The area of study known as the history of mathematics is primarily an investigation into the origin of new discoveries in mathematics, to a lesser extent an investigation into the standard mathematical methods and notation of the past. Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala ( Arabic for "The Compendious Book on Calculation by Completion and Balancing" Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and

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. The most ancient mathematical texts available are Plimpton 322 (Babylonian mathematics ca. Of the approximately half million Babylonian Clay tablets excavated since the beginning of the 19th century several thousand are of a mathematical nature Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (ancient Iraq) from the days of the early Sumerians to the fall of 1900 BC), the Moscow Mathematical Papyrus (Egyptian mathematics ca. The Moscow Mathematical Papyrus is also called the Golenischev Mathematical Papyrus, after its first owner Egyptologist Vladimir Goleniščev. Egyptian mathematics refers to the style and methods of Mathematics performed in Ancient Egypt. 1850 BC), the Rhind Mathematical Papyrus (Egyptian mathematics ca. The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish 1650 BC), and the Shulba Sutras (Indian mathematics ca. The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. 800 BC). All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry

Egyptian and Babylonian mathematics were then further developed in Greek and Hellenistic mathematics, which is generally considered to be one of the most important for greatly expanding both the method and the subject matter of mathematics. Greek mathematics, as that term is used in this article is the Mathematics written in Greek, developed from the 6th century BC to the 5th century [1] The mathematics developed in these ancient civilizations were then further developed and greatly expanded in Islamic mathematics. Many Greek and Arabic texts on mathematics were then translated into Latin in medieval Europe and further developed there. The Renaissance of the 12th century saw a major search by European scholars for new learning which led them to the Arabic fringes of Europe especially to Islamic

One striking feature of the history of ancient and medieval mathematics is that bursts of mathematical development were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries, were made at an ever increasing pace, and this continues to the present day. The Renaissance (from French Renaissance, meaning "rebirth" Italian: Rinascimento, from re- "again" and nascere Italy (Italia officially the Italian Republic, (Repubblica Italiana is located on the Italian Peninsula in Southern Europe, and on the two largest Exponential growth (including Exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value

## Early mathematics

The Ishango bone, dating to perhaps 18000 to 20000 B.C.

Evidence exists that early counting involved women who kept records of their monthly biological cycles; twenty-eight, twenty-nine, or thirty scratches on bone or stone, followed by a distinctive scratching on the bone or stone, for example. Moreover, hunters had the concepts of one, two, and many, as well as the idea of none or zero, when considering herds of animals. [5][6]

The Ishango Bone, found in the area of the headwaters of the Nile River (northeastern Congo), dates as early as 20,000 BC. The Ishango bone is a Bone tool, dated to the Upper Paleolithic era about 18000 to 20000 BC The Nile (النيل, Ancient Egyptian iteru or Ḥ'pī, Coptic piaro or phiaro) is a major north-flowing River The Democratic Republic of the Congo (République démocratique du Congo often referred to as DR Congo, DRC or RDC, and formerly known or referred to The Upper Paleolithic (or Upper Palaeolithic) is the third and last subdivision of the Paleolithic or Old Stone Age as it is understood in Europe Africa One common interpretation is that the bone is the earliest known demonstration[7] of sequences of prime numbers and Ancient Egyptian multiplication. In Mathematics, a sequence is an ordered list of objects (or events In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 Ancient Egyptian multiplication is a systematic method for multiplying two numbers that does not require the Multiplication table, only the ability to multiply and divide Predynastic Egyptians of the 5th millennium BC pictorially represented geometric spatial designs. The Predynastic Period of Egypt (prior to 3100 BC is traditionally the period between the Early Neolithic and the beginning of the Pharaonic monarchy beginning with King Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another It has been claimed that Megalithic monuments in England and Scotland from the 3rd millennium BC, incorporate geometric ideas such as circles, ellipses, and Pythagorean triples in their design. England is a Country which is part of the United Kingdom. Its inhabitants account for more than 83% of the total UK population whilst its mainland Scotland ( Gaelic: Alba) is a Country in northwest Europethat occupies the northern third of the island of Great Britain. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a A Pythagorean triple consists of three positive Integers a, b, and c, such that a 2 +  b 2 =  [8]

## Ancient Near East (c. 1800-500 BC)

### Mesopotamia

In contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Egyptian mathematics refers to the style and methods of Mathematics performed in Ancient Egypt. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework.

The earliest evidence of written mathematics dates back to the ancient Sumerians, who built the earliest civilization in Mesopotamia. Sumer ( Sumerian: sux-Latn [[Ki (earth ki]]-[[EN (cuneiform en]]-'''ĝir15''', Akkadian: Šumeru; possibly Biblical Shinar They developed a complex system of metrology from 3000 BC. Metrology (from Ancient Greek metron (measure and logos (study of is the Science of Measurement. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. The earliest traces of the Babylonian numerals also date back to this period. [10]

The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics which include fractions, algebra, quadratic and cubic equations, and the calculation of Pythagorean triples (see Plimpton 322). A Pythagorean triple consists of three positive Integers a, b, and c, such that a 2 +  b 2 =  Of the approximately half million Babylonian Clay tablets excavated since the beginning of the 19th century several thousand are of a mathematical nature [11] The tablets also include multiplication tables, trigonometry tables and methods for solving linear and quadratic equations. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. The Babylonian tablet YBC 7289 gives an approximation to √2 accurate to five decimal places.

Babylonian mathematics were written using a sexagesimal (base-60) numeral system. Sexagesimal ( base-sixty) is a Numeral system with sixty as the base. A numeral system (or system of numeration) is a Mathematical notation for representing numbers of a given set by symbols in a consistent manner From this we derive the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle. Babylonian advances in mathematics were facilitated by the fact that 60 has many divisors. Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values, much as in the decimal system. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. They lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.

### Egypt

Main article: Egyptian mathematics

The oldest mathematical text discovered so far is the Moscow papyrus, which is an Egyptian Middle Kingdom papyrus dated c. The Moscow Mathematical Papyrus is also called the Golenischev Mathematical Papyrus, after its first owner Egyptologist Vladimir Goleniščev. Ancient Egypt was an Ancient Civilization in eastern North Africa, concentrated along the lower reaches of the Nile River in what is now 2000—1800 BC. Like many ancient mathematical texts, it consists of what are today called "word problems" or "story problems", which were apparently intended as entertainment. One problem is considered to be of particular importance because it gives a method for finding the volume of a frustum: "If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top. Elements special cases and related concepts Each plane section is a base of the frustum You are to square this 4, result 16. You are to double 4, result 8. You are to square 2, result 4. You are to add the 16, the 8, and the 4, result 28. You are to take one third of 6, result 2. You are to take 28 twice, result 56. See, it is 56. You will find it right. "

The Rhind papyrus (c. The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish 1650 BC [3]) is another major Egyptian mathematical text, an instruction manual in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge (see [4]), including composite and prime numbers; arithmetic, geometric and harmonic means; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory (namely, that of the number 6)[5]. A composite number is a positive Integer which has a positive Divisor other than one or itself In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics and Statistics, the arithmetic Mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided The geometric mean in Mathematics, is a type of Mean or Average, which indicates the central tendency or typical value of a set of numbers In Mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of Average. In Mathematics, the Sieve of Eratosthenes is a simple ancient Algorithm for finding all Prime numbers up to a specified integer In mathematics a perfect number is defined as a positive integer which is the sum of its proper positive Divisors that is the sum of the positive divisors excluding It also shows how to solve first order linear equations [6] as well as arithmetic and geometric series [7]. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members In Mathematics, a geometric series is a series with a constant ratio between successive terms.

Also, three geometric elements contained in the Rhind papyrus suggest the simplest of underpinnings to analytical geometry: (1) first and foremost, how to obtain an approximation of π accurate to within less than one percent; (2) second, an ancient attempt at squaring the circle; and (3) third, the earliest known use of a kind of cotangent. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry Squaring the circle is a problem proposed by ancient Geometers.

Finally, the Berlin papyrus (c. The Berlin Papyrus 6619 commonly known as the Berlin Papyrus is an Ancient Egyptian papyrus document from the Middle Kingdom. 1300 BC [8] [9]) shows that ancient Egyptians could solve a second-order algebraic equation [10]. In Mathematics, an algebraic equation over a given field is an Equation of the form P = Q where P and Q

## Ancient Indian mathematics (c. 900 BC — AD 200)

Main article: Indian mathematics

Vedic mathematics began in the early Iron Age, with the Shatapatha Brahmana (c. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. The Shatapatha Brahmana (sa शतपथ ब्राह्मण śatapatha brāhmaṇa, " Brahmana of one-hundred paths" abbreviated ŚB 9th century BC), which approximates the value of π to 2 decimal places. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems [11], and the Sulba Sutras (c. The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the 800-500 BC) were geometry texts that used irrational numbers, prime numbers, the rule of three and cube roots; computed the square root of 2 to five decimal places; gave the method for squaring the circle; solved linear equations and quadratic equations; developed Pythagorean triples algebraically and gave a statement and numerical proof of the Pythagorean theorem. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics, a cube root of a number denoted \sqrt{x} or x1/3 is a number a such that a 3 =  x In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose Squaring the circle is a problem proposed by ancient Geometers. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. A Pythagorean triple consists of three positive Integers a, b, and c, such that a 2 +  b 2 =  In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry

Pāṇini (c. Pāṇini ( IAST: Pāṇini Dēvanāgarī: sa पाणिनि a Patronymic meaning "descendant of {{IAST|Paṇi}} " was an ancient 5th century BC) formulated the grammar rules for Sanskrit. The Sanskrit grammar has a complex verbal system rich nominal Declension, and extensive use of Compound nouns It was studied and codified by Sanskrit (sa संस्कृता वाक् saṃskṛtā vāk, for short sa संस्कृतम् saṃskṛtam) is a historical His notation was similar to modern mathematical notation, and used metarules, transformations, and recursions with such sophistication that his grammar had the computing power equivalent to a Turing machine. Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition Computing is usually defined like the activity of using and developing Computer technology Computer hardware and software. Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm Pingala (roughly 3rd-1st centuries BC) in his treatise of prosody uses a device corresponding to a binary numeral system. Pingala ( पिङ्गल piṅgalá) was an ancient Indian writer famous for his work the Chandas Shastra ( chandaḥ-śāstra The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. His discussion of the combinatorics of meters, corresponds to the binomial theorem. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects Meter or metre is a concept related to an underlying division of time characteristic of western music In Mathematics, the binomial theorem is an important Formula giving the expansion of powers of Sums Its simplest version says Pingala's work also contains the basic ideas of Fibonacci numbers (called mātrāmeru). In Mathematics, the Fibonacci numbers are a Sequence of numbers named after Leonardo of Pisa, known as Fibonacci The Brāhmī script was developed at least from the Maurya dynasty in the 4th century BC, with recent archeological evidence appearing to push back that date to around 600 BC. The Maurya Empire ( 322 – 185 BCE) ruled by the Mauryan dynasty was a geographically extensive and powerful political and military The Brahmi numerals date to the 3rd century BC. The Brahmi numerals are an indigenous Indian numeral system attested from the 3rd century BCE (somewhat later in the case of most of the tens

Between 400 BC and AD 200, Jaina mathematicians began studying mathematics for the sole purpose of mathematics. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. They were the first to develop transfinite numbers, set theory, logarithms, fundamental laws of indices, cubic equations, quartic equations, sequences and progressions, permutations and combinations, squaring and extracting square roots, and finite and infinite powers. Transfinite numbers are Cardinal numbers or Ordinal numbers that are larger than all finite numbers yet not necessarily absolutely infinite. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce The word index is used in variety of senses in Mathematics. In perhaps the most frequent sense an index is a Superscript This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. In Mathematics, a quartic equation is one which can be expressed as a Quartic function equalling zero In Mathematics, a sequence is an ordered list of objects (or events Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness The Bakhshali Manuscript written between 200 BC and AD 200 included solutions of linear equations with up to five unknowns, the solution of the quadratic equation, arithmetic and geometric progressions, compound series, quadratic indeterminate equations, simultaneous equations, and the use of zero and negative numbers. The Bakhshali Manuscript is a mathematical manuscript written on birch bark which was found near the village of Bakhshali in 1881 in what was In Mathematics simultaneous equations are a set of Equations containing multiple variables A negative number is a Number that is less than zero, such as −2 Accurate computations for irrational numbers could be found, which includes computing square roots of numbers as large as a million to at least 11 decimal places.

## Greek and Hellenistic mathematics (c. 550 BC—AD 300)

Main article: Greek mathematics
Pythagoras of Samos

Greek mathematics refers to mathematics written in Greek between about 600 BCE and 450 CE. Greek mathematics, as that term is used in this article is the Mathematics written in Greek, developed from the 6th century BC to the 5th century Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly [12] Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. Alexander the Great ( or, Mégas Aléxandros; July 20 356 BC June 10 or June 11 323 BC also known as Alexander III of Macedon (el Ἀλέξανδρος Γ'

Thales of Miletus

Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of pre-Greek mathematics show the use of inductive reasoning, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to derive conclusions from definitions and axioms. [13]

Greek mathematics is thought to have begun with Thales (c. Thales of Miletus According to Bertrand Russell, "Philosophy begins with Thales 624—c. 546 BC) and Pythagoras (c. "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. 582—c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by the ideas of Egypt, Mesopotamia and perhaps India. Egyptian mathematics refers to the style and methods of Mathematics performed in Ancient Egypt. Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (ancient Iraq) from the days of the early Sumerians to the fall of Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. According to legend, Pythagoras travelled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.

Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Pythagoras is credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry [14] In his commentary on Euclid, Proclus states that Pythagoras expressed the theorem that bears his name and constructed Pythagorean triples algebraically rather than geometrically. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Proclus Lycaeus ( February 8, c 411 &ndash April 17, 485) called "The Successor" or "Diadochos" ( Greek Próklos A Pythagorean triple consists of three positive Integers a, b, and c, such that a 2 +  b 2 =  The Academy of Plato had the motto "let none unversed in geometry enter here". For the Raphael painting see The School of Athens The Academy (Ἀκαδήμεια was founded by Plato in ca

The Pythagoreans discovered the existence of irrational numbers. Pythagoreanism is a term used for the Esoteric and metaphysical beliefs held by Pythagoras and his followers the Pythagoreans who were much influenced Eudoxus (408 —c. 355 BC) developed the method of exhaustion, a precursor of modern integration. The method of exhaustion is a method of finding the Area of a Shape by inscribing inside it a sequence of Polygons whose areas converge to the The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Aristotle (384—c. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. 322 BC) first wrote down the laws of logic. Logic is the study of the principles of valid demonstration and Inference. Euclid (c. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry 300 BC) is the earliest example of the format still used in mathematics today, definition, axiom, theorem, proof. He also studied conics. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface His book, Elements, was known to all educated people in the West until the middle of the 20th century. Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek [15] In addition to the familiar theorems of geometry, such as the Pythagorean theorem, Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry The Sieve of Eratosthenes (ca. In Mathematics, the Sieve of Eratosthenes is a simple ancient Algorithm for finding all Prime numbers up to a specified integer 230 BC) was used to discover prime numbers.

Some say the greatest of Greek mathematicians, if not of all time, was Archimedes (287—212 BC) of Syracuse. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer Syracuse (Siracusa Sicilian: Sarausa, Classical Greek: / transliterated Syrakousai) is a historic City in He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of Pi. The method of exhaustion is a method of finding the Area of a Shape by inscribing inside it a sequence of Polygons whose areas converge to the Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems [16] He also defined the spiral bearing his name, formulas for the volumes of surfaces of revolution and an ingenious system for expressing very large numbers. The Archimedean spiral (also known as the arithmetic spiral) is a Spiral named after the 3rd century BC Greek Mathematician The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically A surface of revolution is a Surface created by rotating a Curve lying on some plane (the Generatrix) around a Straight line (the Axis

## Classical Chinese mathematics (c. 500 BC—AD 1300)

The Nine Chapters on the Mathematical Art.
Main article: Chinese mathematics

In China(212 BC), the Emperor Qin Shi Huang (Shi Huang-ti) commanded that all books outside of Qin state to be burned. Mathematics in China emerged independently by the 11th century BC China ( Wade-Giles ( Mandarin) Chung¹kuo² is a cultural region, an ancient Civilization, and depending on perspective a National Qin Shi Huang ( (259 BC – September 10 210 BC personal name Yíng Zhèng, was king of the Chinese State of Qin from 247 BCE to 221 BCE (during the While this order was not universally obeyed, as a consequence little is known with certainty about ancient Chinese mathematics.

From the Western Zhou Dynasty (from 1046 BC), the oldest mathematical work to survive the book burning is the I Ching, which uses the 8 binary 3-tuples (trigrams) and 64 binary 6-tuples (hexagrams) for philosophical, mathematical, and/or mystical purposes. The Zhou Dynasty ( POJ: Chiu Tiau 1122 BC to 256 BC was preceded by the Shang Dynasty and followed by the Qin Dynasty in China. Book burning (a category of biblioclasm or book destruction is the practice of destroying often ceremoniously, one or more copies of a book or other written material The I Ching ( Wade-Giles) or “Yì Jīng” ( Pinyin) also called “Classic of Changes” or “Book of Changes” is one of the oldest of the In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple The binary tuples are composed of broken and solid lines, called yin 'female' and yang 'male' respectively (see King Wen sequence). The King Wen sequence (文王卦序 of the Yi Jing (易經 is a series of sixty-four binary figures ( Hexagrams, each composed of 6 lines either unbroken

The oldest existent work on geometry in China comes from the philosophical Mohist canon of c. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Mohism or Moism ( was a Chinese philosophy developed by the followers of Mozi (also referred to as Mo Di 470 &ndashc 330 BC, compiled by the followers of Mozi (470 BC-390 BC). Mozi ( Lat as Micius, ca 470 BCE&ndashca 391 BCE was a Philosopher who lived in China during the Hundred Schools of Thought The Mo Jing described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well.

After the book burning, the Han dynasty (202 BC–220 AD) produced works of mathematics which presumably expand on works that are now lost. The Han Dynasty ( 206 BC–220 AD followed the Qin Dynasty and preceded the Three Kingdoms in China. The most important of these is The Nine Chapters on the Mathematical Art, the full title of which appeared by 179 AD, but existed in part under other titles beforehand. The Nine Chapters on the Mathematical Art ( is a Chinese Mathematics book composed by several generations of scholars from the 10th&ndash2nd century BC and It consists of 246 word problems, involving agriculture, business, employment of geometry to figure height spans and dimension ratios for Chinese pagoda towers, engineering, surveying, and includes material on right triangles and π. The Chinese Pagoda is a Landmark in Birmingham, England. It is a stone carving of a Chinese pagoda, carved in Fujian, China Surveying is the technique and science of accurately determining the terrestrial or three-dimensional space Position of points and the distances and angles between Two types of special right triangles appear commonly in geometry the "angle based" and the "side based" (or Pythagorean Triangles The former are characterised IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems It also made use of Cavalieri's principle on volume more than a thousand years before Cavalieri would propose it in the West. Bonaventura Francesco Cavalieri (in Latin, Cavalerius) ( 1598 - November 30, 1647) was an Italian mathematician It created mathematical proof for Pythagoras' Pythagorean theorem, and mathematical formula for Gaussian elimination. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry In Linear algebra, Gaussian elimination is an efficient Algorithm for solving systems of linear equations, to find the rank of a matrix The work was commented on by Liu Hui in the 3rd century AD. Liu Hui ( fl 3rd century) was a Chinese Mathematician who lived in the Wei Kingdom.

In addition, the mathematical works of the Han astronomer and inventor Zhang Heng (78-139 AD) had a formulation for pi as well, which differed from Liu Hui's calculation. Zhang Heng ( (CE 78–139 was an astronomer, mathematician, inventor, geographer, cartographer, artist, poet IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems Zhang Heng used his formula of pi to find spherical volume. There was also the written work of the mathematician and music theorist Jing Fang (78–37 BC); by using the Pythagorean comma, Jing observed that 53 just fifths approximates to 31 octaves. Music theory is the field of study that deals with the Mechanics of music and how Music works Jing Fang ( 78&ndash37 BC born Li Fang (李房 Courtesy name Junming (君明 was a Chinese music theorist, Mathematician In Music, when ascending from an initial (low pitch by a cycle of justly tuned perfect fifths (ratio 32 ( leapfrogging twelve times one eventually reaches a The perfect fifth ( is the Musical interval between a note and the note seven Semitones above it on the musical scale In Music, an octave ( is the the use of which is "common in most musical systems This would later lead to the discovery of 53 equal temperament, and was not calculated precisely elsewhere until the German Nicholas Mercator did so in the 17th century. In music 53 equal temperament, called 53-TET 53- EDO, or 53-ET is the tempered scale derived by dividing the octave into fifty-three equally large steps Nicholas ( Nikolaus) Mercator (c 1620 Eutin -1687 Versailles) also known by his Germanic name Kauffmann, was a 17th-century mathematician

The Chinese also made use of the complex combinatorial diagram known as the 'magic square and magic circles which was described in ancient times and perfected by Yang Hui (1238–1398 AD). In Recreational mathematics, a magic square of order n is an arrangement of n ² numbers usually distinct Integers in a square, such Magic circles were invented by the Song Dynasty (960&ndash1279 Chinese mathematician Yang Hui (c Yang Hui ( ca 1238–1298 Courtesy name Qianguang (谦光 was a Chinese Mathematician from Qiantang (modern Hangzhou

Zhang Heng (78-139)

Zu Chongzhi (5th century) of the Southern and Northern Dynasties computed the value of π to seven decimal places, which remained the most accurate value of π for almost 1000 years. Zhang Heng ( (CE 78–139 was an astronomer, mathematician, inventor, geographer, cartographer, artist, poet Zu Chongzhi ( 429–500 Courtesy name Wenyuan (文遠 was a prominent Chinese mathematician and astronomer during the Liu This article is about the Southern and Northern Dynasties in China.

In the thousand years following the Han dynasty, starting in the Tang dynasty and ending in the Song dynasty, Chinese mathematics thrived at a time when European mathematics did not exist. The Tang Dynasty ( Middle Chinese: dhɑng (June 18 618&ndashJune 4 907 was an imperial dynasty of China preceded by the Sui Dynasty and followed by The Song Dynasty ( Wade-Giles: Sung Ch'ao was a ruling dynasty in China between 960&ndash1279 CE it succeeded the Five Dynasties and Ten Kingdoms Developments first made in China, and only much later known in the West, include negative numbers, the binomial theorem, matrix methods for solving systems of linear equations and the Chinese remainder theorem. The term Western world, the West or the Occident ( Latin: occidens -sunset -west as distinct from the Orient) can have multiple meanings A negative number is a Number that is less than zero, such as −2 In Mathematics, the binomial theorem is an important Formula giving the expansion of powers of Sums Its simplest version says In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable The Chinese remainder theorem is a result about congruences in Number theory and its generalizations in Abstract algebra. The Chinese also developed Pascal's triangle and the rule of three long before it was known in Europe. \begin{matrix}&&&&&1\\&&&&1&&1\\&&&1&&2&&1\\&&1&&3&&3&&1\\&1&&4&&6&&4&&1\end{matrix Besides Zu Chongzhi, some of the most important figures of Chinese mathematics during this period include Yi Xing, Shen Kuo, Qin Jiushao, Zhu Shijie, and others. Yi Xing ( 683–727 born Zhang Sui (张遂 was a Chinese Astronomer, Mathematician, mechanical engineer, and Buddhist monk Shen Kuo or Shen Kua ( (1031&ndash1095 style name Cunzhong and pseudonym Mengqi Weng, was a Polymathic Chinese Qin Jiushao ( ca 1202–1261 Courtesy name Daogu (道古 was a Chinese Mathematician born in Ziyang Sichuan, his Zhu Shijie ( fl 13th century) Courtesy name Hanqing (汉卿 Pseudonym Songting (松庭 was one of the greatest Chinese The scientist Shen Kuo used problems involving calculus, trigonometry, metrology, permutations, and once computed the possible amount of terrain space that could be used with specific battle formations, as well as the longest possible military campaign given the amount of food carriers could bring for themselves and soldiers. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Metrology (from Ancient Greek metron (measure and logos (study of is the Science of Measurement. In several fields of Mathematics the term permutation is used with different but closely related meanings

Even after European mathematics began to flourish during the Renaissance, European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline, until the Jesuit missionaries such as Matteo Ricci carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries. The Renaissance (from French Renaissance, meaning "rebirth" Italian: Rinascimento, from re- "again" and nascere The Society of Jesus ( Latin: Societas Iesu, SJ and SI or SJ, SI) is a Catholic religious order Matteo Ricci SJ ( October 6 1552 &ndash May 11 1610;; Courtesy name: 西泰 Xītài was an Italian Jesuit priest

## Classical Indian mathematics (c. 400—1600)

Main article: Indian mathematics
See also: History of the Hindu-Arabic numeral system

The Surya Siddhanta (c. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. The Hindu-Arabic numeral system is a Place-value numeral system the value of a digit depends on the place where it appears the '2' in 205 is ten times greater than Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics The Surya Siddhanta is a treatise of Indian astronomy. Later Indian mathematicians and astronomers such as Aryabhata and Varahamihira 400) introduced the trigonometric functions of sine, cosine, and inverse sine, and laid down rules to determine the true motions of the luminaries, which conforms to their actual positions in the sky. The cosmological time cycles explained in the text, which was copied from an earlier work, corresponds to an average sidereal year of 365. The sidereal year is the time taken for the Sun to return to the same position with respect to the Stars of the Celestial sphere. 2563627 days, which is only 1. 4 seconds longer than the modern value of 365. 25636305 days. This work was translated to Arabic and Latin during the Middle Ages. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome.

Aryabhata in 499 introduced the versine function, produced the first trigonometric tables of sine, developed techniques and algorithms of algebra, infinitesimals, differential equations, and obtained whole number solutions to linear equations by a method equivalent to the modern method, along with accurate astronomical calculations based on a heliocentric system of gravitation. Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics The versed sine, also called the versine and in Latin, the sinus versus ("flipped sine" or the sagitta ("arrow" is a Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study In Astronomy, heliocentrism is the theory that the Sun is at the center of the Solar System. Gravitation is a natural Phenomenon by which objects with Mass attract one another An Arabic translation of his Aryabhatiya was available from the 8th century, followed by a Latin translation in the 13th century. Arabic (ar الْعَرَبيّة (informally ar عَرَبيْ) in terms of the number of speakers is the largest living member of the Semitic language He also computed the value of π to the fourth decimal place as 3. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems 1416. Madhava later in the 14th century computed the value of π to the eleventh decimal place as 3. Mādhava of Sangamagrama (born as Irinjaatappilly Madhavan Namboodiri) (c 14159265359.

In the 7th century, Brahmagupta identified the Brahmagupta theorem, Brahmagupta's identity and Brahmagupta's formula, and for the first time, in Brahma-sphuta-siddhanta, he lucidly explained the use of zero as both a placeholder and decimal digit and explained the Hindu-Arabic numeral system. Brahmagupta ( (598–668 was an Indian mathematician and astronomer. Brahmagupta's theorem is a result in Geometry. It states that if a Cyclic quadrilateral has Perpendicular diagonals then the perpendicular to a side from In Geometry, Brahmagupta 's formula finds the Area of any Quadrilateral given the lengths of the sides and some of their angles The main work of Brahmagupta, Brahmasphuta-siddhanta (The Opening of the Universe, written in the year 628, contains some remarkably advanced ideas including In Mathematics and Computer science, a digit is a symbol (a number symbol e The Hindu-Arabic numeral system is a Positional Decimal Numeral system first documented in the ninth century It was from a translation of this Indian text on mathematics (around 770) that Islamic mathematicians were introduced to this numeral system, which they adapted as Arabic numerals. For other meanings including people named 'Islam' see Islam (disambiguation. The arabic numerals (often capitalized are the ten Digits (0 1 2 3 4 5 6 7 8 9 which—along with the system Islamic scholars carried knowledge of this number system to Europe by the 12th century, and it has now displaced all older number systems throughout the world. In the 10th century, Halayudha's commentary on Pingala's work contains a study of the Fibonacci sequence and Pascal's triangle, and describes the formation of a matrix. Halayudha (हलायुद्ध was a 10th century Indian Mathematician who wrote a commentary on Pingala 's Chandah-shastra Pingala ( पिङ्गल piṅgalá) was an ancient Indian writer famous for his work the Chandas Shastra ( chandaḥ-śāstra In Mathematics, the Fibonacci numbers are a Sequence of numbers named after Leonardo of Pisa, known as Fibonacci \begin{matrix}&&&&&1\\&&&&1&&1\\&&&1&&2&&1\\&&1&&3&&3&&1\\&1&&4&&6&&4&&1\end{matrix In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally

In the 12th century, Bhaskara first conceived differential calculus, along with the concepts of the derivative, differential coefficient and differentiation. Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change He also stated Rolle's theorem (a special case of the mean value theorem), studied Pell's equation, and investigated the derivative of the sine function. In Calculus, a branch of Mathematics, Rolle's theorem essentially states that a differentiable function, which attains equal values at two points must In Calculus, the mean value theorem states roughly that given a section of a smooth curve there is at least one point on that section at which the Derivative Pell's equation is any Diophantine equation of the form x^2-ny^2=1\ where n is a nonsquare integer and x From the 14th century, Madhava and other Kerala School mathematicians, further developed his ideas. Mādhava of Sangamagrama (born as Irinjaatappilly Madhavan Namboodiri) (c They developed the concepts of mathematical analysis and floating point numbers, and concepts fundamental to the overall development of calculus, including the mean value theorem, term by term integration, the relationship of an area under a curve and its antiderivative or integral, tests of convergence, iterative methods for solutions of non-linear equations, and a number of infinite series, power series, Taylor series and trigonometric series. Analysis has its beginnings in the rigorous formulation of Calculus. In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Mathematics, the integral test for convergence is a method used to test infinite series of Non-negative terms for Convergence. In Computational mathematics, an iterative method attempts to solve a problem (for example an equation or system of equations by finding successive Approximations This article describes the use of the term nonlinearity in mathematics In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In the 16th century, Jyeshtadeva consolidated many of the Kerala School's developments and theorems in the Yuktibhasa, the world's first differential calculus text, which also introduced concepts of integral calculus. Jyestadeva (ജ്യേഷ്ഠദേവ(ന് (1500 &ndash 1575 was an astronomer of the Kerala school founded by Madhava of Sangamagrama and The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Mathematical progress in India became stagnant from the late 16th century onwards due to subsequent political turmoil.

## Islamic mathematics (c. 800—1500)

Main article: Islamic mathematics
See also: History of the Hindu-Arabic numeral system

The Islamic Arab Empire established across the Middle East, Central Asia, North Africa, Iberia, and in parts of India in the 8th century made significant contributions towards mathematics. The Hindu-Arabic numeral system is a Place-value numeral system the value of a digit depends on the place where it appears the '2' in 205 is ten times greater than For other meanings including people named 'Islam' see Islam (disambiguation. The Middle East is a Subcontinent with no clear boundaries often used as a synonym to Near East, in opposition to Far East. Central Asia is a region of Asia from the Caspian Sea in the west to central China in the east and from southern Russia in the north to northern Pakistan in the south North Africa or Northern Africa is the Northernmost Region of the African Continent, separated by the Sahara from Sub-Saharan The Iberian Peninsula, or Iberia, is located in the extreme southwest of Europe, and includes modern day Spain, Portugal, Andorra This article is about the history of South Asia prior to the Partition of British India in 1947 Although most Islamic texts on mathematics were written in Arabic, they were not all written by Arabs, since much like the status of Greek in the Hellenistic world, Arabic was used as the written language of non-Arab scholars throughout the Islamic world at the time. Arabic (ar الْعَرَبيّة (informally ar عَرَبيْ) in terms of the number of speakers is the largest living member of the Semitic language The araB gene Promoter is a bacterial promoter activated by e L-arabinose binding Some of the most important Islamic mathematicians were Persian. layout and formatting it should ensure no clashes with the top of the infobox

Muḥammad ibn Mūsā al-Ḵwārizmī, a 9th century Persian mathematician and astronomer to the Caliph of Baghdad, wrote several important books on the Hindu-Arabic numerals and on methods for solving equations. A caliphate (from the Arabic خلافة or khilāfa) is the political leadership of the Muslim community in classical and medieval Islamic history His book On the Calculation with Hindu Numerals, written about 825, along with the work of the Arab mathematician Al-Kindi, were instrumental in spreading Indian mathematics and Indian numerals to the West. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. The Hindu-Arabic numeral system is a Positional Decimal Numeral system first documented in the ninth century The word algorithm is derived from the Latinization of his name, Algoritmi, and the word algebra from the title of one of his works, Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing). In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala ( Arabic for "The Compendious Book on Calculation by Completion and Balancing" Al-Khwarizmi is often called the "father of algebra", for his preservation of ancient algebraic methods and for his original contributions to the field. [17] Further developments in algebra were made by Abu Bakr al-Karaji (953—1029) in his treatise al-Fakhri, where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. (or) (c 953 in Karaj or Karkh &ndash c 1029 was a 10th century Persian Muslim mathematician and engineer. In the 10th century, Abul Wafa translated the works of Diophantus into Arabic and developed the tangent function. Diophantus of Alexandria ( Greek: b between 200 and 214 d between 284 and 298 AD sometimes called "the father of Algebra " a title some claim should

The first known proof by mathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that (or) (c 953 in Karaj or Karkh &ndash c 1029 was a 10th century Persian Muslim mathematician and engineer. In Mathematics, the binomial theorem is an important Formula giving the expansion of powers of Sums Its simplest version says \begin{matrix}&&&&&1\\&&&&1&&1\\&&&1&&2&&1\\&&1&&3&&3&&1\\&1&&4&&6&&4&&1\end{matrix The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Arithmetic and Algebra, the cube of a number n is its third power &mdash the result of multiplying it by itself three times [18] The historian of mathematics, F. See also History An historian is an individual who studies and writes about History, and is regarded as an Authority on it Woepcke,[19] praised Al-Karaji for being "the first who introduced the theory of algebraic calculus. The word theory has many distinct meanings in different fields of Knowledge, depending on their methodologies and the context of discussion. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives " Ibn al-Haytham was the first mathematician to derive the formula for the sum of the fourth powers, and using the method of induction, he developed a method for determining the general formula for the sum of any integral powers, which was fundamental to the development of integral calculus. TemplateInfobox Muslim scholars --> ( Arabic: ابو علی، حسن بن حسن بن هيثم Latinized In Arithmetic and Algebra, the fourth power of a number n is the result of multiplying n by itself four times The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives [20]

Other achievements of Muslim mathematicians during this period include the development of algebra and algorithms (see Muhammad ibn Mūsā al-Khwārizmī), the development of spherical trigonometry,[21] the addition of the decimal point notation to the Arabic numerals, the discovery of all the modern trigonometric functions besides sine, al-Kindi's introduction of cryptanalysis and frequency analysis, al-Karaji's introduction of algebraic calculus and proof by mathematical induction, the development of analytic geometry and the earliest general formula for infinitesimal and integral calculus by Ibn al-Haytham, the beginning of algebraic geometry by Omar Khayyam, the first refutations of Euclidean geometry and the parallel postulate by Nasīr al-Dīn al-Tūsī, the first attempt at a non-Euclidean geometry by Sadr al-Din, and numerous other advances in algebra, arithmetic, calculus, cryptography, geometry, number theory and trigonometry. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Spherical trigonometry is a part of Spherical geometry that deals with Polygons (especially Triangles on the Sphere and explains how to find relations In a positional Numeral system, the decimal separator is a Symbol used to mark the boundary between the integral and the fractional The arabic numerals (often capitalized are the ten Digits (0 1 2 3 4 5 6 7 8 9 which—along with the system ( أبو يوسف يعقوب إبن إسحاق الكندي) (c Cryptanalysis (from the Greek kryptós, "hidden" and analýein, "to loosen" or "to untie" is the study of methods for In Cryptanalysis, frequency analysis is the study of the frequency of letters or groups of letters in a Ciphertext. (or) (c 953 in Karaj or Karkh &ndash c 1029 was a 10th century Persian Muslim mathematician and engineer. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space TemplateInfobox Muslim scholars --> ( Arabic: ابو علی، حسن بن حسن بن هيثم Latinized Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with For the Thoroughbred racehorse see Omar Khayyam (horse Ghiyās od-Dīn Abol-Fath Omār ibn Ebrāhīm Khayyām Neyshābūri (غیاث الدین Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.

During the time of the Ottoman Empire (from the 15th century) the development of Islamic mathematics became stagnant. The Ottoman Empire (1299–1923 ( Old Ottoman Turkish: دولتْ علیّه عثمانیّه Devlet-i Âliye-yi Osmâniyye, Late Ottoman and Modern Turkish This parallels the stagnation of mathematics when the Romans conquered the Hellenistic world.

John J. O'Connor and Edmund F. Robertson wrote in the MacTutor History of Mathematics archive:

"Recent research paints a new picture of the debt that we owe to Islamic mathematics. The MacTutor History of Mathematics archive is an award-winning website maintained by John J Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians four centuries earlier. In many respects, the mathematics studied today is far closer in style to that of Islamic mathematics than to that of Hellenistic mathematics. "

## Medieval European mathematics (c. 500—1400)

Medieval European interest in mathematics was driven by concerns quite different from those of modern mathematicians. One driving element was the belief that mathematics provided the key to understanding the created order of nature, frequently justified by Plato's Timaeus and the biblical passage that God had "ordered all things in measure, and number, and weight" (Wisdom 11:21). Biography Early life Birth and family Plato was born in Athens Greece

### Early Middle Ages (c. 500—1100)

Boethius provided a place for mathematics in the curriculum when he coined the term "quadrivium" to describe the study of arithmetic, geometry, astronomy, and music. Anicius Manlius Severinus Boethius (480&ndash524 or 525 was a Christian philosopher of the 6th century The quadrivium comprised the four subjects or arts taught in Medieval universities after the trivium. He wrote De institutione arithmetica, a free translation from the Greek of Nicomachus's Introduction to Arithmetic; De institutione musica, also derived from Greek sources; and a series of excerpts from Euclid's Geometry. Nicomachus (Νικόμαχος (c 60 &ndash c 120 was an important mathematician in the ancient world and is best known for his works Introduction to Arithmetic Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works. [22][23]

### Rebirth of mathematics in Europe (1100—1400)

In the 12th century, European scholars travelled to Spain and Sicily seeking scientific Arabic texts, including al-Khwarizmi's al-Jabr wa-al-Muqabilah, translated into Latin by Robert of Chester, and the complete text of Euclid's Elements, translated in various versions by Adelard of Bath, Herman of Carinthia, and Gerard of Cremona. The Renaissance of the 12th century saw a major search by European scholars for new learning which led them to the Arabic fringes of Europe especially to Islamic Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala ( Arabic for "The Compendious Book on Calculation by Completion and Balancing" Robert of Chester ( Latin: Robertus Castrensis) was an English Arabist who flourished around 1150. Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek Adelard of Bath ( Latin: Adelardus Bathensis) (c 1080 &ndash c Herman of Carinthia or Herman Dalmatin (also known in Latin as Sclavus Dalmata Secundus was a Philosopher, Astronomer, Astrologer, Mathematician Gerard of Cremona ( Italian: Gerardo da Cremona; Latin: Gerardus Cremonensis; c [24][25]

These new sources sparked a renewal of mathematics. Fibonacci, writing in the Liber Abaci, in 1202 and updated in 1254, produced the first significant mathematics in Europe since the time of Eratosthenes, a gap of more than a thousand years. Leonardo of Pisa (c 1170 – c 1250 also known as Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or most commonly simply Fibonacci Liber Abaci (1202 also spelled as Liber Abbaci) is an historic book on Arithmetic by Leonardo of Pisa known later by his nickname Fibonacci Eratosthenes of Cyrene ( Greek; 276 BC - 194 BC was a Greek Mathematician, Poet, athlete, Geographer and The work introduced Hindu-Arabic numerals to Europe, and discussed many other mathematical problems. The arabic numerals (often capitalized are the ten Digits (0 1 2 3 4 5 6 7 8 9 which—along with the system The fourteenth century saw the development of new mathematical concepts to investigate a wide range of problems. [26] One important area that contributed to the development of mathematics concerned the analysis of local motion.

Thomas Bradwardine proposed that speed (V) increases in arithmetic proportion as the ratio of force (F) to resistance (R) increases in geometric proportion. Thomas Bradwardine (c 1290 &ndash 26 August 1349 often called "the Profound Doctor" was an English scholar and courtier and very briefly Archbishop of Canterbury Bradwardine expressed this by a series of specific examples, but although the logarithm had not yet been conceived, we can express his conclusion anachronistically by writing: V = log (F/R). [27] Bradwardine's analysis is an example of transferring a mathematical technique used by al-Kindi and Arnald of Villanova to quantify the nature of compound medicines to a different physical problem. ( أبو يوسف يعقوب إبن إسحاق الكندي) (c Arnaldus de Villa Nova (also called Arnaldus de Villanueva Arnaldus Villanovanus Arnaud de Ville-Neuve or Arnau de Vilanova ( 1235 Valencia –1311 Genoa) [28]

One of the 14th-century Oxford Calculators, William Heytesbury, lacking differential calculus and the concept of limits, proposed to measure instantaneous speed "by the path that would be described by [a body] if . The Oxford Calculators were a group of 14th-century thinkers almost all associated with Merton College, Oxford, who took a strikingly logico-mathematical William Heytesbury (a 1313 – 1372 / 1373) philosopher and logician is best known as one of the Oxford Calculators of Merton College Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" . . it were moved uniformly at the same degree of speed with which it is moved in that given instant". [29]

Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (which we would solve by a simple integration), stating that "a moving body uniformly acquiring or losing that increment [of speed] will traverse in some given time a [distance] completely equal to that which it would traverse if it were moving continuously through the same time with the mean degree [of speed]". The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space [30]

Nicole Oresme at the University of Paris and the Italian Giovanni di Casali independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance traveled. Nicole Oresme, also known as Nicolas Oresme, Nicholas Oresme, or Nicolas d'Oresme (c The historic University of Paris (Université de Paris first appeared in the second half of the 13th century Giovanni (or Johannes) di Casali (or Casale) was a Friar in the Franciscan Order a natural philosopher and a theologian [31] In a later mathematical commentary on Euclid's Geometry, Oresme made a more detailed general analysis in which he demonstrated that a body will acquire in each successive increment of time an increment of any quality that increases as the odd numbers. Since Euclid had demonstrated the sum of the odd numbers are the square numbers, the total quality acquired by the body increases as the square of the time. [32]

## Early modern European mathematics (c. 1400—1600)

In Europe at the dawn of the Renaissance, mathematics was still limited by the cumbersome notation using Roman numerals and expressing relationships using words, rather than symbols: there was no plus sign, no equal sign, and no use of x as an unknown. The Renaissance (from French Renaissance, meaning "rebirth" Italian: Rinascimento, from re- "again" and nascere Roman numerals are a Numeral system originating in ancient Rome, adapted from Etruscan numerals.

In 16th century European mathematicians began to make advances without precedent anywhere in the world, so far as is known today. The first of these was the general solution of cubic equations, generally credited to Scipione del Ferro circa 1510, but first published by Johannes Petreius in Nuremberg in Gerolamo Cardano's Ars magna, which also included the solution of the general quartic equation from Cardano's student Lodovico Ferrari . This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. Scipione del Ferro ( February 6 1465 &ndash November 5, 1526) was an Italian mathematician who first discovered a Johann(es Petreius aka Hans Peterlein (c 1497 near Bad Kissingen - March 18 1550, Nuremberg) was a German printer In Mathematics, a quartic equation is one which can be expressed as a Quartic function equalling zero Lodovico Ferrari ( February 2, 1522 – October 5, 1565) was an Italian Mathematician.

From this point on, mathematical developments came swiftly, contributing to and benefiting from contemporary advances in the physical sciences. Physical science is an encompassing term for the branches of Natural science and Science that study non-living systems in contrast to the biological sciences This progress was greatly aided by advances in printing. Printing is a process for reproducing text and image typically with ink on Paper using a printing press The earliest mathematical books printed were Peurbach's Theoricae nova planetarum 1472 followed by a book on commercial arithmetic, the 1478 Treviso Arithmetic and then the first real mathematics book Euclid's Elements printed and published by Ratdolt 1482. Georg von Peuerbach (also Purbach, Peurbach, Purbachius, his real Surname is unknown (born May 30, 1423 in Peuerbach The Treviso Arithmetic, or Arte dell'Abbaco, is an anonymous textbook in commercial arithmetic written in vernacular Venetian and published in Treviso, Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry

Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. "Pitiscus" redirects here For the crater see Pitiscus (crater. Regiomontanus' table of sines and cosines was published in 1533. [33]

By century's end, thanks to Regiomontanus (1436—1476) and François Vieta (1540—1603), among others, mathematics was written using Hindu-Arabic numerals and in a form not too different from the notation used today. Johannes Müller von Königsberg ( June 6, 1436 &ndash July 6, 1476) known by his Latin Pseudonym Regiomontanus François Viète (or Vieta) seigneur de la Bigotière ( 1540 - February 13, 1603) generally known as Franciscus Vieta,

## 17th century

The 17th century saw an unprecedented explosion of mathematical and scientific ideas across Europe. Galileo, an Italian, observed the moons of Jupiter in orbit about that planet, using a telescope based on a toy imported from Holland. Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher Tycho Brahe, a Dane, had gathered an enormous quantity of mathematical data describing the positions of the planets in the sky. Tycho Brahe, born Tyge Ottesen Brahe ( December 14 1546 &ndash October 24 1601) was a Danish nobleman His student, Johannes Kepler, a German, began to work with this data. Johannes Kepler (ˈkɛplɚ ( December 27 1571 &ndash November 15 1630) was a German Mathematician, Astronomer In part because he wanted to help Kepler in his calculations, John Napier, in Scotland, was the first to investigate natural logarithms. For other people with the same name see John Napier (disambiguation. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational Kepler succeeded in formulating mathematical laws of planetary motion. The analytic geometry developed by René Descartes (1596-1650), a French mathematician and philosopher, allowed those orbits to be plotted on a graph, in Cartesian coordinates. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane Building on earlier work by many mathematicians, Isaac Newton, an Englishman, discovered the laws of physics explaining Kepler's Laws, and brought together the concepts now known as calculus. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements In Astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of Planets in the Solar System. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Independently, Gottfried Wilhelm Leibniz, in Germany, developed calculus and much of the calculus notation still in use today. Science and mathematics had become an international endeavor, which would soon spread over the entire world. [34]

In addition to the application of mathematics to the studies of the heavens, applied mathematics began to expand into new areas, with the correspondence of Pierre de Fermat and Blaise Pascal. Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the Blaise Pascal (blɛz paskal (June 19 1623 &ndash August 19 1662 was a French Mathematician, Physicist, and religious Philosopher Pascal and Fermat set the groundwork for the investigations of probability theory and the corresponding rules of combinatorics in their discussions over a game of gambling. Probability theory is the branch of Mathematics concerned with analysis of random phenomena Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects Pascal, with his wager, attempted to use the newly developing probability theory to argue for a life devoted to religion, on the grounds that even if the probability of success was small, the rewards were infinite. Pascal's Wager (or Pascal's Gambit) is a suggestion posed by the French Philosopher Blaise Pascal that even though the Existence of God In some sense, this foreshadowed the development of utility theory in the 18th-19th century. In Economics, utility is a measure of the relative satisfaction from or desirability of Consumption of various Goods and services.

## 18th century

Leonhard Euler by Emanuel Handmann. Emanuel Handmann (born 1718 in Basel, died 1781 in Bern) was a Swiss painter.

As we have seen, knowledge of the natural numbers, 1, 2, 3,. . . , as preserved in monolithic structures, is older than any surviving written text. The earliest civilizations -- in Mesopotamia, Egypt, India and China -- knew arithmetic.

One way to view the development of the various number systems of modern mathematics is to see new numbers studied and investigated to answer questions about arithmetic performed on older numbers. In prehistoric times, fractions answered the question: what number, when multiplied by 3, gives the answer 1? In India and China, and much later in Germany, negative numbers were developed to answer the question: what do you get when you subtract a larger number from a smaller?

Another natural question is: what kind of a number is the square root of two? The Greeks knew that it was not a fraction, and this question may have played a role in the development of continued fractions. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} But a better answer came with the invention of decimals, developed by John Napier (1550 - 1617) and perfected later by Simon Stevin. For other people with the same name see John Napier (disambiguation. Simon Stevin (1548/49 &ndash 1620 was a Flemish Mathematician and Engineer. Using decimals, and an idea that anticipated the concept of the limit, Napier also studied a new constant, which Leonhard Euler (1707 - 1783) named e. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line

Euler was very influential in the standardization of other mathematical terms and notations. He named the square root of minus 1 with the symbol i. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation He also popularized the use of the Greek letter π to stand for the ratio of a circle's circumference to its diameter. He then derived one of the most remarkable identities in all of mathematics:

$e^{i \pi} +1 = 0 \,$

(see Euler's Identity. In Mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation e^{i \pi} + 1 = 0 \\! where )

## 19th century

Behavior of lines with a common perpendicular in each of the three types of geometry

Throughout the 19th century mathematics became increasingly abstract. In this century lived Carl Friedrich Gauss (1777 - 1855). Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Leaving aside his many contributions to science, in pure mathematics he did revolutionary work on functions of complex variables, in geometry, and on the convergence of series. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with He gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law. In Mathematics, the Fundamental theorem of algebra states that every non-constant single-variable Polynomial with complex coefficients has at The law of quadratic reciprocity is a theorem from Modular arithmetic, a branch of Number theory, which shows a remarkable relationship between the solvability

This century saw the development of the two forms of non-Euclidean geometry, where the parallel postulate of Euclidean geometry no longer holds. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival, the Hungarian mathematician Janos Bolyai, independently discovered hyperbolic geometry, where uniqueness of parallels no longer holds. Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский ( December 1 1792 &ndash February 24 1856 ( N János Bolyai ( December 15, 1802 – January 27, 1860) was a Hungarian Mathematician, known for his work in Non-Euclidean In In this geometry the sum of angles in a triangle add up to less than 180°. Elliptic geometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°. Elliptic geometry (sometimes known as Riemannian geometry) is a Non-Euclidean geometry, in which given a line L and a point Riemann also developed Riemannian geometry, which unifies and vastly generalizes the three types of geometry, and he defined the concept of a manifold, which generalize the ideas of curves and surfaces. Elliptic geometry is also sometimes called Riemannian geometry. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. Also in the nineteenth century William Rowan Hamilton developed noncommutative algebra. Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those

In addition to new directions in mathematics, older mathematics were given a stronger logical foundation, especially in the case of calculus, in work by Augustin-Louis Cauchy and Karl Weierstrass. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician

A new form of algebra was developed in the nineteenth century called Boolean algebra, named after the British mathematician George Boole. Boolean logic is a complete system for Logical operations It was named after George Boole, who first defined an algebraic system of George Boole (buːl ( November 2, 1815 &ndash December 8, 1864) was a British Mathematician and Philosopher. It was a system in which the only numbers were 0 and 1, a system which today has important applications in computer science. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their

Also, for the first time, the limits of mathematics were explored. Niels Henrik Abel, a Norwegian, and Évariste Galois, a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four. Niels Henrik Abel (August 5 1802 &ndash April 6 1829 was a noted Norwegian Mathematician who proved the impossibility of solving the Quintic equation Other 19th century mathematicians utilized this in their proofs that straightedge and compass alone are not sufficient to trisect an arbitrary angle, to construct the side of a cube twice the volume of a given cube, nor to construct a square equal in area to a given circle. The problem of trisecting the angle is a classic problem of Compass and straightedge constructions of ancient Greek mathematics. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks.

Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of group theory, and the associated fields of abstract algebra. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In the 20th century physicists and other scientists have seen group theory as the ideal way to study symmetry. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or

Towards the end of the 19th century, Georg Cantor invented the set theory, which has become the common language of different mathematical branches. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. The introduction of infinite set set off a debate on foundations of mathematics. In Set theory, an infinite set is a set that is not a Finite set. Foundations of mathematics is a term sometimes used for certain fields of Mathematics, such as Mathematical logic, Axiomatic set theory, Proof theory

The 19th century also saw the founding of the first mathematical societies: the London Mathematical Society in 1865, the Société Mathématique de France in 1872, the Circolo Mathematico di Palermo in 1884, the Edinburgh Mathematical Society in 1883, and the American Mathematical Society in 1888. The London Mathematical Society ( LMS) is the leading mathematical society in England. The Société Mathématique de France (SMF is the main professional society of French Mathematicians. The Edinburgh Mathematical Society is the leading mathematical society in Scotland. The American Mathematical Society (AMS is an association of professional Mathematicians dedicated to the interests of mathematical research and scholarship which

Before the 20th century, there were very few creative mathematicians in the world at any one time. For the most part, mathematicians were either born to wealth, like Napier, or supported by wealthy patrons, like Gauss. There were a few who found meager livelihoods teaching at a university, like Fourier. Niels Henrik Abel, unable to obtain a position, died in poverty of malnutrition and tuberculosis at the age of twenty-six. Niels Henrik Abel (August 5 1802 &ndash April 6 1829 was a noted Norwegian Mathematician who proved the impossibility of solving the Quintic equation

## 20th century

A map illustrating the Four Color Theorem

The profession of mathematician became much more important in the 20th century. The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country Every year, hundreds of new Ph. D. s in mathematics are awarded, and jobs are available both in teaching and industry. Mathematical development has grown at an exponential rate, with too many new developments for a survey to even touch on any but a few of the most profound.

In 1900, David Hilbert presented a list of 23 unsolved problems in mathematics at the International Congress of Mathematicians. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Hilbert's problems are a list of twenty-three problems in Mathematics put forth by German Mathematician David Hilbert at the Paris The International Congress of Mathematicians (ICM is the largest congress in the Mathematics community These problems spanned many areas of mathematics and have formed a central focus for much of 20th century mathematics. Today ten have been resolved, seven are partially resolved and two problems are still open. The remaining four are too loose to be stated as resolved or not.

In the 1910s, Srinivasa Aiyangar Ramanujan (1887-1920) developed over 3000 theorems, including properties of highly composite numbers, the partition function and its asymptotics, and mock theta functions. A highly composite number ( HCN) is a positive Integer with more Divisors than any smaller positive integer In Number theory, a partition of a positive Integer n is a way of writing n as a Sum of positive integers In pure and Applied mathematics, particularly the Analysis of algorithms, real analysis and engineering asymptotic analysis is a method of describing In Mathematics, the Ramanujan theta function generalizes the form of the Jacobi Theta functions while capturing their general properties He also made major breakthroughs and discoveries in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory. In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function In Mathematics, a modular form is a (complex Analytic function on the Upper half-plane satisfying a certain kind of Functional equation and In Mathematics, a divergent series is an Infinite series that is not convergent, meaning that the infinite Sequence of the Partial sums In Mathematics, a hypergeometric series is a Power series in which the ratios of successive Coefficients k is a Rational function

In 1931, Kurt Gödel published his two incompleteness theorems which state the limit of mathematical logic. Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most It put an end to David Hilbert's dream of a complete and consistent mathematical system.

Famous conjectures of the past yielded to new and more powerful techniques. Wolfgang Haken and Kenneth Appel used a computer to prove the four color theorem in 1976. Kenneth Appel (born 1932 is a mathematician who in 1976 with colleague Wolfgang Haken at the University of Illinois at Urbana-Champaign, solved one of the most famous The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country Andrew Wiles, working alone in his office for years, proved Fermat's last theorem in 1995. Sir Andrew John Wiles KBE FRS (born 11 April 1953 is a British Mathematician and a professor at Princeton University Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like Mathematical collaborations of unprecedented size and scope took place. The classification of finite simple groups (also called the "enormous theorem") spanned tens of thousands of pages in 500-odd journal articles written by about 100 authors, published mostly between 1955 and 1983. The classification of the finite simple groups, also called the enormous theorem is believed to classify all finite simple groups.

Entire new areas of mathematics such as mathematical logic, topology, complexity theory, and game theory changed the kinds of questions that could be answered by mathematical methods. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Game theory is a branch of Applied mathematics that is used in the Social sciences (most notably Economics) Biology, Engineering,

The French Bourbaki Group attempted to bring all of mathematics into a coherent rigorous whole, publishing under the pseudonym Nicolas Bourbaki. Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition A pseudonym is a fictitious alternative to a person's legal name (see Alias) Their extensive work had a controversial influence on mathematical education. [35]

There were also new investigations of limitations to mathematics. Kurt Gödel proved that in any mathematical system that includes the integers, there are true statements that cannot be proved. Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most Paul Cohen proved the independence of the continuum hypothesis from the standard axioms of set theory. Paul Joseph Cohen ( April 2, 1934 &ndash March 23, 2007) was an American Mathematician best known for his proof of In Mathematical logic, a sentence &sigma is called independent of a given first-order theory T if T neither proves nor In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common

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• Eves, Howard, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0-03-029558-0,
• Hoffman, Paul, The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. Paul Hoffman (born 1956 is a prominent author and host of the PBS television series Great Minds of Science. Paul Erdős ( Hungarian: Erdős Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 &ndash New York: Hyperion, 1998 ISBN 0-7868-6362-5.
• Grattan-Guinness, Ivor (2003). Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. The Johns Hopkins University Press. ISBN 0801873975.
• van der Waerden, B. L. , Geometry and Algebra in Ancient Civilizations, Springer, 1983, ISBN 0387121595.
• O'Connor, John J. and Robertson, Edmund F. The MacTutor History of Mathematics Archive. (See also MacTutor History of Mathematics archive. The MacTutor History of Mathematics archive is an award-winning website maintained by John J ) This website contains biographies, timelines and historical articles about mathematical concepts; at the School of Mathematics and Statistics, University of St. Andrews, Scotland. The University of St Andrews is the oldest University in Scotland and third oldest in the English-speaking world, having been founded between (Or see the alphabetical list of history topics. )
• Stigler, Stephen M. (1990). Stephen Mack Stigler is Ernest DeWitt Burton Distinguished Service Professor at the Department of Statistics of the University of Chicago[http //chronicle The History of Statistics: The Measurement of Uncertainty before 1900. Belknap Press. ISBN 0-674-40341-X.
• Bell, E. T. (1937). Men of Mathematics. Simon and Schuster.
• Gillings, Richard J. (1972). Mathematics in the time of the pharaohs. Cambridge, MA: M. I. T. Press.
• Heath, Sir Thomas (1981). A History of Greek Mathematics. Dover. ISBN 0-486-24073-8.
• Menninger, Karl W. (1969). Number Words and Number Symbols: A Cultural History of Numbers. MIT Press. ISBN 0-262-13040-8.
• Burton, David M. The History of Mathematics: An Introduction. McGraw Hill: 1997.
• Katz, Victor J. A History of Mathematics: An Introduction, 2nd Edition. Addison-Wesley: 1998. Addison-Wesley is a Book publishing imprint of Pearson PLC, best known for computer books
• Kline, Morris. Mathematical Thought from Ancient to Modern Times.