Mathematics
(from Greek μάθημα máthēma,
"knowledge, study, learning") includes the study of such topics
as quantity (number theory), structure (algebra), space (geometry),[1] and change (mathematical analysis). It
has no generally accepted definition.
Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity
of conjectures 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 logic,
mathematics developed from counting, calculation, measurement, and the systematic study of
the shapes and motions of physical objects. 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 mathematics,
most notably 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 customary to
view mathematical research as establishing truth by rigorous deduction from
appropriately chosen axioms and definitions. Mathematics developed at a
relatively slow pace until the Renaissance, when 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.[11]
Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and 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
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