Many problems that mathematicians study arise from within mathematics itself, or with some inspiration from physics. This is often the situation with pure mathematical areas such as algebra, analysis and geometry. While in the natural sciences physical theories are tested by experiments, mathematical statements are verified objectively by proofs. At an abstract level, mathematicians deal with a collection of assumptions and explore the corresponding consequences.

Mathematicians in the fields of science (for example, physics, chemistry, engineering), social science (for example, psychology, sociology, political science) and business (for example, finance, actuarial science, risk analysis). Through their work, they may:

- apply established methods of numerical analysis, statistics and other known techniques
- develop new and more efficient methods of dealing with mathematical processes.

For example, mathematicians may develop mathematical models of the ocean floor, of changing demographics or of blood flow. Or they may apply number theory to computer-security problems or to the analysis of seismological survey results.

When confronted with any problem that has some mathematical basis, an organization may consult a mathematician for advice about:

- the existence of possible solutions and their accuracy and cost
- ways to better understand the observed behaviour of a system
- the possible behaviours that a configuration might exhibit.

Mathematicians may work in manydifferent areas:

**Applied mathematicians** use mathematical theories and techniques to solve practical problems in business, engineering and the sciences. For example, they may investigate the cost of establishing a new business or the mathematical aspects of drilling for oil.
**Biomathematicians** use mathematical techniques and tools to model natural and biological processes. For example, they may investigate the effectiveness of vaccination programs in preventing epidemics.
**Theoretical mathematicians** seek to advance mathematical science by developing new principles and new relationships between existing principles.
**Mathematical consultants** assist with business and research projects that demand advanced knowledge of mathematics. They may work on mathematical problems in a variety of fields (for example, 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 example, they may use mathematical models to help understand and manage risk.
**Statisticians **and** actuaries** use mathematics to solve problems in finance, business and government.
**Biostatisticians **use statistics to measure the reliability of treatment methods and the performance of drugs.