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Tokens and tokens in clay envelops were found with their impressions on the outside. These tokens were probably used for accounting to claim resources. This eventually created the need for a written number system where symbols were used to represent numbers.
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Sargon of Akkad conquers Mesopotamia including Sumer. The Sexagesimal Place-value system (SPVS) with base 60 was developed during this time and would be used throughout Mesopotamia and Greece
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Gutians invade Akkadian Empire in 2250BC, but Sumerians defeat the Gutians around 2112BC. They develop City States that are each ruled mini-empires.
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Ur III is one of the strongest Sumerian city state. Scribes are the people learning to do mathematics at this time. They focus on applied problems with very recipe like procedures in the context of real world problems. Ur-Namma, the last king of Ur III establishes equity by standardizing weight and measurement.
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During this period, scribes are trained in school like in Mesopotamia. There are many Egyptian mathematical texts with writing in hieroglyphs. They developed a system of symbols to represent numbers, and they developed a way to express fractions in the form of reciprocals.
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Isin: 2017-1763BC. Larsa 1897-1763BC. These are two more strong Sumerian city states. A tablet by the name of Pimplton 322 which uses the SPVS base 60 number system was found to have probably come from this time.
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Babylon started as an Akkadian town and then became another Sumerian City State. Hammurabi ruled as king until 1749BC for what is now known as Old Babylonian Empire. He created a code known as Hammurabi's Law Code which helped develop administration of empires. Eventually, the Hittites took over the Old Babylonian Empire.
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Oracle Bones have been found from this time period which show evidence for a Chinese decimal, positional number system.
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Indo-Aryan Nomads settling in Northern India and establishing small kingdoms. They established the Caste System. Brahmanas (priests) are doing math handed down from father to son. They have math texts like Sulbusutras on understanding the sacred Vedas. They also have texts on mathematics relating to building fire alters using geometry.
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Counting mats first appear as a way to do basic arithmetic. Private schools are string to appear.
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At this time, there is a lot of conquest of empires. This is the Neo-Assyrian Empire. People come in and out of rule at this time
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Thales is a Greek philosopher who had a larger impact on the development of mathematics. He is known as the father of deductive reasoning or deductive proofs. He used this in his theorems for geometry.
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This period births many Greek philosophers. They focus on natural philosophy like substance and change. Most philosophers deduce conclusions with models and did not use experiments.
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Pythagoras is a cult leader who established rules for living which could influence one's fate of death. He tales a lot of stories that modernly we don't believe. The pythagoreans were followers of Pythagoras who more definitely studied math.
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During this time, the land was under rule of Neo-Babylonian Empire.
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This is a Babylonian center of astronomy and math. We start to see the rise of Babylonian astronomy where people make connections to things in the heavens to on Earth. Procedural texts and lists begin to be created of the heavens in which people will study patterns and use arithmetic to try and predict what will happen.
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He is one of the people of this time to actually perform experiments to prove harmonic correspondences. He describes ratios in music with whole numbers.
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This time period started the rise of larger Kingdoms in India amd the first empire, Mauryan Empire. There is a rise of Buddhism and Jainism religions. Ashoka introduces Brahmi Script which is an additive/decimal numerical system. Pengala does work on combinatorics with poetic meter. These texts are in sutra style. Brahmanas or priests, Buddhist monks, and merchants perform most of the mathematics for commerce, religious study, and exploration.
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This time period is the birth of more philosophers in Greece like Plato, Aristotle, and Eudoxus.
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He was a philosopher who believed the world is the product og harmonious things fitting together. He describes the unlimited with the limited
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Archytas was a philosopher who believed every magnitude to be whole number ratios known as commensurability. He showed it is impossible to divide basic musical intervals "in half."
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Eudoxus applies a lot of Aristotle's thoughts to mathematics. He comes up with a resolution for the crisis of incommensurability with his work in ratios. He uses a method of exhaustion that enables different proofs in geometry. Eudoxus also creates a geocentric model of the universe.
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Plato was an aristocrat philosopher who works with only perfect world theoretical mathematics. He describes the world of forms and the world of materials. Plato believes knowledge comes from remembering forms.
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Aristotle was a student of Plato who, unlike Plato, believes knowledge comes from experience and the abstraction from those experiences via reason. He uses reasoning via deductive reasoning and makes claims based on "earlier" claims on top a foundation of claims. He also explains the difference between potential and actual infinity.
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Ptolemy is one of Alexander the Great's friends that rules part of his land after he dies. In this land is the capital, Alexandria. This is the city of economic, political, and cultural power. It houses a museum and library of Alexandria with many texts that scholars go to to study mathematics.
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Alexander the Great rules the land under the Greeks until death in 323BC. After the his death, his friends each rule different parts of his land. Seleucus rules the Seleucid Empire. They continue to study astronomy through the use of texts and lists. They use the base 60 system to measure time of day, and they use the zodiac to measure time of year.
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This is a period in Greece after the death of Alexander the Great
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Euclid studied in Alexandria and his goal was to organize and unite all known theoretical mathematics. His most famous work was the book, "The Elements" which include principles or geometry, proportions, number theory, and 3D geometry.
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Aristarchus is a mathematician and astronomer from the island of Samos. He used hypotheses to measure the distance of the sun and moon via geometry.
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Eratosthenes uses Euclid's ideas in "The Elements" to measure. For example, he used similar triangle to measure the Earth. He then combines the size of the Earth and travel logs to create maps.
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The period has an extensive use of bureaucracy. There is an establishment of an extensive Academy and other provincial schools. This time period marks the beginnings of trade with the west which evolves into the Silk Road. Zhou bi is an anonymous author texts written on calendrical astronomy. Suan Shu Shu is a text on basic calculations. Nine Chapters is a manual of bureaucratic math. Bureaucrats are mostly doing math now for accounting, assessment, astronomy, and calendars.
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There is a continuation of large kingdoms including rise of Gupta Empire. The Silk Road, a trade route, starts to flourish. Astronomical texts by Siddhantas, trigonometrical sine tables by Aryabhata, systems of congruences called pulverizer by Brahmagupta are all developed. The first solid evidence for zero is found with a positional system in Bakshali Manuscript. Royally Patronized scholars and Buddist monks are doing mathematics and are teaching others in universities.
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Hipparchus is from modern day Turkey. He studies mathematics with astronomy. He used theoretical based mathematics to do applied math.
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Still under Greek rule, they continue to study Babylonian Astronomy and use arithmetic to create texts and predict what will happen. They believe there is a connection between the things in the sky to things on Earth.
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Claudius Ptolemy used Hipparchus' and Euclid's work in the Roman Period (31BC to 395AD) to measure things. He is credited to using parts of circles and measuring its chord length. This was the first trigonometric function: the chord of an angle.
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This is a very complicated time period for Ancient China as many people come in and out of power. Mathematical developments during this period include Liu Hui's adds commentating to Nine Chapters wit explanations to the problems. Zu Chongzhi creates a daming calendar, makes an approximation on pi and volume of sphere
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This time period includes the establishing of institution of civil service exams for placement in the bureaucracy. A major math text studied during this time is the Ten Mathematical Manuals. Math is used for bureaucratic work like assessment, measurement, astronomy, and calendar making.
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The Abbasids conquered the Umayyad Caliphate in 750. The Abbasids rule for a long time. They establish a new capital, Baghdad, with access to trade routes and encourage scholarly work. In 900s, the power of caliphs starts to erode from powerful families in provinces establishing Dynasties. 1258 is the end of the end of Abbasid Caliphate when Helugu Khan in the Mongol Invasion kills the last caliphs.
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Under the Abbasid Caliphate, the The House of Wisdom, a cultural and scholarly center, was established in 762 and flourishes under the rule of al-ma'mun. Many mathematicians study and do work in the House of Wisdom.
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Al-Kindi works at the House of Wisdom under Abbasid Caliphate. He does work with mathematical cryptology and language studies by inventing frequency analysis.
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Al-Khwarizmi is a mathematician studying in House of Wisdom. He does work in equation solving, translated many Indian texts, he introduce Hindu numbers, and work in classifying the quadratics. He is also appointed by al-Ma'mun to measure the Earth in which he leads a team into desert to measuring one degree.
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Thabit ibn Qurra works at the House of Wisdom. He is a translator of Greek mathematical works (Euclid, Ptolemy, etc.) He also does work in trigonometry with translating chord tables into sines and paved the way to real numbers by writing Hellenistic Greek ratios as fractions.
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Abu al-Wafa works at House of Wisdom. He is associated with work in trigonometry and spherical trigonometry (spherical law of sines). He made a handbook for artisans on dissections with tiling.
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When the Abbasids overthrew Umayyads, one prince made to Spain and establishes the Emirate of Cordoba. In 929, this is renamed to the CALIPHATE OF CORDOBA. Abd al-Rahman III is the ruler/patron, and he establishes the University of Cordoba.
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Al-Uqlidisi works at House of Wisdom. He works with base ten system and decimal fractions. He also does development with inheriting and using Hindu-Arabic system
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Egypt is part of the Abbasid Caliphate for a long time until it is taken over by the FATIMID CALIPHATE. This Caliphate found a capital, the city of Cairo where there is established the Al-Azhar University. Ibn
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al-Haytham is a mathematician under the Fatmid Caliphate. He does work in physics, optics, geometry of reflections, and volume of paraboloids. He also participates in the critiques of Ptolemy saying that his data measurements and model don't agree. al-Haytham solves the fourth degree polynomial by using two conic sections in a way that proceeds Omar Khayyam.
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GHAZNAVID EMPIRE is part of the Abbasid Caliphate.
The Mahmud of Ghazni is the ruler/patron. al-Biruni is a notable mathematician doing wok in this empire. He shows interests in geography, measures circumference of Earth using mountains and law of sines, and is famous for the qibla problem to find direction of Mecca using spherical trigonometry. -
The Caliphate fractures into small kingdoms called Taifas due to Al-Hakem's death leaving his young son to rule. It is seized power of Almonzor who orders al-Hakem's library to be expunged of all objectionable material. The land then goes back and fourth between being run by a bunch of small kingdoms and then overrun by Dynasties from the south.
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SELJUK EMPIRE is one of the empires to start to taker over the Abbasid Caliphate. Malik Shah I is the ruler/patron. Omar Khayyam is a mathematician working in this empire. He does work in algebra and geometry especially with cubic equations. He is also famous for his work in classifying cubics.
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Al-Samawal is a mathematician who works still under the Abassid Caliphate but while the region is under Seljuks rule. He does work with decimal fractions, polynomials, and systematizing exponents includes negative exponents.
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Europeans are less interested in scholarly work. They had large churches in big cities with schools attached, but math is largely overlooked. Merchants are large driving force of mathematics with focus of practical math for commercial work where they picked up new ideas and techniques. The influx of Greek and Arabic texts renewed interest in math. Frederick II is an outlier with his weird science but also believes in doing similar things with universities and translators.
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Leonardo of Pisa, also known as Fibonacci, was a merchant in Western Europe who encountered various systems of numbers and arithmetic. He, after 3 prior fails, was an advocate for the Hindu-Arabic System. In 1202, he wrote a book, Liber Abaci. This book helped adopt the Hindu-Arabic numeral system and laid how how to use it. He is also known for the Rabbits problem, an example from his book, that derives the Fibonacci Sequence.
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MONGOL INVASION is a full scale invasion of Central Asia. This creates giant empire, but eventually breaks up into 4 Khanates.
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CHAGATAI KHANATE is 1/4 khanates as result from Mongol Invasion. They are not as conducive to mathematics as other Khanates because Central Asia was so destroyed. They are not as interested in Urban centers, Persian or Islamic cultures, and they kept to their nomadic ways and religion.
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ILKHANATE is another 1/4 khanates as result of Mongol Invasion.
al-Tusi is a mathematician doing work at this time period. He is credited with the creation of observatory at Maragha which brings together scholars from Islamic World. This is done under the patronage of Helugu Khan. -
Ulugh Beg is the ruler/patron of the TIMURID EMPIRE. This empire established power shortly after the Mongol Invasion when conditions were unstable. Al-Kashi is a major mathematician in this empire. He does work in trigonometry like producing similar version of law of cosines useful for surveying. He also created many indigenous devices for measurement and calculation and created precise sine tables.