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Mathematicians seek and use patterns [8] [9] to formulate new conjectures ; they resolve the truth or falsity of such by mathematical proof.

When mathematical structures are good models of real phenomena, mathematical reasoning can be used to provide insight or predictions about nature. Through the use of abstraction and logicmathematics developed from countingcalculationmeasurementand the systematic study of the shapes AAn motions of physical objects.

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Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematicsmost notably in Euclid 's Elements. Mathematics developed at a relatively slow pace until the Renaissancewhen mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

Mathematics is essential in many fields, including natural scienceengineeringmedicinefinanceand the social sciences.

Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics mathematics for its own sake without having any application in mind, but practical applications for what began as pure mathematics are often discovered later. The history of mathematics can be seen as an ever-increasing series of abstractions.

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The first abstraction, which is shared by many animals, [14] was probably that of numbers: the realization that a collection of two apples and a collection of two oranges for example have something in common, namely quantity of their members. As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also recognized how to count abstract quantities, like time—days, seasons, or years. The Babylonians also possessed a place-value system, and used a sexagesimal numeral system [19] which is still in use today for measuring angles and time.

Beginning in the 6th century BC with the Pythagoreansthe Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics.

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His textbook Elements is widely considered the most successful and influential textbook of all time. 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 AD in India and were transmitted to the Western world via Islamic mathematics. The most notable achievement of Islamic mathematics Svience the development of algebra.]