Algebra began as bone-setting, then became equation craft
The Arabic word behind algebra once named a surgical fix for broken bones. In the ninth century a Persian mathematician turned that word into a method for completing and balancing equations, and only much later did the subject expand into today's abstract structures.
School algebra is arithmetic with placeholders: variables stand for unknown or unspecified quantities, and you rewrite equations until the unknown is isolated. Linear algebra pushes the same spirit into systems of linear equations, hunting values that satisfy every equation at once. Abstract algebra steps further still, studying sets equipped with operations—groups, rings, fields—defined by axioms rather than by a fixed domain of numbers. Universal algebra and category theory then look across those types for shared patterns.
Ancient geometry already used algebraic-style problem solving, but the ninth-century work of Muḥammad ibn Mūsā al-Khwārizmī treated the craft as its own discipline, separate from geometry. His treatise title used al-jabr—the same Arabic root that once meant bonesetting—for a method of transforming equations; Latin rendered the book Liber Algebrae et Almucabola, and English borrowed the word in the sixteenth century via Italian, Spanish, and medieval Latin. For a long stretch algebra meant the art of manipulating polynomial equations. Symbolic notation tightened in the sixteenth and seventeenth centuries; by the mid-nineteenth century the subject had outgrown equation-solving alone to cover many operations and axiom systems.
Even vocabulary flexes. Sometimes "algebra" means only school algebra or only the abstract branch. As a countable noun, an algebra can mean a vector space with a bilinear product—or a Lie algebra, or an associative algebra, depending on context. Polynomials still sit near the historical core: degree-two equations have a quadratic formula, degrees three and four have cubic and quartic formulas, and the Abel–Ruffini theorem ruled out a general radical solution for higher degrees. The fundamental theorem of algebra guarantees at least one complex root for a non-constant univariate polynomial with real or complex coefficients, yet it does not hand you a method to compute that root.
Source: Algebra
