What does algebraically mean?
"Algebraically" is a formal adverb meaning "by means of algebra" — that is, using the symbols, equations, and manipulative techniques of that branch of mathematics. It most often appears when contrasting methods: a student might solve an equation algebraically rather than graphically, prove a theorem algebraically rather than geometrically, or show that two expressions are algebraically equivalent. The word carries a precise, technical register and is found chiefly in mathematics education, research writing, and scientific discourse. Its root, "algebra," traces back to the Arabic term al-jabr, meaning "the reunion of broken parts," introduced to Europe through medieval translations of mathematical treatises. Knowing this word is useful because it names not just a technique but a whole way of thinking about problems — one grounded in symbolic manipulation rather than pictures or estimates. It is a single-sense adverb with no plural form, and its closest grammatical relatives are the adjective "algebraic" and the noun "algebra."
adverbIn a manner relating to, or by means of, algebra; using the methods or notation of algebra rather than other approaches such as geometry or numerical computation.
- By means of, or in terms of, algebra — solving, proving, or expressing something using algebraic equations and symbols.
- In a manner characteristic of algebra as a branch of mathematics, as opposed to geometric, numerical, or graphical approaches.
"The problem can be solved algebraically by setting up an equation for the unknown quantity."
"Rather than estimating the answer graphically, she solved the system algebraically."
"The two expressions are algebraically equivalent, even though they look quite different."
The word 'algebra' comes from the Arabic 'al-jabr', meaning 'the reunion of broken parts' — so every time you solve something algebraically, you're literally reassembling what's broken.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of algebraically
Algebraically derives from "algebraic," the adjective form of "algebra," which itself entered European languages from Arabic. The Arabic term al-jabr, meaning "the reunion of broken parts," appeared in the title of the 9th-century treatise by the mathematician al-Khwarizmi and was transmitted into Latin during the medieval translation movement. From there it developed into the modern English noun "algebra" and its adjective "algebraic(al)," from which the adverb was formed with the standard "-ly" suffix. Cognate terms sharing this lineage include "algorithm" (from the related name al-Khwarizmi) and "algebraist," one who specializes in algebra.
Related word forms
How algebraically is actually used
A formal, technical adverb used almost exclusively in mathematical, scientific, and educational contexts. It typically signals a contrast with other methods of attack on a problem — geometrically, numerically, or graphically — and is common in textbooks, proofs, and classroom instruction.
Easily confused with algebraically
Algorithmically means by means of a step-by-step procedure or algorithm, whereas algebraically means using the symbolic methods and equations of algebra.