algebra
Algebra is arithmetic with the brakes off. In arithmetic you work with definite numbers — you add 3 and 5 and get 8. In algebra you also work with letters that stand in for numbers you do not yet know, or for any number at all, so that a single sentence can describe a whole family of calculations at once.
More precisely, algebra is the branch of mathematics that studies the rules for combining symbols. The symbols usually represent numbers, and the rules — like the order of operations and the laws that let you rearrange a sum or a product — are the same whether the symbol is a 7 or an x. This lets you state truths such as “a + b equals b + a” that hold for every choice of a and b, and to solve for an unknown by manipulating both sides of an equation until the unknown stands alone.
The word comes from the Arabic al-jabr, meaning roughly “the reunion of broken parts,” from a 9th-century book by al-Khwarizmi describing how to balance and restore equations. What started as a method for solving equations grew, over centuries, into the study of abstract structures — groups, rings, and fields — but every advanced layer still rests on the basic idea of reasoning about quantities through symbols.
Arithmetic says 2 + 3 = 5. Algebra says x + 3 = 5, and asks: for which x is this true? Answer: x = 2.
The leap from a fixed sum to an unknown to be found is the heart of algebra.
“Algebra” names both a school subject (elementary algebra: variables, expressions, equations) and a vast research area (abstract or modern algebra: groups, rings, fields). When someone says they are “learning algebra,” they almost always mean the first.