Algebra

Algebra is often referred to as a generalization of arithmetic. As such, it is a collection of rules: rules for translating words into the symbolic notation of mathematics, rules for formulating mathematical statements using symbolic notation, and rules for rewriting mathematical statements in a manner that leaves their truth unchanged.

The power of elementary algebra, which grew out of a desire to solve problems in arithmetic, stems from its use of variables to represent numbers. This allows the generalization of rules to whole sets of numbers. For example, the solution to a problem may be the variable x or a rule such as ab=ba can be stated for all numbers represented by the variables a and b.

Elementary algebra is concerned with expressing problems in terms of mathematical symbols and establishing general rules for the combination and manipulation of those symbols. There is another type of algebra, however, called abstract algebra, which is a further generalization of elementary algebra, and often bears little resemblance to arithmetic. Abstract algebra begins with a few basic assumptions about sets whose elements can be combined under one or more binary operations, and derives theorems that apply to all sets, satisfying the initial assumptions.