The study of systems consisting of arbitrary sets of elements of unspecified type, together with certain operations satisfying prescribed lists of axioms. Abstract algebra has been developed since the mid-1920s and has become a basic idiom of contemporary mathematics. In contrast to the earlier algebra, which was highly computational and was confined to the study of specific systems generally based on real and complex numbers, abstract algebra is conceptual and axiomatic. A combination of the theories of abstract algebra with the computational speed and complexity of computers has led to significant applications in such areas as information theory and algebraic geometry (Illustration).

One of the four fundamental operations of arithmetic and algebra. The symbol + of addition is thought to be a ligature for “et,” used in a German manuscript of 1456 to denote addition. Its first printed appearance is in Johann Widman's Behennede und hüpsche Rechnung, Leipzig, 1489. As a symbol of operation, the plus sign appeared in algebra before arithmetic, and now the term addition, together with its symbol, is applied to many kinds of objects other than numbers. For example, two vectors x, y are added to produce a third vector z obtained from them by the “parallelogram” law, and two sets A, B are added to form a third set C consisting of all the elements of A and of B. See also: Calculus of vectors

The branch of mathematics dealing with the solution of equations. These equations involve unknowns, or variables, along with fixed numbers from a specified system (Fig. 1). The origins of algebra were based on the need to develop equations that modeled real-world problems. From this came a very extensive theory based on the need to find the values that can be successfully used in the equations.

The study of zero sets of polynomial equations. Examples of zero sets of polynomial equations studied in algebraic geometry are the parabola y − x² = 0, thought of as sitting in the (x, y)-plane, and the locus of all points (t, t², t³, t⁴), which is defined by Eqs. (1),

(1)

A statistical technique that partitions the total variation in experimental data into components assignable to specific sources. Analysis of variance is applicable to data for which (1) effects of sources are additive, (2) uncontrolled or unexplained experimental variations (which are grouped as experimental errors) are independent of other sources of variation, (3) variance of experimental errors is homogeneous, and (4) experimental errors follow a normal distribution. When data depart from these assumptions, one must exercise extreme care in interpreting the results of an analysis of variance. Statistical tests indicate the contribution of the components to the observed variation. See also: Experiment; Statistics

A branch of mathematics in which algebra is applied to the study of geometry. Because algebraic methods were first systematically applied to geometry in 1637 by the French philosopher-mathematician René Descartes, the subject is also called Cartesian geometry. The basis for an algebraic treatment of geometry is provided by the existence of a one-to-one correspondence between the elements, “points” of a directed line g, and the elements, “numbers,” that form the set of all real numbers. Such a correspondence establishes a coordinate system on g, and the number corresponding to a point of g is called its coordinate. The point O of g with coordinate zero is the origin of the coordinate system. A coordinate system on g is Cartesian provided that for each point P of g, its coordinate is the directed distance . Then all points of g on one side of O have positive coordinates (forming the positive half of g) and all points on the other side have negative coordinates. The point with coordinate 1 is called the unit point. Since the relation is clearly valid for each two points P, Q, of directed line g, then , where p and q are the coordinates of P and Q, respectively. Those points of g between P and Q, together with P, Q, form a line segment. In analytic geometry it is convenient to direct segments, writing PQ or QP accordingly as the segment is directed from P to Q or from Q to P, respectively. To find the coordinate of the point P that divides the segment P₁P₂ in a given ratio r, put . Then (x − x₁)/(x − x₂) = r, where x₁, x₂, x are the coordinates of P₁, P₂, P, respectively, and solving for x gives x = (x₁ − rx₂)/(1 − r). Clearly r is negative for each point between P₁, P₂ and is positive for each point of g external to the segment. The midpoint of the segment divides it in the ratio −1, and hence its coordinate x = (x₁ + x₂)/2. See also: Mathematics

A framework for solving a problem. The analytic hierarchy process is a systematic procedure for representing the elements of any problem. It organizes the basic rationality by breaking down a problem into its smaller constituents and then calls for only simple pairwise comparison judgments, to develop priorities in each level.

The superficial contents of a geometrical figure of two dimensions. The area of any rectangle or square is the product of two adjacent sides, one of which may be called the base and the other the altitude. In general, any line segment that partially bounds a plane geometric figure may be called a base if its line does not separate the figure, and a perpendicular drawn to the base line from one of its points at greatest distance may be called the altitude. The area of a parallelogram is equal to the product of its base times its altitude. The area of a triangle is one-half the product of its base times its altitude. The area of a trapezoid is equal to one-half the product of the sum of its parallel sides (bases) times its altitude. Some area formulas are given in the table. See also: Euclidean geometry

A branch of mathematics dealing with numbers, operations on numbers, and computation. Arithmetic is useful in solving many practical problems, such as buying, selling, budgets, sports statistics, and measurement. The usual numbers of arithmetic are whole numbers, fractions, decimals, and percents. Beyond the numbers of arithmetic are negative numbers, rational numbers, and irrational numbers. The rational and irrational numbers together constitute the real numbers.

If the graph of a function y = f (x) goes out to infinity, each “end” of the graph may get closer and closer to a straight line, which is then called an asymptote. In other words, some straight line may well approximate the graph as the function values of y or the values of x become very large. In calculus, asymptotes are traditionally classified as either horizontal, vertical, or slant.