What is algebra?

Colin Maclaurin in his 1748 algebra text defined it like this:

“Algebra is a general method of computation by certain signs and symbols which have been contrived for this purpose, and found convenient. It is called a universal arithmetic, and proceeds by operations and rules similar to those in common arithmetic, founded upon the same principles.” (Katz & Barton, 2007, p. 185).

Leonhard Euler, in his own algebra text in 1770, defined algebra as:

“the science which teaches how to determine unknowns’ quantities by means of those that are known.” (ibid., p. 185).

Katz and Barton (2007) categorize the historical development of algebra in four stages: the rhetorical stage, the syncopated stage, the symbolic stage and the purely abstract stage, but they also name another four conceptual stages:

“the geometric stage, where most of the concepts of algebra are geometric; the static equation-solving stage, where the goal is to find numbers satisfying certain relationship; the dynamic function stage, where motion seems to be an underlying idea; and finally the abstract stage, where structure is the goal” (p. 186).

As an old science, algebra has a complicated historical background according to Katz and Barton (2007). Algebraic procedures have developed slowly. There are different opinions about where the evolution of the term “algebra” started. It is commonly believed that algebra first appeared among the Egyptians, the Babylonians, the Greeks or the Arabs. The geometrical influence on algebraic reasoning was strong in ancient Greece. However, the word algebra originated in Baghdad, where the Arabic scientist al-Khwarizmi (A.D. 780-850) published a short book about calculating with the help of al-jabr (restoration) and al-muqabala (reduction). Today’s algebra has its root in Arabic algebra. Western mathematics tended to turn algebraic operations into symbols and later developed abstract algebra. The process of algebra development was slow and the whole history lasted 4000 years (Katz & Barton, 2007).