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Mathematics is the study of topics such as quantity (numbers),[2] structure,[3] space,[2] and change.[4][5][6]
There is a range of views among mathematicians and philosophers as to the exact scope
and definition of mathematics.[7][8]

Mathematicians seek out patterns[9][10] and use them to formulate new conjectures. Mathematicians resolve
the truth or falsity of conjectures by mathematical proof.

When mathematical structures are good models of real phenomena, then mathematical reasoning
can 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 for as far back as written records exist.
The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind.
There is no clear line separating pure and applied mathematics, and practical applications
for what began as pure mathematics are often discovered.[18]