The mathematical sciences investigate the relationships between things that can be measured or quantified in either a real or abstract form. Pure mathematics differs from other sciences because it deals solely with logic, rather than with nature's underlying laws. However, because it can be used to solve so many scientific problems, mathematics is usually considered to be a science itself.
Central to mathematics is arithmetic, the use of numbers for calculation. In arithmetic, mathematicians combine specific numbers to produce a result. A separate branch of mathematics, called algebra, works in a similar way, but uses general expressions that apply to numbers as a whole. For example, if there are three separate items on a restaurant bill, simple arithmetic produces the total amount to be paid. But the total can also be calculated by using an algebraic formula. A powerful and flexible tool, algebra enables mathematicians to solve highly complex problems in every branch of science.
Geometry investigates objects and the spaces around them. In its simplest form, it deals with objects in two or three dimensions, such as lines, circles, cubes, and spheres. Geometry can be extended to cover abstractions, including objects in many dimensions. Although we cannot perceive these extra dimensions ourselves, the logic of geometry still holds.