Algebra is the branch of mathematics that uses symbols to represent arithmetic operations. One of the earliest mathematical concepts was to represent a number by a symbol and to represent rules for manipulating numbers in symbolic form as equations. For example, we can represent the numbers 2 and 3 by the symbols x and y. From observation we know that it does not matter in which order we add the numbers (2 + 3 = 3 + 2), and we can represent this equivalence as the equation x + y = y + x. The equation is valid no matter what numbers x and y represent. Because algebra uses symbols rather than numbers, it can produce general rules that apply to all numbers. What most people commonly think of as algebra involves the manipulation of equations and the solving of equations.

An area of mathematics research is also called algebra, or modern algebra. It developed after the discovery that laws such as the commutative law (x + y = y + x) held true not only for the addition of real numbers (rational and irrational numbers) but could extend to more complex operations and objects. Interest eventually focused on the concepts themselves and the conclusions that could be drawn about sets of objects with certain properties. Among the objects studied by modern algebra are groups, rings, and fields. Algebra also can be combined with other areas of pure mathematics such as geometry and a branch of geometry called topology.