgraha which gives systematic rules for expressing a fraction as the sum of unit fractions.
• 895 – Syria, Thābit 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 – Egypt, Abu Kamil had begun to understand what we would write in symbols as ${\displaystyle x^{n}\cdot x^{m}=x^{m+n}}$
• 940 – Iran, Abu al-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 – Persia, 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 ${\displaystyle x}$, ${\displaystyle x^{2}}$, ${\displaystyle x^{3}}$, ... and ${\displaystyle 1/x}$, ${\displaystyle 1/x^{2}}$, ${\displaystyle 1/x^{3}}$, ... 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 – Mesopotamia, al-Battani extended the Indian concepts of sine and cosine to other trigonometrical ratios, like tangent, secant and their inverse functions. Derived the formulae: ${\displaystyle \sin \alpha =\tan \alpha /{\sqrt {1+\tan ^{2}\alpha }}}$ and ${\displaystyle \cos \alpha =1/{\sqrt {1+\tan ^{2}\alpha }}}$.

## Symbolic stage

### 1000–1500

#### 15th century

• 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 π correct to 11 decimal places.
• c. 1400 Jamshid al-Kashi "contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real numbers such as π. 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 Ibn al-Banna' al-Marrakushi and Abu'l-Hasan ibn Ali al-Qalasadi introduced symbolic notation for algebra and for mathematics in general. [12]
• 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 π to sixteen decimal places using inscribed and circumscribed polygons.
• 1427 Jamshid 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.
• 1478 – An anonymous author writes the Treviso Arithmetic .
• 1494 Luca Pacioli writes Summa de arithmetica, geometria, proportioni et proportionalità ; introduces primitive symbolic algebra using "co" (cosa) for the unknown.

[19]

## Related Research Articles

The history of mathematics deals with the origin of discoveries in mathematics and 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. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars.

Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers, or defined as generalizations of the integers.

A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called numerals; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number. The most common numeral system is the Hindu–Arabic numeral system, which allows for the representation of any non-negative integer using a combination of ten fundamental numeric symbols, called digits. In addition to their use in counting and measuring, numerals are often used for labels, for ordering, and for codes. In common usage, a numeral is not clearly distinguished from the number that it represents.

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics, therefore, excludes topics in "continuous mathematics" such as calculus and analysis.

The timeline below shows the date of publication of possible major scientific breakthroughs, theories and discoveries, along with the discoverer. This article discounts mere speculation as discovery, although imperfect reasoned arguments, arguments based on elegance/simplicity, and numerically/experimentally verified conjectures qualify. The timeline begins at the Bronze Age, as it is difficult to give even estimates for the timing of events prior to this, such as of the discovery of counting, natural numbers and arithmetic.

Mathematics emerged independently in China by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system, algebra, geometry, number theory and trigonometry.

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.

In mathematics, a proof of impossibility is a proof that demonstrates that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general. Such a case is also known as a negative proof, proof of an impossibility theorem, or negative result. Proofs of impossibility often are the resolutions to decades or centuries of work attempting to find a solution, eventually proving that there is no solution. Proving that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a proof that works in general, rather than to just show a particular example. Impossibility theorems are usually expressible as negative existential propositions or universal propositions in logic.

The history of mathematical notation includes the commencement, progress, and cultural diffusion of mathematical symbols and the conflict of the methods of notation confronted in a notation's move to popularity or inconspicuousness. Mathematical notation comprises the symbols used to write mathematical equations and formulas. Notation generally implies a set of well-defined representations of quantities and symbols operators. The history includes Hindu–Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a host of symbols invented by mathematicians over the past several centuries.

In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.

A timeline of number theory.

A timeline of numerals and arithmetic.

The following is a timeline of key developments of geometry:

The Story of Maths is a four-part British television series outlining aspects of the history of mathematics. It was a co-production between the Open University and the BBC and aired in October 2008 on BBC Four. The material was written and presented by University of Oxford professor Marcus du Sautoy. The consultants were the Open University academics Robin Wilson, professor Jeremy Gray and June Barrow-Green. Kim Duke is credited as series producer.

In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length, no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.

Mathematics is a field of study that investigates topics such as number, space, structure, and change.

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