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3000 BCE - 2023

Story of Mathematics



The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BCE the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars.


The earliest mathematical texts available are from Mesopotamia and Egypt – Plimpton 322 (Babylonian c. 2000 – 1900 BCE),[1] the Rhind Mathematical Papyrus (Egyptian c. 1800 BCE)[2] and the Moscow Mathematical Papyrus (Egyptian c. 1890 BCE). All of these texts mention the so-called Pythagorean triples, so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.


The study of mathematics as a "demonstrative discipline" began in the 6th century BCE with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction".[3] Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics.[4] Although they made virtually no contributions to theoretical mathematics, the ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering, bookkeeping, creation of lunar and solar calendars, and even arts and crafts. Chinese mathematics made early contributions, including a place value system and the first use of negative numbers.[5] The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium CE in India and were transmitted to the Western world via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī.[6] Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations.[7] Contemporaneous with but independent of these traditions were the mathematics developed by the Maya civilization of Mexico and Central America, where the concept of zero was given a standard symbol in Maya numerals.


Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century onward, leading to further development of mathematics in Medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation.[8] Beginning in Renaissance Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus during the course of the 17th century.

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Ancient Egyptian Mathematics
Egyptian measurement unit of the cubit. ©HistoryMaps
3000 BCE Jan 1 - 300 BCE

Ancient Egyptian Mathematics

Egypt

Ancient Egyptian mathematics was developed and used in Ancient Egypt c. 3000 to c. 300 BCE, from the Old Kingdom of Egypt until roughly the beginning of Hellenistic Egypt. The ancient Egyptians utilized a numeral system for counting and solving written mathematical problems, often involving multiplication and fractions. Evidence for Egyptian mathematics is limited to a scarce amount of surviving sources written on papyrus. From these texts it is known that ancient Egyptians understood concepts of geometry, such as determining the surface area and volume of three-dimensional shapes useful for architectural engineering, and algebra, such as the false position method and quadratic equations.


Written evidence of the use of mathematics dates back to at least 3200 BCE with the ivory labels found in Tomb U-j at Abydos. These labels appear to have been used as tags for grave goods and some are inscribed with numbers.[18] Further evidence of the use of the base 10 number system can be found on the Narmer Macehead which depicts offerings of 400,000 oxen, 1,422,000 goats and 120,000 prisoners.[19] Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa.[20] Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs.[20]


The earliest true mathematical documents date to the 12th Dynasty (c. 1990–1800 BCE). The Moscow Mathematical Papyrus, the Egyptian Mathematical Leather Roll, the Lahun Mathematical Papyri which are a part of the much larger collection of Kahun Papyri and the Berlin Papyrus 6619 all date to this period. The Rhind Mathematical Papyrus which dates to the Second Intermediate Period (c. 1650 BCE) is said to be based on an older mathematical text from the 12th dynasty.[22]

Sumerian Mathematics
Ancient Sumer ©Anonymous
3000 BCE Jan 1 - 2000 BCE

Sumerian Mathematics

Iraq

The ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BCE. From 2600 BCE onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.[9]

Abacus
Julius Caesar as a Boy, Learning to Count Using an Abacus. ©Peter Jackson
2700 BCE Jan 1 - 2300 BCE

Abacus

Mesopotamia, Iraq

The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, millennia before the adoption of the Hindu-Arabic numeral system.[127] The exact origin of the abacus has not yet emerged. It consists of rows of movable beads, or similar objects, strung on a wire. They represent digits. One of the two numbers is set up, and the beads are manipulated to perform an operation such as addition, or even a square or cubic root. The Sumerian abacus appeared between 2700 and 2300 BCE. It held a table of successive columns which delimited the successive orders of magnitude of their sexagesimal (base 60) number system.[128]

Old Babylonian Mathematics
Ancient Mesopotamia ©Anonymous
2000 BCE Jan 1 - 1600 BCE

Old Babylonian Mathematics

Babylon, Iraq

Babylonian mathematics were written using a sexagesimal (base-60) numeral system.[12] From this derives the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 × 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. It is likely the sexagesimal system was chosen because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30.[12] Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a place-value system, where digits written in the left column represented larger values, much as in the decimal system.[13] The power of the Babylonian notational system lay in that it could be used to represent fractions as easily as whole numbers; thus multiplying two numbers that contained fractions was no different from multiplying integers, similar to modern notation.[13] The notational system of the Babylonians was the best of any civilization until the Renaissance,[14] and its power allowed it to achieve remarkable computational accuracy; for example, the Babylonian tablet YBC 7289 gives an approximation of √2 accurate to five decimal places.[14] The Babylonians lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.[13] By the Seleucid period, the Babylonians had developed a zero symbol as a placeholder for empty positions; however it was only used for intermediate positions.[13] This zero sign does not appear in terminal positions, thus the Babylonians came close but did not develop a true place value system.[13]


Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of regular numbers, and their reciprocal pairs.[15] The tablets also include multiplication tables and methods for solving linear, quadratic equations and cubic equations, a remarkable achievement for the time.[16] Tablets from the Old Babylonian period also contain the earliest known statement of the Pythagorean theorem.[17] However, as with Egyptian mathematics, Babylonian mathematics shows no awareness of the difference between exact and approximate solutions, or the solvability of a problem, and most importantly, no explicit statement of the need for proofs or logical principles.[13]


They also used a form of Fourier analysis to compute an ephemeris (table of astronomical positions), which was discovered in the 1950s by Otto Neugebauer.[11] To make calculations of the movements of celestial bodies, the Babylonians used basic arithmetic and a coordinate system based on the ecliptic, the part of the heavens that the sun and planets travel through.

Thales's Theorem
©Gabriel Nagypal
600 BCE Jan 1

Thales's Theorem

Babylon, Iraq

Greek mathematics allegedly began with Thales of Miletus (c. 624–548 BCE). Very little is known about his life, although it is generally agreed that he was one of the Seven Wise Men of Greece. According to Proclus, he traveled to Babylon from where he learned mathematics and other subjects, coming up with the proof of what is now called Thales' Theorem.[23]


Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.[30]

Pythagoras
Detail of Pythagoras with a tablet of ratios, from The School of Athens by Raphael. Vatican Palace, Rome, 1509. ©Raphael Santi
580 BCE Jan 1

Pythagoras

Samos, Greece

An equally enigmatic figure is Pythagoras of Samos (c. 580–500 BCE), who supposedly visited Egypt and Babylon,[24] and ultimately settled in Croton, Magna Graecia, where he started a kind of brotherhood. Pythagoreans supposedly believed that "all is number" and were keen in looking for mathematical relations between numbers and things.[25] Pythagoras himself was given credit for many later discoveries, including the construction of the five regular solids.


Almost half of the material in Euclid's Elements is customarily attributed to the Pythagoreans, including the discovery of irrationals, attributed to Hippasus (c. 530–450 BCE) and Theodorus (fl. 450 BCE).[26] It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The greatest mathematician associated with the group, however, may have been Archytas (c. 435-360 BCE), who solved the problem of doubling the cube, identified the harmonic mean, and possibly contributed to optics and mechanics.[26] Other mathematicians active in this period, not fully affiliated with any school, include Hippocrates of Chios (c. 470–410 BCE), Theaetetus (c. 417–369 BCE), and Eudoxus (c. 408–355 BCE).

Discovery of Irrational Numbers
Pythagoreans' Hymn to the Rising Sun. ©Fyodor Bronnikov
400 BCE Jan 1

Discovery of Irrational Numbers

Metapontum, Province of Matera

The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum),[39] who probably discovered them while identifying sides of the pentagram.[40] The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. Hippasus, in the 5th century BCE, however, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction.


Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans 'for having produced an element in the universe which denied the... doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.'[41] Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that number and geometry were inseparable–a foundation of their theory.

Plato
Plato's Academy mosaic – from the Villa of T. Siminius Stephanus in Pompeii. ©Anonymous
387 BCE Jan 1

Plato

Athens, Greece

Plato is important in the history of mathematics for inspiring and guiding others.[31] His Platonic Academy, in Athens, became the mathematical center of the world in the 4th century BCE, and it was from this school that the leading mathematicians of the day, such as Eudoxus of Cnidus, came.[32] Plato also discussed the foundations of mathematics,[33] clarified some of the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions.[34] The analytic method is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his name.[32]

Chinese Geometry
©HistoryMaps
330 BCE Jan 1

Chinese Geometry

China

The oldest existent work on geometry in China comes from the philosophical Mohist canon c. 330 BCE, compiled by the followers of Mozi (470–390 BCE). The Mo Jing described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well.[77] It also defined the concepts of circumference, diameter, radius, and volume.[78]

Chinese Decimal System
©Anonymous
305 BCE Jan 1

Chinese Decimal System

Hunan, China

The Tsinghua Bamboo Slips, containing the earliest known decimal multiplication table (although ancient Babylonians had ones with a base of 60), is dated around 305 BCE and is perhaps the oldest surviving mathematical text of China.[68] Of particular note is the use in Chinese mathematics of a decimal positional notation system, the so-called "rod numerals" in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten.[69] Thus, the number 123 would be written using the symbol for "1", followed by the symbol for "100", then the symbol for "2" followed by the symbol for "10", followed by the symbol for "3". This was the most advanced number system in the world at the time, apparently in use several centuries before the common era and well before the development of the Indian numeral system.[76] Rod numerals allowed the representation of numbers as large as desired and allowed calculations to be carried out on the suan pan, or Chinese abacus. It is presumed that officials used the multiplication table to calculate land surface area, yields of crops and the amounts of taxes owed.[68]

Hellenistic Greek Mathematics
©Aleksandr Svedomskiy
300 BCE Jan 1

Hellenistic Greek Mathematics

Greece

The Hellenistic era began in the late 4th century BCE, following Alexander the Great's conquest of the Eastern Mediterranean, Egypt, Mesopotamia, the Iranian plateau, Central Asia, and parts of India, leading to the spread of the Greek language and culture across these regions. Greek became the lingua franca of scholarship throughout the Hellenistic world, and the mathematics of the Classical period merged with Egyptian and Babylonian mathematics to give rise to Hellenistic mathematics.[27]


Greek mathematics and astronomy reached its acme during the Hellenistic and early Roman periods, and much of the work represented by authors such as Euclid (fl. 300 BCE), Archimedes (c. 287–212 BCE), Apollonius (c. 240–190 BCE), Hipparchus (c. 190–120 BCE), and Ptolemy (c. 100–170 CE) was of a very advanced level and rarely mastered outside a small circle.


Several centers of learning appeared during the Hellenistic period, of which the most important one was the Mouseion in Alexandria, Egypt, which attracted scholars from across the Hellenistic world (mostly Greek, but also Egyptian, Jewish, Persian, among others).[28] Although few in number, Hellenistic mathematicians actively communicated with each other; publication consisted of passing and copying someone's work among colleagues.[29]

Euclid
Detail of Raphael's impression of Euclid, teaching students in The School of Athens (1509–1511) ©Raffaello Santi
300 BCE Jan 1

Euclid

Alexandria, Egypt

In the 3rd century BCE, the premier center of mathematical education and research was the Musaeum of Alexandria.[36] It was there that Euclid (c. 300 BCE) taught, and wrote the Elements, widely considered the most successful and influential textbook of all time.[35]


Considered the "father of geometry", Euclid is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics.


The Elements introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.[37] In addition to the familiar theorems of Euclidean geometry, the Elements was meant as an introductory textbook to all mathematical subjects of the time, such as number theory, algebra and solid geometry,[37] including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also wrote extensively on other subjects, such as conic sections, optics, spherical geometry, and mechanics, but only half of his writings survive.[38]


The Euclidean algorithm is one of the oldest algorithms in common use.[93] It appears in Euclid's Elements (c. 300 BCE), specifically in Book 7 (Propositions 1–2) and Book 10 (Propositions 2–3). In Book 7, the algorithm is formulated for integers, whereas in Book 10, it is formulated for lengths of line segments. Centuries later, Euclid's algorithm was discovered independently both in India and in China,[94] primarily to solve Diophantine equations that arose in astronomy and making accurate calendars.

Archimedes
©Anonymous
287 BCE Jan 1

Archimedes

Syracuse, Free municipal conso

Archimedes of Syracuse is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time,[42] Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems.[43] These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.[44]


Archimedes' other mathematical achievements include deriving an approximation of pi, defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics. Archimedes' achievements in this area include a proof of the law of the lever,[45] the widespread use of the concept of center of gravity,[46] and the enunciation of the law of buoyancy or Archimedes' principle.


Archimedes died during the siege of Syracuse, when he was killed by a Roman soldier despite orders that he should not be harmed.

Apollonius's Parabola
©Domenico Fetti
262 BCE Jan 1

Apollonius's Parabola

Aksu/Antalya, Türkiye

Apollonius of Perga (c. 262–190 BCE) made significant advances to the study of conic sections, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone.[47] He also coined the terminology in use today for conic sections, namely parabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond").[48] His work Conics is one of the best known and preserved mathematical works from antiquity, and in it he derives many theorems concerning conic sections that would prove invaluable to later mathematicians and astronomers studying planetary motion, such as Isaac Newton.[49] While neither Apollonius nor any other Greek mathematicians made the leap to coordinate geometry, Apollonius' treatment of curves is in some ways similar to the modern treatment, and some of his work seems to anticipate the development of analytical geometry by Descartes some 1800 years later.[50]

Nine Chapters on the Mathematical Art
©Luo Genxing
200 BCE Jan 1

Nine Chapters on the Mathematical Art

China

In 212 BCE, the Emperor Qin Shi Huang commanded all books in the Qin Empire other than officially sanctioned ones be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date. After the book burning of 212 BCE, the Han dynasty (202 BCE–220 CE) produced works of mathematics which presumably expanded on works that are now lost. 


After the book burning of 212 BCE, the Han dynasty (202 BCE–220 CE) produced works of mathematics which presumably expanded on works that are now lost. The most important of these is The Nine Chapters on the Mathematical Art, the full title of which appeared by CE 179, but existed in part under other titles beforehand. 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.[79] It created mathematical proof for the Pythagorean theorem,[81] and a mathematical formula for Gaussian elimination.[80] The treatise also provides values of π,[79] which Chinese mathematicians originally approximated as 3 until Liu Xin (d. 23 CE) provided a figure of 3.1457 and subsequently Zhang Heng (78–139) approximated pi as 3.1724,[82] as well as 3.162 by taking the square root of 10.[83]


Negative numbers appear for the first time in history in the Nine Chapters on the Mathematical Art but may well contain much older material.[84] The mathematician Liu Hui (c. 3rd century) established rules for the addition and subtraction of negative numbers.

Hipparchus & Trigonometry
“Hipparchus in the observatory of Alexandria.” Ridpath's history of the world. 1894. ©John Clark Ridpath
190 BCE Jan 1

Hipparchus & Trigonometry

İznik, Bursa, Türkiye

The 3rd century BCE is generally regarded as the "Golden Age" of Greek mathematics, with advances in pure mathematics henceforth in relative decline.[51] Nevertheless, in the centuries that followed significant advances were made in applied mathematics, most notably trigonometry, largely to address the needs of astronomers.[51] Hipparchus of Nicaea (c. 190–120 BCE) is considered the founder of trigonometry for compiling the first known trigonometric table, and to him is also due the systematic use of the 360 degree circle.[52]

Almagest of Ptolemy
©Anonymous
100 Jan 1

Almagest of Ptolemy

Alexandria, Egypt

In the 2nd century CE, the Greco-Egyptian astronomer Ptolemy (from Alexandria, Egypt) constructed detailed trigonometric tables (Ptolemy's table of chords) in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today. Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval Byzantine, Islamic, and, later, Western European worlds. Ptolemy is also credited with Ptolemy's theorem for deriving trigonometric quantities, and the most accurate value of π outside of China until the medieval period, 3.1416.[63]

Chinese Remainder Theorem
©张文新
200 Jan 1

Chinese Remainder Theorem

China

In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). The earliest known statement of the theorem is by the Chinese mathematician Sun-tzu in the Sun-tzu Suan-ching in the 3rd century CE.

Diophantine Analysis
©Tom Lovell
200 Jan 1

Diophantine Analysis

Alexandria, Egypt

Following a period of stagnation after Ptolemy, the period between 250 and 350 CE is sometimes referred to as the "Silver Age" of Greek mathematics.[53] During this period, Diophantus made significant advances in algebra, particularly indeterminate analysis, which is also known as "Diophantine analysis".[54] The study of Diophantine equations and Diophantine approximations is a significant area of research to this day. His main work was the Arithmetica, a collection of 150 algebraic problems dealing with exact solutions to determinate and indeterminate equations.[55] The Arithmetica had a significant influence on later mathematicians, such as Pierre de Fermat, who arrived at his famous Last Theorem after trying to generalize a problem he had read in the Arithmetica (that of dividing a square into two squares).[56] Diophantus also made significant advances in notation, the Arithmetica being the first instance of algebraic symbolism and syncopation.[55]

Story of Zero
©HistoryMaps
224 Jan 1

Story of Zero

India

Ancient Egyptian numerals were of base 10. They used hieroglyphs for the digits and were not positional. By the middle of the 2nd millennium BCE, the Babylonian mathematics had a sophisticated base 60 positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a placeholder within its vigesimal (base-20) positional numeral system.


The concept of zero as a written digit in the decimal place value notation was developed in India.[65] A symbol for zero, a large dot likely to be the precursor of the still-current hollow symbol, is used throughout the Bakhshali manuscript, a practical manual on arithmetic for merchants.[66] In 2017, three samples from the manuscript were shown by radiocarbon dating to come from three different centuries: from CE 224–383, CE 680–779, and CE 885–993, making it South Asia's oldest recorded use of the zero symbol. It is not known how the birch bark fragments from different centuries forming the manuscript came to be packaged together.[67] Rules governing the use of zero appeared in Brahmagupta's Brahmasputha Siddhanta (7th century), which states the sum of zero with itself as zero, and incorrectly division by zero as:


A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.
Hypatia
©Julius Kronberg
350 Jan 1

Hypatia

Alexandria, Egypt

The first woman mathematician recorded by history was Hypatia of Alexandria (CE 350–415). She wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria had her stripped publicly and executed. Her death is sometimes taken as the end of the era of the Alexandrian Greek mathematics, although work did continue in Athens for another century with figures such as Proclus, Simplicius and Eutocius.[57] Although Proclus and Simplicius were more philosophers than mathematicians, their commentaries on earlier works are valuable sources on Greek mathematics. The closure of the neo-Platonic Academy of Athens by the emperor Justinian in 529 CE is traditionally held as marking the end of the era of Greek mathematics, although the Greek tradition continued unbroken in the Byzantine empire with mathematicians such as Anthemius of Tralles and Isidore of Miletus, the architects of the Hagia Sophia.[58] Nevertheless, Byzantine mathematics consisted mostly of commentaries, with little in the way of innovation, and the centers of mathematical innovation were to be found elsewhere by this time.[59]

Play button
505 Jan 1

Indian Trigonometry

Patna, Bihar, India

The modern sine convention is first attested in the Surya Siddhanta (showing strong Hellenistic influence)[64], and its properties were further documented by the 5th century (CE) Indian mathematician and astronomer Aryabhata.[60] The Surya Siddhanta describes rules to calculate the motions of various planets and the moon relative to various constellations, diameters of various planets, and calculates the orbits of various astronomical bodies. The text is known for some of earliest known discussion of sexagesimal fractions and trigonometric functions.[61]

Play button
510 Jan 1

Indian Decimal System

India

Around 500 CE, Aryabhata wrote the Aryabhatiya, a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration.[62] Though about half of the entries are wrong, it is in the Aryabhatiya that the decimal place-value system first appears.

Play button
780 Jan 1

Muhammad ibn Musa al-Khwarizmi

Uzbekistan

In the 9th century, the mathematician Muḥammad ibn Mūsā al-Khwārizmī wrote an important book on the Hindu–Arabic numerals and one on methods for solving equations. His book On the Calculation with Hindu Numerals, written about 825, along with the work of Al-Kindi, were instrumental in spreading Indian mathematics and Indian numerals to the West. 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). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,[87] and he was the first to teach algebra in an elementary form and for its own sake.[88] He also discussed the fundamental method of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as al-jabr.[89] His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."[90]

Abu Kamil
©Davood Diba
850 Jan 1

Abu Kamil

Egypt

Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ was a prominent Egyptian mathematician during the Islamic Golden Age. He is considered the first mathematician to systematically use and accept irrational numbers as solutions and coefficients to equations.[91] His mathematical techniques were later adopted by Fibonacci, thus allowing Abu Kamil an important part in introducing algebra to Europe.[92]

Mayan Mathematics
©Louis S. Glanzman
900 Jan 1

Mayan Mathematics

Mexico

In the Pre-Columbian Americas, the Maya civilization that flourished in Mexico and Central America during the 1st millennium CE developed a unique tradition of mathematics that, due to its geographic isolation, was entirely independent of existing European, Egyptian, and Asian mathematics.[92] Maya numerals used a base of twenty, the vigesimal system, instead of a base of ten that forms the basis of the decimal system used by most modern cultures.[92] The Maya used mathematics to create the Maya calendar as well as to predict astronomical phenomena in their native Maya astronomy.[92] While the concept of zero had to be inferred in the mathematics of many contemporary cultures, the Maya developed a standard symbol for it.[92]

Al-Karaji
©Osman Hamdi Bey
953 Jan 1

Al-Karaji

Karaj, Alborz Province, Iran

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

 

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

Chinese Algebra
©Anonymous Chinese artist of the Song Dynasty
960 Jan 1 - 1279

Chinese Algebra

China

The high-water mark of Chinese mathematics occurred in the 13th century during the latter half of the Song dynasty (960–1279), with the development of Chinese algebra. The most important text from that period is the Precious Mirror of the Four Elements by Zhu Shijie (1249–1314), dealing with the solution of simultaneous higher order algebraic equations using a method similar to Horner's method.[70] The Precious Mirror also contains a diagram of Pascal's triangle with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100.[71] The Chinese also made use of the complex combinatorial diagram known as the magic square and magic circles, described in ancient times and perfected by Yang Hui (CE 1238–1298).[71]


Japanese mathematics, Korean mathematics, and Vietnamese mathematics are traditionally viewed as stemming from Chinese mathematics and belonging to the Confucian-based East Asian cultural sphere.[72] Korean and Japanese mathematics were heavily influenced by the algebraic works produced during China's Song dynasty, whereas Vietnamese mathematics was heavily indebted to popular works of China's Ming dynasty (1368–1644).[73] For instance, although Vietnamese mathematical treatises were written in either Chinese or the native Vietnamese Chữ Nôm script, all of them followed the Chinese format of presenting a collection of problems with algorithms for solving them, followed by numerical answers.[74] Mathematics in Vietnam and Korea were mostly associated with the professional court bureaucracy of mathematicians and astronomers, whereas in Japan it was more prevalent in the realm of private schools.[75]

Hindu-Arabic Numerals
The Scholars ©Ludwig Deutsch
974 Jan 1

Hindu-Arabic Numerals

Toledo, Spain

Europeans learned of Arabic numerals about the 10th century, though their spread was a gradual process. Two centuries later, in the Algerian city of Béjaïa, the Italian scholar Fibonacci first encountered the numerals; his work was crucial in making them known throughout Europe. European trade, books, and colonialism helped popularize the adoption of Arabic numerals around the world. The numerals have found worldwide use significantly beyond the contemporary spread of the Latin alphabet, and have become common in the writing systems where other numeral systems existed previously, such as Chinese and Japanese numerals. The first mentions of the numerals from 1 to 9 in the West are found in the Codex Vigilanus of 976, an illuminated collection of various historical documents covering a period from antiquity to the 10th century in Hispania.[68]

Leonardo Fibonacci
Portrait of Medieval Italian Man ©Vittore Carpaccio
1202 Jan 1

Leonardo Fibonacci

Pisa, Italy

In the 12th century, European scholars traveled to Spain and Sicily seeking scientific Arabic texts, including al-Khwārizmī's The Compendious Book on Calculation by Completion and Balancing, 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.[95] These and other new sources sparked a renewal of mathematics.


Leonardo of Pisa, now known as Fibonacci, serendipitously learned about the Hindu–Arabic numerals on a trip to what is now Béjaïa, Algeria with his merchant father. (Europe was still using Roman numerals.) There, he observed a system of arithmetic (specifically algorism) which due to the positional notation of Hindu–Arabic numerals was much more efficient and greatly facilitated commerce. He soon realised the many advantages of the Hindu-Arabic system, which, unlike the Roman numerals used at the time, allowed easy calculation using a place-value system. Leonardo wrote Liber Abaci in 1202 (updated in 1254) introducing the technique to Europe and beginning a long period of popularizing it. The book also brought to Europe what is now known as the Fibonacci sequence (known to Indian mathematicians for hundreds of years before that)[96] which Fibonacci used as an unremarkable example.

Infinite Series
©Veloso Salgado
1350 Jan 1

Infinite Series

Kerala, India

Greek mathematician Archimedes produced the first known summation of an infinite series with a method that is still used in the area of calculus today. 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 π.[86] The Kerala school has made a number of contributions to the fields of infinite series and calculus.

Probability Theory
Gerolamo Cardano ©R. Cooper
1564 Jan 1

Probability Theory

Europe

The modern mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century (for example the "problem of points").[105] Christiaan Huygens published a book on the subject in 1657.[106] In the 19th century, what is considered the classical definition of probability was completed by Pierre Laplace.[107]


Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory.


This culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933. This became the mostly undisputed axiomatic basis for modern probability theory; but, alternatives exist, such as the adoption of finite rather than countable additivity by Bruno de Finetti.[108]

Logarithms
Johannes Kepler ©August Köhler
1614 Jan 1

Logarithms

Europe

The 17th century saw an unprecedented increase of mathematical and scientific ideas across Europe. Galileo observed the moons of Jupiter in orbit about that planet, using a telescope based Hans Lipperhey's. Tycho Brahe had gathered a large quantity of mathematical data describing the positions of the planets in the sky. By his position as Brahe's assistant, Johannes Kepler was first exposed to and seriously interacted with the topic of planetary motion. Kepler's calculations were made simpler by the contemporaneous invention of logarithms by John Napier and Jost Bürgi. Kepler succeeded in formulating mathematical laws of planetary motion. The analytic geometry developed by René Descartes (1596–1650) allowed those orbits to be plotted on a graph, in Cartesian coordinates.

Cartesian Coordinate System
René Descartes ©Frans Hals
1637 Jan 1

Cartesian Coordinate System

Netherlands

The Cartesian refers to the French mathematician and philosopher René Descartes, who published this idea in 1637 while he was resident in the Netherlands. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery.[109] The French cleric Nicole Oresme used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat.[110]


Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes's work.[111]


The development of the Cartesian coordinate system would play a fundamental role in the development of the calculus by Isaac Newton and Gottfried Wilhelm Leibniz.[112] The two-coordinate description of the plane was later generalized into the concept of vector spaces.[113]


Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three-dimensional space.

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1670 Jan 1

Calculus

Europe

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.[97]


Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz.[98] Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing. Today, calculus has widespread uses in science, engineering, and social science.


Isaac Newton developed the use of calculus in his laws of motion and universal gravitation.

These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. Unlike Newton, Leibniz put painstaking effort into his choices of notation.[99]


Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today.[100] The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, emphasizing that differentiation and integration are inverse processes, second and higher derivatives, and the notion of an approximating polynomial series.

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1736 Jan 1

Graph Theory

Europe

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics.


The paper written by Leonhard Euler on the Seven Bridges of Königsberg and published in 1736 is regarded as the first paper in the history of graph theory.[114] This paper, as well as the one written by Vandermonde on the knight problem, carried on with the analysis situs initiated by Leibniz. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy[115] and L'Huilier,[116] and represents the beginning of the branch of mathematics known as topology.

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1738 Jan 1

Normal Distribution

France

In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.[124] Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as measurement errors, often have distributions that are nearly normal.[125] Some authors[126] attribute the credit for the discovery of the normal distribution to de Moivre, who in 1738 published in the second edition of his "The Doctrine of Chances" the study of the coefficients in the binomial expansion of (a + b)n.

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1740 Jan 1

Euler's Formula

Berlin, Germany

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". When x = π, Euler's formula may be rewritten as eiπ + 1 = 0 or eiπ = -1, which is known as Euler's identity.

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1763 Jan 1

Bayes' Theorem

England, UK

In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event.[122] For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately by conditioning it relative to their age, rather than simply assuming that the individual is typical of the population as a whole.


In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event.[122] For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately by conditioning it relative to their age, rather than simply assuming that the individual is typical of the population as a whole.

Gauss's Law
Carl Friedrich Gauss ©Christian Albrecht Jensen
1773 Jan 1

Gauss's Law

France

In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it states that the flux of the electric field out of an arbitrary closed surface is proportional to the electric charge enclosed by the surface, irrespective of how that charge is distributed. Even though the law alone is insufficient to determine the electric field across a surface enclosing any charge distribution, this may be possible in cases where symmetry mandates uniformity of the field. Where no such symmetry exists, Gauss's law can be used in its differential form, which states that the divergence of the electric field is proportional to the local density of charge.


The law was first[101] formulated by Joseph-Louis Lagrange in 1773,[102] followed by Carl Friedrich Gauss in 1835,[103] both in the context of the attraction of ellipsoids. It is one of Maxwell's equations, which forms the basis of classical electrodynamics. Gauss's law can be used to derive Coulomb's law,[104] and vice versa.

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1800 Jan 1

Group Theory

Europe

In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.


The early history of group theory dates from the 19th century. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete classification of finite simple groups.

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1807 Jan 1

Fourier Analysis

Auxerre, France

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.


The subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term Fourier analysis often refers to the study of both operations.


The decomposition process itself is called a Fourier transformation. Its output, the Fourier transform, is often given a more specific name, which depends on the domain and other properties of the function being transformed. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis. Each transform used for analysis (see list of Fourier-related transforms) has a corresponding inverse transform that can be used for synthesis.

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1850 Jan 1 - 1870

Maxwell's Equations

Cambridge University, Trinity

Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar, etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formulation is credited to Oliver Heaviside.


The equations have two major variants. The microscopic equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges and currents in materials at the atomic scale. The macroscopic equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic-scale charges and quantum phenomena like spins. However, their use requires experimentally determined parameters for a phenomenological description of the electromagnetic response of materials. The term "Maxwell's equations" is often also used for equivalent alternative formulations. Versions of Maxwell's equations based on the electric and magnetic scalar potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics. The covariant formulation (on spacetime rather than space and time separately) makes the compatibility of Maxwell's equations with special relativity manifest. Maxwell's equations in curved spacetime, commonly used in high-energy and gravitational physics, are compatible with general relativity. In fact, Albert Einstein developed special and general relativity to accommodate the invariant speed of light, a consequence of Maxwell's equations, with the principle that only relative movement has physical consequences.


The publication of the equations marked the unification of a theory for previously separately described phenomena: magnetism, electricity, light, and associated radiation. Since the mid-20th century, it has been understood that Maxwell's equations do not give an exact description of electromagnetic phenomena, but are instead a classical limit of the more precise theory of quantum electrodynamics.

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1870 Jan 1

Set Theory

Germany

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole.


The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of naive set theory. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied.


Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity, and has various applications in computer science (such as in the theory of relational algebra), philosophy and formal semantics. Its foundational appeal, together with its paradoxes, its implications for the concept of infinity and its multiple applications, have made set theory an area of major interest for logicians and philosophers of mathematics. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

Game Theory
John von Neumann ©Anonymous
1927 Jan 1

Game Theory

Budapest, Hungary

Game theory is the study of mathematical models of strategic interactions among rational agents.[117] It has applications in all fields of social science, as well as in logic, systems science and computer science. The concepts of game theory are used extensively in economics as well.[118] The traditional methods of game theory addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by the losses and gains of other participants. In the 21st century, the advanced game theories apply to a wider range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers.


Game theory did not exist as a unique field until John von Neumann published the paper On the Theory of Games of Strategy in 1928.[119] Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern.[120] The second edition of this book provided an axiomatic theory of utility, which reincarnated Daniel Bernoulli's old theory of utility (of money) as an independent discipline. Von Neumann's work in game theory culminated in this 1944 book. This foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. Subsequent work focused primarily on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.[121]

Appendices



APPENDIX 1

The History of Mathematics and Its Applications


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APPENDIX 2

The Map of Mathematics


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Footnotes



  1. Friberg, J. "Methods and traditions of Babylonian mathematics. Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations", Historia Mathematica, 8, 1981, pp. 277-318.
  2. Neugebauer, Otto (1969) [1957]. The Exact Sciences in Antiquity. Acta Historica Scientiarum Naturalium et Medicinalium. Vol. 9 (2 ed.). Dover Publications. pp. 1-191. ISBN 978-0-486-22332-2. PMID 14884919. Chap. IV "Egyptian Mathematics and Astronomy", pp. 71-96.
  3. Turnbull (1931). "A Manual of Greek Mathematics". Nature. 128 (3235): 5. Bibcode:1931Natur.128..739T. doi:10.1038/128739a0. S2CID 3994109.
  4. Heath, Thomas L. (1963). A Manual of Greek Mathematics, Dover, p. 1: "In the case of mathematics, it is the Greek contribution which it is most essential to know, for it was the Greeks who first made mathematics a science."
  5. Joseph, George Gheverghese (1991). The Crest of the Peacock: Non-European Roots of Mathematics. Penguin Books, London, pp. 140-48.
  6. Kaplan, Robert (1999). The Nothing That Is: A Natural History of Zero. Allen Lane/The Penguin Press, London.
  7. Juschkewitsch, A. P. (1964). Geschichte der Mathematik im Mittelalter. Teubner, Leipzig.
  8. Eves, Howard (1990). History of Mathematics, 6th Edition, "After Pappus, Greek mathematics ceased to be a living study, ..." p. 185; "The Athenian school struggled on against growing opposition from Christians until the latter finally, in A.D. 529, obtained a decree from Emperor Justinian that closed the doors of the school forever." p. 186; "The period starting with the fall of the Roman Empire, in the middle of the fifth century, and extending into the eleventh century is known in Europe as the Dark Ages ... . Schooling became almost nonexistent." p. 258.
  9. Duncan J. Melville (2003). Third Millennium Chronology, Third Millennium Mathematics. St. Lawrence University.
  10. Maor, Eli (1998). Trigonometric Delights. Princeton University Press. p. 20. ISBN 0-691-09541-8.
  11. Prestini, Elena (2004). The evolution of applied harmonic analysis: models of the real world. Birkhauser. ISBN 978-0-8176-4125-2., p. 62
  12. Boyer, C.B. (1991) [1989], A History of Mathematics (2nd ed.), New York: Wiley , ISBN 978-0-471-54397-8, p. 25.
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  14. Boyer 1991, "Mesopotamia" p. 27.
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  73. Volkov 2009, pp. 154-55
  74. Volkov 2009, pp. 156-57
  75. Volkov 2009, p. 155
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  77. Needham, Joseph; Wang, Ling (1995) [1959], Science and Civilization in China: Mathematics and the Sciences of the Heavens and the Earth, vol. 3, Cambridge: Cambridge University Press, ISBN 978-0-521-05801-8, pp. 91-92
  78. Needham & Wang 1995, p. 94
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  80. Straffin, Philip D. (1998), "Liu Hui and the First Golden Age of Chinese Mathematics", Mathematics Magazine, 71 (3): 163-81, doi:10.1080/0025570X.1998.11996627, p. 164.
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  82. Needham & Wang 1995, p. 99-100
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Further Reading

  • Aaboe, Asger (1964). Episodes from the Early History of Mathematics. New York: Random House.
  • Bell, E.T. (1937). Men of Mathematics. Simon and Schuster.
  • Burton, David M. The History of Mathematics: An Introduction. McGraw Hill: 1997.
  • Corry, Leo (2015), A Brief History of Numbers, Oxford University Press, ISBN 978-0198702597
  • Gillings, Richard J. (1972). Mathematics in the Time of the Pharaohs. Cambridge, MA: MIT Press.
  • Grattan-Guinness, Ivor (2003). Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. The Johns Hopkins University Press. ISBN 978-0-8018-7397-3.
  • Heath, Sir Thomas (1981). A History of Greek Mathematics. Dover. ISBN 978-0-486-24073-2.
  • Hoffman, Paul (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. Hyperion. ISBN 0-7868-6362-5.
  • Kline, Morris. Mathematical Thought from Ancient to Modern Times.
  • Menninger, Karl W. (1969). Number Words and Number Symbols: A Cultural History of Numbers. MIT Press. ISBN 978-0-262-13040-0.
  • Stigler, Stephen M. (1990). The History of Statistics: The Measurement of Uncertainty before 1900. Belknap Press. ISBN 978-0-674-40341-3.
  • Struik, D.J. (1987). A Concise History of Mathematics, fourth revised edition. Dover Publications, New York.
  • van der Waerden, B.L., Geometry and Algebra in Ancient Civilizations, Springer, 1983, ISBN 0-387-12159-5.