In lower mathematics, it means the study of some well-known algebraic operations and their algebraic properties.

Most of the algebra in school is concerned with the operations +, -, * and / on natural, integral, rational and real numbers.

By extending + and * to work on vectors of numbers, we get linear algebra, which models natural manipulations on objects in n-dimensional space, such as rotation, translation and scaling.


An algebra is a set with one or more operations defined on it; an operation in this context is a function whose arguments and results are all in the set.

The subject of algebra is the study of combinatorial equivalences of these operations. It isn't at all interested in what the set or the operation(s) represent, but only in the mathematical laws that hold in combining them, and the theory which follows from that.

Group theory for instance is the branch of algebra concerned with groups, a very simple class of algebras that arises often in practice.