Algebra, from the Arabic al-jebr, meaning “reunion of broken parts”, is one of the wide-ranging parts of mathematics, combined with number theory, geometry, and analysis. Algebra can basically be considered as doing computations much like that of arithmetic with non-numerical mathematical objects. Initially, these objects were variables that represented either numbers that weren’t yet known or unspecified numbers, permitting one to state and prove properties that are true no matter which numbers are involved. Presently, algebra is split up into several subareas including linear algebra, group theory, ring theory, and combinatorics.
Elementary algebra is a part of algebra that is normally taught in elementary courses of mathematics. Abstract algebra is the name that is normally given to the study of the algebraic structures, such as rings, fields, groups, and algebras, themselves.
Algebra is also the name of a variety of specific mathematical structures that occur in algebra.
The adjective “algebraic” normally is in reference to the relation to algebra, as in “algebraic structure”. For historical reasons, it might also be in reference to the relation with the roots of polynomial equations, like in algebraic numbers, algebraic extension, or algebraic expression. This comes from the fact that, until the end of the 19th century, algebra was basically the same area as the theory of equations. A witness of that is the fundamental theorem of algebra, which nowadays, isn’t considered to belong to algebra.
Image Caption: The general formula for quadratic equations (ax^2+bx+c=0). Credit: Jennifer Ledwith/Wikipedia