algebraically

Algebraically is an adverb used to indicate that something is done or described using algebraic methods, structures, or properties. It denotes reasoning or results derived from algebra rather than from numerical approximation, geometry, or empirical observation.

In everyday mathematical contexts, solving a problem algebraically means manipulating symbols—factoring expressions, expanding products, collecting like terms, and applying the basic laws of arithmetic and algebra—to obtain an exact answer. For example, one might solve for a variable by isolating it through a sequence of symbolic steps. The phrase also appears in more advanced contexts, such as proving statements algebraically or describing objects that satisfy algebraic relations rather than analytic or geometric ones.

In higher mathematics, “algebraically” appears in phrases such as algebraically closed, algebraically independent, or algebraic equations, where the emphasis is on polynomial relations rather than transcendental or numerical properties.

Etymology: the term derives from algebra, itself from the Arabic al-jabr, via Latin and French, with the adverb formed by adding -ly to the adjective algebraic.

See also:

- algebra

- algebraic

- algebraic geometry

- algebraic number theory

- algebraic topology

- algebraic closure