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The original meaning of the word ‘algebra’ was the solution, by mathematical manipulation, of simple equations (of the form now called linear equations, where only the unknowns appear and not their squares or cubes or other powers). It was a study originated by the Arabs, and the word itself is a corruption of the name of Al Jabr, an 8th-century Arab mathematician whose book was the first on the subject to be translated into Western languages. Algebra in this sense always aims to discover the number called the unknown and usually denoted by ‘x’ (though there can be more than one unknown) through arithmetical operations on a system of equations, which the unknown is supposed to satisfy all at once (this is why such systems are often known as simultaneous equations).
Algebra was able to develop because of the simplification of arithmetic caused by the development of a place dependent number system and the introduction of the number zero by Indian mathematicians. From this origin, algebra took two main turns.
The first is known as linear algebra, and is the study of systems of linear equations in greater generality than that described above; the aim is to say when systems of equations have any solutions and, if they do, when the solutions are unique. This is closely related to the concept of the vector in the Cartesian co-ordinate system, and led to the formulation of the concept of the vector space and of dimension. Much of the theory in this area has become associated with analysis and topology, in which context it is known as functional analysis, and has turned out to be a simple way to describe quantum mechanics mathematically. One of the most important concepts in this field is that of linear transformations, which are functions which preserve addition (in the sense that f(a + b) = f(a) + f(b)). (See linearity).
The second area is more diverse, and is generally given the title abstract algebra, though axiomatic algebra would perhaps be a better name. Here, those properties of number which are used in linear algebra to solve equations are treated as axioms (see axiomatization), and the properties of these systems of axioms are investigated. An example would be the commutativity of multiplication, the fact that for any two numbers ab = ba; it would be possible to study the properties common to all mathematical structures that have an operation which is commutative. The different branches of this kind of algebra are usually named after the system of axioms used; so we have group theory, ring theory and field theory (in which the objects of study are groups, rings and fields respectively). One of the main subjects of enquiry in this type of algebra, the homomorphism, is similar to the linear transformation in linear algebra. A homomorphism is a function from one group, ring or field to another, which preserves the properties of the operation(s) in the same way that a linear transformation preserves addition. A relatively new branch of study is that of category theory, which looks more generally at the properties of these algebraic objects; the objects of study here are classes (examples would be groups or rings) and functors, which are the homomorphisms between members of the same class. The results in the field are very powerful, but particularly difficult mathematically, because of their extremely general nature.
Algebra is important in mathematics because its use has revolutionized many other areas of the subject. For example, there are several branches of mathematics devoted to the use of algebra in other contexts, such as algebraic topology, algebraic geometry and algebraic number theory. It introduces standard notation to many other problems, and by doing this shows up similarities between branches of mathematics—a process which greatly eases the search for solutions to the problems. So its importance lies in the applications it has in other areas of mathematics. SMcL
Further reading L. Nový, Origins of Modern Algebra. |
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