The term algebra comes from Arabic (al-jabr) and is nothing but a branch of mathematics that allows the expression of the properties of operations (additions, subtractions, multiplications and divisions...) as well as the treatment of equations, leading to the study of algebraic structures. Originally, and this is very telling, the Arabic word al-jabr means "the reduction of a fracture", "the reunion or reunification (of pieces)", or even "the reconstruction", "the connection", or even "the restoration". As for the mathematical context, in this specific case, the term algebra refers to the transformation of an equation by the addition of a term.
The word "algebra" is simply derived from the title of a work written by a Persian mathematician around 825, a title which meant "A Summary of Calculations by Restoration and Comparison". This book had very clear practical objectives and symbolizes the advent of complex calculations, which allowed for the development of human societies and civilizations: calculations related to inheritance, surveying, commercial exchanges, etc.
At the time, algebra naturally fit into the chronology of the development of Islamic science and technology. Depending on the period and the level of education considered, this branch of mathematics called algebra can be described in several ways:
- It can be a generalized arithmetic, extending the usual operations on numbers to different objects or quantities.
- It can also be the theory of equations and polynomials.
- Finally, since the beginning of the 20th century, it is the study of algebraic structures (in this case, we speak of general or abstract algebra).
It is good to know that the scope of algebra ranges from arithmetic problems (dealing with numbers themselves...) to geometric problems such as analytic geometry or complex numbers. This is why algebra occupies a special place that plays more or less the role of a hinge, an axis, between arithmetic and geometry, allowing for the extension and unification of the numerical domain.
To be complete on the subject, and without going into details that have their place in theses on the subject, it should be noted that this qualifier of "algebraic" is also given to other parts of mathematics whose objects or methods involve algebra.
- For example, this may be algebraic topology, which applies algebraic tools to the study of topological spaces, while seeking to naturally associate algebraic invariants with associated topological structures.
- It may also be algebraic geometry, which is the part of geometry that studies algebraic curves or varieties, i.e. curves or varieties defined by polynomial equations, with techniques themselves often derived from algebra.
- Finally, it may be algebraic analysis, a branch of mathematics that uses, among other things, complex analysis to study the properties of hyperfunctions.