He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. It has been claimed that megalithic monuments in England and Scotland , dating from the 3rd millennium BC, incorporate geometric ideas such as circles , ellipses , and Pythagorean triples in their design.
Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia modern Iraq from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity. Later under the Arab Empire , Mesopotamia, especially Baghdad , once again became an important center of study for Islamic mathematics. In contrast to the sparsity of sources in Egyptian mathematics , our knowledge of Babylonian mathematics is derived from more than clay tablets unearthed since the s. Some of these appear to be graded homework.
The earliest evidence of written mathematics dates back to the ancient Sumerians , who built the earliest civilization in Mesopotamia. They developed a complex system of metrology from BC. From around BC 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. Babylonian mathematics were written using a sexagesimal base numeral system. 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 Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of regular reciprocal pairs.
Egyptian mathematics refers to mathematics written in the Egyptian language. From the Hellenistic period , Greek replaced Egyptian as the written language of Egyptian scholars. Mathematical study in Egypt later continued under the Arab Empire as part of Islamic mathematics , when Arabic became the written language of Egyptian scholars.
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The most extensive Egyptian mathematical text is the Rhind papyrus sometimes also called the Ahmes Papyrus after its author , dated to c. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge,  including composite and prime numbers ; arithmetic , geometric and harmonic means ; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory namely, that of the number 6.
Another significant Egyptian mathematical text is the Moscow papyrus , also from the Middle Kingdom period, dated to c.
One problem is considered to be of particular importance because it gives a method for finding the volume of a frustum truncated pyramid. Finally, the Berlin Papyrus c. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures.
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All surviving records of pre-Greek mathematics show the use of inductive reasoning, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to derive conclusions from definitions and axioms, and used mathematical rigor to prove them. Greek mathematics is thought to have begun with Thales of Miletus c. Although the extent of the influence is disputed, they 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. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem.
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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. The Pythagoreans are credited with the first proof of the Pythagorean theorem ,  though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers. Eudoxus —c. Though he made no specific technical mathematical discoveries, Aristotle —c. In the 3rd century BC, the premier center of mathematical education and research was the Musaeum of Alexandria.
Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.
Euclid also wrote extensively on other subjects, such as conic sections , optics , spherical geometry , and mechanics, but only half of his writings survive. Archimedes c. Apollonius of Perga c. Around the same time, Eratosthenes of Cyrene c. AD 90— , a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years. Following a period of stagnation after Ptolemy, the period between and AD is sometimes referred to as the "Silver Age" of Greek mathematics. His main work was the Arithmetica , a collection of algebraic problems dealing with exact solutions to determinate and indeterminate equations.
He is known for his hexagon theorem and centroid theorem , as well as the Pappus configuration and Pappus graph. His Collection is a major source of knowledge on Greek mathematics as most of it has survived.
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The first woman mathematician recorded by history was Hypatia of Alexandria AD — She succeeded her father Theon of Alexandria as Librarian at the Great Library [ citation needed ] and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria had her stripped publicly and executed. The closure of the neo-Platonic Academy of Athens by the emperor Justinian in AD 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.
Although ethnic Greek mathematicians continued under the rule of the late Roman Republic and subsequent Roman Empire , there were no noteworthy native Latin mathematicians in comparison. Using calculation, Romans were adept at both instigating and detecting financial fraud , as well as managing taxes for the treasury. The creation of the Roman calendar also necessitated basic mathematics. The first calendar allegedly dates back to 8th century BC during the Roman Kingdom and included days plus a leap year every other year.
At roughly the same time, the Han Chinese and the Romans both invented the wheeled odometer device for measuring distances traveled, the Roman model first described by the Roman civil engineer and architect Vitruvius c. With each revolution, a pin-and-axle device engaged a tooth cogwheel that turned a second gear responsible for dropping pebbles into a box, each pebble representing one mile traversed.
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. 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. 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.
The date of the invention of the suan pan is not certain, but the earliest written mention dates from AD , in Xu Yue 's Supplementary Notes on the Art of Figures. The oldest existent work on geometry in China comes from the philosophical Mohist canon c. The Mo Jing described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well.
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 BC, the Han dynasty BC— AD 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 AD , but existed in part under other titles beforehand.
It consists of 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. The high-water mark of Chinese mathematics occurred in the 13th century during the latter half of the Song dynasty — , with the development of Chinese algebra.
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The most important text from that period is the Precious Mirror of the Four Elements by Zhu Shijie — , dealing with the solution of simultaneous higher order algebraic equations 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 such as Matteo Ricci carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries, though at this point far more mathematical ideas were entering China than leaving. 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. The earliest civilization on the Indian subcontinent is the Indus Valley Civilization mature phase: to 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. As in China, there is a lack of continuity in Indian mathematics; significant advances are separated by long periods of inactivity.
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. Around AD, Aryabhata wrote the Aryabhatiya , a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology. 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 for the first time, in Brahma-sphuta-siddhanta , he lucidly explained the use of zero as both a placeholder and decimal digit , and explained the Hindu—Arabic numeral system. Islamic scholars carried knowledge of this number system to Europe by the 12th century, and it has now displaced all older number systems throughout the world.