History of Mathematics

• Deals with the origin of discoveries in mathematics and the mathematical methods of the past.
• Before the modern age, written examples of new mathematical developments have come to light only in a few locales.
• From 3000 BC, Mesopotamian states and Ancient Egypt used arithmetic, algebra, and geometry for taxation, commerce, trade, and astronomy to record time and formulate calendars.
• Earliest available mathematical texts are from Mesopotamia and Egypt.
• Plimpton 322 is Babylonian, c. 2000–1900 BC.
• Rhind Mathematical Papyrus is Egyptian, c. 1800 BC.
• Moscow Mathematical Papyrus is Egyptian, c. 1890 BC.
• These texts mention the Pythagorean triples, suggesting the Pythagorean theorem was an ancient and widespread mathematical development post basic arithmetic and geometry.
• Mathematics as a "demonstrative discipline" began in the 6th century BC with the Pythagoreans.
• Pythagoreans coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction".
• Ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering, bookkeeping, lunar and solar calendars, arts, and crafts.
• Chinese mathematics made early contributions, including a place value system and the first use of negative numbers.
• Hindu–Arabic numeral system and the rules for its operations evolved over the course of the first millennium AD in India.
• The system was transmitted to the Western world via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī.
• Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations.
• The Maya civilization of Mexico and Central America developed mathematics independently, giving zero a standard symbol in Maya numerals.
• Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century onward, furthering mathematics in Medieval Europe.
• Mathematical discoveries were often followed by centuries of stagnation through the Middle Ages.
• Beginning in Renaissance Italy in the 15th century, new mathematics interacted with new scientific discoveries, continuing through the present day.
• This includes the groundbreaking work of Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus, and discoveries of German mathematicians like Carl Friedrich Gauss and David Hilbert.

Prehistoric

• Origins of mathematical thought are in concepts of number, patterns in nature, magnitude, and form.
• Modern studies of animal cognition show these concepts are not unique to humans.
• Concepts would have been part of the everyday life in hunter-gatherer societies.
• The “number” concept evolving gradually is supported by languages preserving the distinction between "one", "two", and "many", but not larger numbers.
• Use of yarn by Neanderthals some 40,000 years ago at a site in France suggests they knew basic concepts in mathematics.
• Ishango bone is more than 20,000 years old, found near the Nile, with marks carved in three columns.
• Common interpretations state the Ishango bone shows either a tally of the earliest known sequences of prime numbers or a six-month lunar calendar.
• Rudman argues prime numbers could only come about after division, which he dates to after 10,000 BC, with primes not being understood until about 500 BC.
• Marshack suggests the Ishango bone may have influenced the development of math as Egyptian arithmetic also used multiplication by 2; this however, is disputed.
• Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs.
• Megalithic monuments in England and Scotland from the 3rd millennium BC incorporate geometric ideas such as circles, ellipses, and Pythagorean triples in their design, however, it is disputed.
• Oldest undisputed mathematical documents are from Babylonian and dynastic Egyptian sources.

Babylonian

• Refers to any mathematics of the peoples of Mesopotamia (modern Iraq) from the days of the early Sumerians to almost the dawn of Christianity.
• Majority of Babylonian mathematical work comes from two periods: millennium BC (Old Babylonian period) and centuries of the first millennium BC (Seleucid period).
• Termed Babylonian mathematics due to the integral role of Babylon as a place of subject matter study.
• Later under Arab Empire, Mesopotamia, especially Baghdad, once again became an study center for Islamic mathematics.
• Derive knowledge from more than 400 clay tablets unearthed since the 1850s, as opposed to Egyptian mathematics.
• Written in Cuneiform script, the clay was moist when tablets were inscribed and baked hard by the heat of the sun or oven. Some were graded homework.
• Earliest evidence of written mathematics dates back to the ancient Sumerians, who built the Mesopotamian civilization.
• Developed a metrology system from 3000 BC concerned with administrative/financial counting, such as grain allotments, workers, weights of silver, or liquids.
• From around 2500 BC onward, Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems.
• Earliest traces of Babylonian numerals date back to this period.
• Wrote using a sexagesimal (base-60) numeral system. Modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a circle derived from it.
• Sexagesimal system thought to be initially used by Sumerian scribes because 60 could be evenly divided by factors such as 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30.
• Unlike Egyptians, Greeks, and Romans, Babylonians had a place-value system, where digits written in the left column represented larger values.
• Power of the Babylonian notational system lay in that it could represent fractions as easily as whole numbers; multiplying two numbers with fractions was no different from multiplying integers.
• Notational system was the best of any civilization until the Renaissance, and its power allowed it to achieve remarkable computational accuracy.
• The Babylonian tablet YBC 7289 gives an approximation of √2 accurate to five decimal places.
• Lacked an equivalent of the decimal point, so the place value of a symbol often had to be inferred from the context.
• By the Seleucid period, Babylonians had developed a zero symbol to placeholder empty positions; however, it was only used for intermediate positions.
• Other topics include fractions, algebra, quadratic and cubic equations, and the calculation of regular numbers, and their reciprocal pairs.
• Tablets include multiplication tables and methods for solving linear, quadratic, and cubic equations.
• Old Babylonian period tablets contain the earliest known statement of the Pythagorean theorem.
• Babylonian mathematics shows no awareness of the difference between exact and approximate solutions, or the solvability of a problem, and no explicit statement of the need for proofs or logical principles.

Egyptian

• Refers to mathematics written in the Egyptian language.
• From the Hellenistic period, Greek replaced Egyptian as the written language of Egyptian scholars.
• Under Arab Empire as part of Islamic mathematics, Arabic became the written language of Egyptian scholars.
• Archaeological data suggested Ancient Egyptian counting system had origins in Sub-Saharan Africa.
• Fractal geometry designs, widespread among Sub-Saharan African cultures, are also found in Egyptian architecture and cosmological signs.
• Most extensive Egyptian mathematical text is the Rhind papyrus, dated to c. 1650 BC but likely a copy of an older document from the Middle Kingdom of about 2000-1800 BC.
• Manual for students in arithmetic and geometry giving methods for area formulas and calculating via multiplication, division, and fractions.
• Also contains evidence of other mathematical knowledge, including composite and prime numbers, arithmetic, geometric, and harmonic means; simplistic understandings of both the Sieve of Eratosthenes and perfect number theory (namely, that of the number 6), as well as how to solve first order linear equations and number sequences.
• Moscow papyrus also comes from the Middle Kingdom period c. 1890 BC, consists of word or story problems, which were apparently entertainment.
• Gives a method for the volume of a frustum (truncated pyramid), considered to be of special importance.
• Berlin Papyrus 6619 (c. 1800 BC) reveals ancient Egyptians could solve a second-order algebraic equation.

Greek

• Refers to mathematics written in the Greek language from the time of Thales of Miletus (~600 BC) to the closure of the Academy of Athens in 529 AD.
• Greek mathematicians lived in cities spread across the Eastern Mediterranean, from Italy to North Africa. The culture and language united them.
• Greek mathematics of the period after Alexander the Great is called Hellenistic mathematics.
• Was more sophisticated than those by earlier cultures.
• Pre-Greek records show inductive reasoning use - repeated observations to establish rules of thumb.
• Mathematicians used deductive reasoning with logic to derive conclusions from definitions and axioms, and mathematical rigor to prove them.
• Thought to have begun with Thales of Miletus (c. 624-c.546 BC) and Pythagoras of Samos (c. 582-c. 507 BC), who were probably inspired by Egyptian and Babylonian mathematics.
• According to legend, Pythagoras traveled 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, credited with the first use of deductive reasoning as applied to deriving corollaries to the 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.
• Pythagoras established the Pythagorean School, whose doctrine was that mathematics rules the universe and whose motto was "All is number".
• The Pythagoreans coined the term "mathematics", and with whom the study of mathematics for its own sake begins, credited with the first proof of the Pythagorean theorem, statement has a long history, and proof of the existence of irrational numbers.
• Nicomachus (60–120 AD) provided one of the earliest Greco-Roman multiplication tables (preceded by the Babylonians, Indians, and the Chinese).
• Wax tablet dated to the 1st century AD contains the oldest extant Greek multiplication table.
• Connection of the Neopythagoreans to the Western invention of the multiplication table is evident in its later Medieval name: the mensa Pythagorica.
• Plato (428/427 BC – 348/347 BC) inspired and guided others ,His Platonic Academy, the mathematical center of the world in the 4th century BC, gave rise to mathematicians such as Eudoxus of Cnidus (c. 390 c. 340 BC), also discussed the foundations of mathematics, and defined (e.g. "breadthless length" for line).
• Eudoxus developed the method of exhaustion, a precursor of modern integration and a theory of ratios that avoided the problem of incommensurable magnitudes.
• The former allowed calculations of areas and volumes of curvilinear figures, and the latter allows advancements in geometry.
• Aristotle (384-c, 322 BC) contributed to mathematics by laying the foundations of logic, though having made no specific technical mathematical discoveries.
• In the 3rd century BC, the premier center of mathematical education and research was the Musaeum of Alexandria, lead by Euclid (c. 300 BC) who wrote the Elements.
• Introduced mathematical rigor through the axiomatic method and is the earliest example of the format of definition, axiom, theorem, and proof.
• Euclid arranged known mathematical items into a coherent logical framework.
• Was known to educated people through the middle of the 20th century and contents taught in geometry, also serves as an introduction to number theory, algebra and solid geometry.
• Euclid also wrote extensively on conic sections, optics, spherical geometry, and mechanics, but only half of his writings survive.
• Archimedes (c. 287-212 BC), considered the greatest mathematician of antiquity.
• Archimedes used the method of exhaustion to calculate the area with infinite series, in a manner similar to modern calculus.
• He obtained the most accurate value of π then known, 3 + 10 < π < 3.
• He studied his eponymous spiral, obtained formulas for the surfaces of revolution , and an exponentiation method for very large numbers.
• Archimedes placed far greater value on his thought and general mathematical principles, than to his contributions in physics and several mechanical devices.
• He regarded his finding of a sphere's surface area and volume as his greatest achievement, obtained by proving these are 2/3 the surface area and volume of a cylinder circumscribing the sphere.
• Apollonius of Perga (c. 262-190 BC) made significant contributions to the study of conic sections.
• He showed that one could see 3 varieties of conic section by varying the angle of the plane that cuts a double-napped cone and coined the terminology in use today for conic sections, namely parabola, ellipse, and hyperbola.
• Apollonius' conics and its theorems regarding conic sections would later prove invaluable in studying planetary motion.
• Apollonius' treatment of curves are similar to the modern treatment.
• Eratosthenes of Cyrene (c. 276-194 BC) thought up the sieve of Eratosthenes for finding prime numbers.
• Third century BC is generally regarded as the "Golden Age" of Greek mathematics, with advances in pure mathematics after that henceforth in relative decline.
• Significant advances were made in applied mathematics (most notably trigonometry) to meet astronomical needs.
• Hipparchus of Nicaea (c. 190-120 BC) is considered the founder of trigonometry, for compiling the first known trigonometric table, systematic use of the 360-degree circle.
• Heron of Alexandria (c. 10–70 AD) is credited with Heron's formula for finding the area of a scalene triangle and with being first with recognizing possession of square roots of negative numbers.
• Menelaus of Alexandria (c. 100 AD) pioneered spherical trigonometry via Menelaus' theorem.
• The most complete and influential trigonometric work of antiquity is the Almagest of Ptolemy (c. AD 90-168), whose trigonometric tables would be used by astronomers for the next thousand years and theorem for deriving trigonometric quantities.
• Ptolemy is also credited with the most accurate value of π outside of China until the medieval period, 3.1416.
• Following a period of stagnation after Ptolemy, the period between 250 and 350 AD is referred to as the "Silver Age" of Greek mathematics.
• Diophantus made contributions in algebra, particularly indeterminate analysis, also known as Diophantine analysis, with his main work being the Arithmetica.
• Pappus of Alexandria (4th century AD) is known for his hexagon theorem and centroid theorem, as well as the Pappus configuration and Pappus graph and his Collection, a major source of knowledge on Greek mathematics, as most of it has survived.
• Hypatia of Alexandria (AD 350-415) succeeded her father (Theon of Alexandria) as Librarian at the Library and wrote works on applied mathematics.
• The Greek tradition which was thought to have ended from her public execution, although work did continue in Athens with figures such as Proclus, Simplicius, and Eutocius, consisted mostly of commentaries with little innovation and centers of innovation were to be found elsewhere,.

Roman

• There were no noteworthy native Latin mathematicians despite ethnic Greek mathematicians continued in Ancient Rome Republic and Roman Empire.
• Romans thought mathematicians and calculators were more interested in applied than in theoretical.
• Numerals derived from the Greek or Etruscan numerals used.
• Romans were skilled at instigating and detecting financial fraud as well as managing taxes for the treasury using calculation.
• Siculus Flaccus, a land surveyor, wrote the Categories of Fields which aided Roman surveyors in measuring surface areas of allotted lands and territories
• Math also helped solve problems in engineering, such as the erection of architecture such as bridges, road-building, and military campaigns.
• Mosaics created illusionist geometric patterns and rich, detailed scenes that required precise measurements for each tessera tile.
• Calendar creation also necessitated math, the first calendar allegedly dating back to the 8th century BC during the Roman Kingdom, which included 356 days plus a leap year every other year.
• This calendar was supplanted by the Julian calendar, a solar calendar organized by Julius Caesar (100–44 BC) and devised by Sosigenes of Alexandria to include a leap day every four years in a 365-day cycle.
• Calendar was later corrected by Gregorian calendar virtually the same solar calendar used in modern times.
• Around the same time, the Han Chinese and the Romans both invented the wheeled odometer device for measuring distances traveled.

Chinese

• An analysis of early Chinese mathematics has demonstrated its unique development compared to other parts of the world, leading scholars to assume an entirely independent development.
• The oldest extant mathematical text from China is the Zhoubi Suanjing, dated to between 1200 BC and 100 BC
• 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 BC
• 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
• The oldest extant work on geometry in China comes from the philosophical Mohist canon c. 330 BC, compiled by the followers of Mozi (470-390 BC) and described various aspects of many fields associated with physical science and a set of geometrical theorems,.
• In 212 BC, Emperor Qin Shi Huang commanded all books in the Qin Empire other than officially sanctioned ones be burned with a consequence of little being known about ancient Chinese mathematics before this date.
• Han dynasty (202 BC-220 AD) produced works of mathematics, the most important of these is The Nine Chapters on the Mathematical Art, including word problems agriculture, business, employment of geometry to figure height spans and dimension ratios for Chinese pagoda towers, engineering, surveying, and includes material on right triangles
• 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 Precious Mirror of the Four Elements by Zhu Shijie (1249-1314), dealt with solving simultaneous higher order algebraic using a method similar to Horner's method.
• Even after European mathematics began to flourish during the Renaissance, European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline from the 13th century onwards, Jesuit missionaries moved mathematical ideas back and forth.
• Japanese mathematics, Korean mathematics, and Vietnamese mathematics stemmed from Chinese mathematics and belonging to the Confucian-based East Asian sphere with all following the Chinses format.

Indian

• Is the Indus Valley civilization (mature second phase: 2600 to 1900 BC) that flourished in the Indus river basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization.
• The oldest extant mathematical records from India are the Sulba Sutras (dated variously between the 8th century BC and the 2nd century AD), appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others.
• Pāṇini (c. 5th century BC) formulated rules for Sanskrit grammar, with metarules, transformations, and recursion, similar to modern mathematical notation,
• Pingala (roughly 3rd-1st centuries BC) in his treatise of prosody uses a device equal to a binary numeral system, His discussion of the combinatorics of meters is of the binomial theorem
• The next significant mathematical documents from India after the Sulba Sutras are the Siddhantas, astronomical treatises from the 4th and 5th centuries AD (Gupta period) showing strong Hellenistic influence, significant in that they contain the first instance of trigonometric relations based on the half-chord. Around 500 AD, Aryabhata wrote the Aryabhatiya, a slim volume, written in verse, for rules of calculation used in astronomy and mathematical mensuration, .
• The decimal place-value system first appears. Several centuries later, the Muslim mathematician Abu Rayhan Biruni described the Aryabhatiya as a "mix of common pebbles and costly crystals"
• In the 7th century, Brahmagupta identified the Brahmagupta theorem, Brahmagupta's identity and Brahmagupta's formula, and lucidly explained the use of zero as both a placeholder and decimal digit, and explained the Hindu-Arabic numeral system
• 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.
• Bhāskara II wrote extensively on all branches of mathematics, objects equivalent to infinitesimals, the mean value theorem and the derivative of the sine function although he did not develop the notion of a derivative.
• Madhava of Sangamagrama, found the Madhava-Leibniz series and obtained from it a transformed series, he used to compute the value of π , also found the Madhava-Gregory is to determine the arctangent, the Madhava-Newton power series to determine sine and cosine and the Taylor approximation for sine and cosine functions.
• The 16th century, Jyesthadeva consolidated many of the Kerala School's developments and theorems in the Yukti-bhāṣā.

Islamic empires

• Islamic Empire established across the Middle East, Central Asia, North Africa, Iberia, and in parts of India in the 8th century contributed towards mathematics and since like Greek in the Hellenistic world, texts written in Arabic written by non-Arab scholars.
• In the 9th century, the Persian Muhammad ibn Mūsā al-Khwārizmī wrote on Hindi-Arabic numerals and on methods for solving equations which books spread with Al-Kindi to the West derived algorithm from the Latinization of his name, Algoritmi, and the word algebra from the title of one of his works.
• Abu Kamil extended algebra to irrational numbers, and techniques to Solve non-linear simultaneous with three known varinables finding many solutions
• Al-Karaji methodology to include integer powers of unknown,
• Woepcke, praised Al-Karaji for being "the first who introduced the theory of algebraic calculus.".
• Abul Wafa translated the works of Diophantus into Arabic with Ibn al-Haytham deriving the formula for the sum of the fourth powers
• Ibn al-Haytham was first to find the formula for the sum of integers
• Omar Khayyam wrote Discussions of the Difficulties in Euclid and also was the first to find the general geometric solution to cubic equations.
• Nasir al-Din Tusi (Nasireddin) made advances in spherical trigonometry.

Maya

• Maya civilization developed a unique tradition of mathematics that was entirely independent of existing Egyptian, European, and Asian mathematics.
• Maya numbers used a base of twenty, which was the vigesimal system, unlike a base of ten for most modern cultures,
• Zero standard symbol.

Medieval Euclidean

• Medieval European interest was belief that math was a key to the understanding or nature from Plato's Timaeus and biblical passages.
• Boethius used math for quadrivium - arithmetic, geometry, astronomy, and music
• 12th Century traveling to spain for scientific Arabic texts sparked math renewal
• Leonardo "Fibonacci" Abaci- Hindu-Arabic numeral system by al-Khwarizmi in Liber Abaci in 1202.
• Thomas Bradwardine speed increases in airthmetic ratio as force increases vs resistance by geometric ratio,
• Oxford Calculators measure speed "by the path that would be described by [a body] if... it were moved uniformly with means of the given speed", Heytesbury distances,
• Nicole Oresme* graph distance for contant accelerations each increment is increased as the square of an odd number

Renaissance

• Development of Mathematics often intertwined with accountings since books by teachers intended for young rich people entering commerce

• 1415, Della Francesca writes books on solid Geomtry and Linear perspective with linear perspectives

• Bookkepping was added to math textbooks in 1494, along with symbols of plus and minus.

• 16th contury: de ferro and Niccolo dis covered solutions of cyclic equations published in 1545 book.

• Bombelli showd how to deal with imagininary by cardano's formula

• navigation needed more accurate math

• 1595 Pitiscus first wrote 'Trigonometria'

Mathematics in Scientific Revolution

• 17th century Johannes Kepler laws of planetary motion, Jost bürgi invention of logorithms. Rene Decarties orbits plotted on a graph in cartesian coordinates. Issac physics to explain kepler laws
• Pierre de Fermat, Blaise Pascal probability. Binary # system of Leibniz, Utility theory
• 18th C - Leanhart Euler - graph theory, I for i as square root of -1 π greek for circumference to radius ratio Topology, graph theroy Jospeh - Lagrange Simonde - laplace

Modern

• 19th Abstract Gauss:
• complex Variables,
• geometry
• series

Non euclid by Lobachevsky

• Hyperbolic; no unique parallel, less than 180 degree angles

Riemann;

• Elliptic no parallel
• Reianman - united to three types Abstract algebra: Vector by William Hamilton george boole boolean:

Niel's heinrich abel. - no general method for polynomial ˃4. Cantlor's theory Math foundations

• *20th major profession Hilbert : areas and appilications Millenium prize problmes

hulk and code to RSA algorithm

RamanuJan self taught high composites and their theta functions for prime number theory

• Erdos # co-op papers than history, Erot's number collaborative.
• Emmy studied algrebas

end c specializations 21st cen new algorithms. .