When you say "Theoretical mathematicians are usually found working at the university level", does that mean they work as professors in colleges?
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Mathematicians are people who specialize in the study of mathematics, the manipulation of numbers, variables, and abstract concepts in equations and theorems. They can work on a variety of topics, from the history of math to theoretical mathematics, and they work in many industries, including computer science and engineering. Mathematicians are also an important part of the research teams which work on issues like geology, demographics, physics, and so forth, and they can be found in a variety of settings.
The study of mathematics is ancient. Humans have been working with numbers for both mundane and theoretical tasks for thousands of years, and they have also used math to learn more about the world and the universe. Mathematicians were responsible for calculating the data which allowed people to understand how the seasons worked, how to build huge structures, and how to perform a variety of other tasks. They also contributed immensely to the advancement of the sciences in many cultures.
Some mathematicians specialize in applied or practical mathematics, which involves the application of mathematics to real world issues. A statistician who compiles employment statistics, for example, could be considered an applied mathematician. These mathematicians often work with existing mathematical knowledge to interpret information. Their skills are useful in a number of settings, from architecture offices to zoological parks.
Theoretical mathematicians work on the theory, rather than the practical application, of math. Some work on notable problems, such as Fermat's last theorem, while others develop new mathematical theorems which have interesting applications or connotations. For example, a mathematician can use math to think of an entirely new way to approach computer science, or to describe the dynamics of crowds. Theoretical mathematicians are usually found working at the university level.
Most mathematicians have a doctoral degree, and many have completed post-doctoral studies as well. They can work in academic settings as well as in offices of government agencies and other companies, and they may utilize a variety of tools in their work, including computer systems which are capable of very complex and detailed calculations. Deductive logic and critical thinking skills are also very important for professionals in this field, as is the ability to make intuitive leaps, in the case of people who work with theoretical math.
Some notable mathematicians include: Lewis Carroll, Pythagoras, Christopher Wren, John Nash, Albert Einstein, Hypatia of Alexandria, and Emmy Noether. Many of these individuals are also noted for other areas of expertise, such as physics, in addition to mathematical skills.
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