Mathematics includes the study of topics such as quantity (number theory), structure (algebra), space (geometry), and change (mathematical analysis).  It does not have a generally accepted definition.  Mathematicians look for and use patterns to formulate new guesses;  they solve the truth or falsity of the conjectures by means of mathematical tests.
When mathematical structures are good models of real phenomena, mathematical reasoning can be used to provide information or predictions about nature.  Through the use of abstraction and logic, mathematics developed from counting, calculating, measuring, and the systematic study of the shapes and movements of physical objects.

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Practical mathematics has been a human endeavor for as long as there are written records.  The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, especially in Euclid's Elements.  Since the pioneering work of Giuseppe Peano (1858-1932), David Hilbert (1862-1943), and others on axiomatic systems in the late 19th century, it has become common to view mathematical research as the establishment of truth by rigorous deduction of  properly chosen axioms and definitions.  


Mathematics developed at a relatively slow rate until the Renaissance, when mathematical innovations that interacted with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to this day.

Mathematics is essential in many fields, including the natural sciences, engineering, medicine, finance, and the social sciences.  Applied mathematics has given rise to entirely new mathematical disciplines, such as statistics and game theory.  

Mathematicians engage in pure mathematics (mathematics itself) without having any applications in mind, but the practical applications of what began as pure mathematics are often discovered later.  

The history of mathematics can be seen as an ever-growing series of abstractions.  The first abstraction, shared by many animals, was probably that of numbers: the finding that a collection of two apples and a collection of two oranges (for example) have something in common, namely the number of their members. 

Found in bone, in addition to recognizing how to count physical objects, prehistoric peoples may also have recognized how to count abstract quantities, such as time: days, seasons, or years.


Evidence for more complex mathematics does not appear until around 3000 BC.  C., when the Babylonians and Egyptians began to use arithmetic, algebra and geometry for taxes and other financial calculations, for construction and astronomy.  

The oldest mathematical texts from Mesopotamia and Egypt.  They are from 2000 to 1800 BC.  Many ancient texts mention Pythagorean triples and therefore, by inference, the Pythagorean theorem appears to be the oldest and most widespread mathematical development after basic arithmetic and geometry.  

It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication and division) first appears.  in the archaeological record.  The Babylonians also possessed a place value system and used a sexagesimal number system that is still used today to measure angles and time.
From the 6th century BC.  C. with the Pythagoreans, the ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics.  Around 300 a.  C., Euclides introduced the axiomatic method that is still used in mathematics today, which consists of definition, axiom, theorem and proof.  

His Elements textbook is widely considered the most successful and influential textbook of all time.  The greatest mathematician of antiquity is often considered to be Archimedes (c. 287-212 BC) of Syracuse.  

He developed formulas to calculate the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the sum of an infinite series, in a way not unlike modern calculus.

Other notable achievements of Greek mathematics are conic sections (Apollonius of Perge, 3rd century BC), trigonometry (Hipparchus of Nicaea (2nd century BC), and the beginnings of algebra.