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Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications.
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions.German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."
In the present day, the distinction between pure and applied mathematics is more a question of personal research aim of mathematicians than a division of mathematics into broad areas. [124] [125] The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure mathematics". [14]
If mathematics has been informally used throughout history, in numerous cultures and continents, then it could be argued that "mathematical practice" is the practice, or use, of mathematics in everyday life. One definition of mathematical practice, as described above, is the "working practices of professional mathematicians".
Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). Mathematics is used throughout the world as an essential tool in many fields, including natural science , engineering , medicine , and the social sciences .
Pure mathematics is mathematics that studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics , [ 12 ] and at variance with the trend towards meeting the needs of navigation , astronomy , physics , economics , engineering ...
Mathematics majors study additional other areas within pure mathematics—and often in applied mathematics—with the requirement of specified advanced courses in analysis and modern algebra. Other topics in pure mathematics include differential geometry , set theory , and topology .
Despite the close relationship between math and physics, they are not synonyms. In mathematics objects can be defined exactly and logically related, but the object need have no relationship to experimental measurements. In physics, definitions are abstractions or idealizations, approximations adequate when compared to the natural world.