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There are two broad areas of opportunity in mathematics: theoretical and applied. In addition, mathematicians may choose to pursue a career in teaching. The duties performed, the processes involved, the work situations encountered, and the equipment used varies considerably, depending on the institutional or organizational setting.
Theoretical mathematicians deal with pure and abstract mathematical concepts rather than the practical application of such concepts to everyday problems. They might teach in a college or university or work in the research department of a businesses or government office. They are concerned with the advancement of mathematical knowledge, the logical development of mathematical systems, and the study and analysis of relationships among mathematical forms. "Pure" mathematicians focus their efforts mainly on problems dealing with mathematical principles and reasoning.
Applied mathematicians develop and apply mathematical knowledge to practical and research problems in the social, physical, life, and earth sciences. Business, industry, and government agencies such as the National Aeronautics and Space Administration (NASA) rely heavily on applied mathematicians, particularly for research and development programs. Therefore, it is necessary for these mathematicians to be knowledgeable about their employer's operations and products as well as their own field. Applied mathematicians work on problems ranging from the stability of rockets to the effects of new drugs on disease.
The applied and theoretical aspects of mathematicians' work are not always clearly separated. Some mathematicians, usually those dealing with the application of mathematics, may become involved in both aspects. In addition to having general knowledge about modern technology, mathematicians need some basic experience in computer programming and operation because of the reliance on computers by almost every industry.
Specialists in the field of applied mathematics include the following:
Computer applications engineers formulate mathematical models and develop computer systems to solve scientific and engineering problems.
Engineering analysts apply logical analysis to scientific, engineering, and other technical problems and convert them to mathematical terms to be solved by computers.
Operations research analysts employ mathematics to solve management and operational problems.
Weight analysts are concerned with weight, balance, loading, and operational functions of space vehicles, ships, aircraft, missiles, research instrumentation, and commercial and industrial products and systems. These mathematicians use computers to analyze weight factors and work with design engineers to coordinate their specifications with product development.
Mathematics teachers instruct students at the middle school and high school levels. In high school, they provide instruction in more complex mathematics such as algebra, geometry, trigonometry, precalculus, and calculus.
College mathematics professors provide instruction to future mathematicians and students in other disciplines. They often teach courses at various levels of difficulty. Professors usually spend less time in the classroom than high school teachers, but they may have many other responsibilities, including advising doctoral candidates, serving on university or mathematical organization committees, and reading mathematical books and journals to stay up to date regarding developments in the field. Some professors are also actively involved in research and in contributing to the development of the field; this often includes writing and submitting articles on their research to mathematical journals.