Algebra

Algebra is a branch of mathematics in which the general properties of numbers are studied by using symbols, usually letters, to represent variables and unknown quantities.

For example, the algebraic statement

( x + y ) ^ 2 = x ^ 2 + 2 x y + y ^ 2

is true for all values of x and y.

An algebraic expression that has one or more variables (denoted by letters) is a polynomial equation . Algebra is used in many areas of mathematics, for example, matrix algebra and Boolean algebra (the latter is used in working out the logic for computers).

In ordinary algebra the same operations are carried on as in arithmetic, but, as the symbols are capable of a more generalized and extended meaning than the figures used in arithmetic, it facilitates calculation where the numerical values are not known, or are inconveniently large or small, or where it is desirable to keep them in an analyzed form.

Within an algebraic equation the separate calculations involved must be completed in a set order. Any elements in brackets should always be calculated first, followed by multiplication, division, addition, and subtraction.

Algebra was originally the name given to the study of equations. In the 9th century, the Arab mathematician Muhammad ibn-Musa al-Khwarizmi used the term al-jabr for the process of adding equal quantities to both sides of an equation. When his treatise was later translated into Latin, al-jabr became algebra and the word was adopted as the name for the whole subject.

The basics of algebra were familiar in ancient Babylonia (c. 18th century BC). Numerous tablets giving sets of problems and their answers, evidently classroom exercises, survive from that period. The subject was also considered by mathematicians in ancient Egypt, China, and India. A comprehensive treatise on the subject, entitled Arithmetica, was written in the 3rd century AD by Diophantus of Alexandria. In the 9th century, al-Khwarizmi drew on Diophantus' work and on Hindu sources to produce his influential work Hisab al-jabr wa'l-muqabalah (Calculation by Restoration and Reduction).

In modern terminology, an algebraic structure consists of a set, A, and one or more binary operations (that is, functions mapping A × A into A) which satisfy prescribed axioms . A typical example is a structure which had been studied from the 18th century onwards and is known as a group. This structure had turned up in the study of the solvability of polynomial equations, but it also appears in numerous other problems (for example, in geometry), and even has applications in modern physics.

The objective of modern algebra is to study each possible structure in turn, in order to establish general rules for each structure which can be applied in any situation in which the structure occurs. Numerous structures have been studied, and since 1930 a greater level of generality has been achieved by the study of universal algebra , which concentrates on properties that are common to all types of algebraic structure.

The above material is quoted from the Hutchinson Family Encyclopedia , available at http://ebooks.whsmithonline.co.uk/ .