Many problems that mathematicians study arise from within mathematics itself, or with some inspiration from physics. This is often the case with areas of pure math such as algebra, analysis, and geometry.
In natural science fields, scientists test physical theories with objective experiments. Mathematicians verify mathematical statements objectively using proofs. At an abstract level, mathematicians deal with assumptions and explore their consequences.
Mathematicians work in science (such as physics, chemistry, and engineering), social science (such as psychology, sociology, and political science), and business (such as finance, actuarial science, and risk analysis). Through their work, they may:
- Apply established methods of analyzing numbers and statistics
- Develop new and more efficient methods of dealing with numerical processes
For example, mathematicians may develop models of the ocean floor, changing demographics, or blood flow. Or they may apply number theory to computer-security problems or seismological (earthquake) survey results.
When confronted with any problem that has a numerical basis, an organization may consult a mathematician for advice about:
- The existence, accuracy, and cost of possible solutions
- Ways to better understand the observed behaviour of a system
- The possible behaviours that a configuration might exhibit
Mathematicians may work in many different areas.
Applied mathematicians use math theories and techniques to solve practical problems in business, engineering, and science. For example, they may explore the cost of creating a new business or certain aspects of drilling for oil.
Biomathematicians use math techniques and tools to model natural and biological processes. For instance, they may research the effectiveness of vaccination programs in preventing epidemics.
Theoretical mathematicians seek to advance mathematical science. They develop new principles and new relationships between existing principles.
Mathematical consultants help with business and research projects. They may work on math problems in fields such as mechanics, electromagnetic theory, economics, communication networks, or the petrochemical industry.
Financial mathematicians develop mathematical and probability models of stock markets, options, and currency futures. For instance, they may use mathematical models to understand and manage risk.
Statisticians and actuaries use math to solve problems in finance, business, and government.
Biostatisticians use statistics to measure how well drugs and other treatment methods perform.