A mathematician uses numbers and symbols in many ways, from creating new theories to translating scientific and technical problems into mathematical terms. Some mathematicians are more focused on pure mathematics. The pure or theoretical mathematician is more likely to teach and do research at universities or other research institutes, while the applied mathematician is likely to work for business, government or industry.
Some mathematicians have their own consulting firms.
A mathematician appreciates beauty, symmetry, and order in nature and in logical and analytical thought. She should have a logical mind, a sense of curiosity, the desire and ability to solve problems, and numerical aptitude. A mathematician cannot be easily discouraged, for solving research problems often requires months of work. Some mathematical problems have remained unsolved for centuries. An applied mathematician must be able to communicate effectively and bring structure and analytical rigor to what is often a morass of confusing information.