Algebra

Algebra, branch of mathematics in which relationships between known and unknown quantities are represented symbolically. For a relationship to satisfy the fundamental theorem of algebra it must consist of a finite number of quantities and must have a solution. An example of such a relationship taken from elementary algebra is: axⁿ+bxⁿ⁻¹ +cxⁿ⁻²+ … +z=0

This is an “n degree” polynomial equation (of order n). Here x is a variable denoting an unknown quantity to be found, and a, b, c … z represent known values. Elementary algebra, the algebraic system most familiar to the general public, uses operations of arithmetic to solve equations from sets of numbers. Abstract algebra developed from elementary algebra by mathematicians attempting to solve specific problems. Mathematical structures such as fields, rings, and groups were devised. Concepts of abstract algebra have been used by theoretical physicists in the development of quantum theory as well as by digital communications engineers in the development of coding theory. Linear algebra is used to solve simultaneous linear equations and is applied extensively in economics and psychology. Manipulations of equations are accomplished through the use of matrices and vectors. Boolean algebra is a symbolic representation of classical logic developed in 1854 by George Boole. Operations such as union and intersection are used. This algebraic system is used in computer science.

Gradual introduction of algebraic symbols occurred between 2000 B.C. and 1550 A.D. Arithmetica, regarded as the first treatise on algebra, was written by Diophantus of Alexandria in the 3rd century A.D. The Arabs became leaders in the field in about the 9th century. It was not until the 16th and 17th centuries in Europe that algebra underwent a complete transformation and became almost completely symbolic, much as it is today. Abstract algebra developed in the early 19th century, with major contributions by Niels Abel and Evariste Galois.