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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…

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…

The study of zero sets of polynomial equations. Examples 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 in the coordinate space with coordinates w,…

Total variation in experimental data is partitioned into components assignable to specific sources by the analysis of variance. This statistical technique is applicable to data for which (1) effects of sources are additive, (2) uncontrolled or…

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…

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…

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…

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…

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…

The application of the properties of the centroid of a set of weighted points to geometry. For certain new operations such as the addition of point and point or of point and vector, it is possible to give their vector equivalents and work within the…

An approach to statistics in which estimates are based on a synthesis of a prior distribution and current sample data. Bayesian statistics is not a branch of statistics in the way that, say, nonparametric statistics is. It is, in fact, a…

By definition the solutions of Bessel's differential equation, Eq. (1). 1Bessel functions, also called cylinder functions, are examples of special functions which are introduced by a differential equation. Bessel functions are of great interest in…

One of the most important algebraic identities, with many applications in a variety of fields. The binomial theorem, discovered by Isaac Newton, expresses the result of multiplying a binomial by itself any number of times, as given in Eq. (1),…

The application of mathematical and statistical methods to describe and analyze data concerning the variation of biological characteristics obtained from either observation or experiment. The concept of a population and the samples derived from it…

The branch of mathematics dealing with two fundamental operations, differentiation and integration, which are carried out on functions. The subject, as traditionally developed in college textbooks, is partly an elementary development of the purely…

The curve formed by an ideal heavy uniform string hanging freely from two points of support. The lowest point A (Fig. 1) is the vertex. The portion AP is an equilibrium under the horizontal tension H at A, the tension F directed along the tangent at…

The curve that is the locus of points in a plane with equal distance from a fixed point. One of the most basic geometric figures, a circle is perfectly round. In elementary mathematics, "circle" often refers to the finite portion of the plane…

A procedure in which the basic problem is to pass a curve through a set of points, representing experimental data, in such a way that the curve shows as well as possible the relationship between the two quantities plotted. It is always possible to…

A member of the class of curves that are intersections of a plane with a cone of revolution. It is obtained (see illustration) when the cutting plane is parallel to an element of the cone. Conic section Parabola as (a) conic section and (b) locus…

An auxiliary variable, functions of which give the coordinates of a curve or surface. The coordinates of a curve are functions of one parameter. A curve in 3-space has parametric equations (1). 1 The coordinates of a surface are functions of two…

A type of mathematical equation used, typically, to represent curves in a plane or in space of three dimensions. In principle, however, there is no limitation to any particular number of dimensions. A parameter is actually an independent variable.…

A mathematical operation performed on functions of more than one variable. In this article only two or three variables are considered; however, the principles apply to functions of n variables, for any positive integer n > 1. If z = f(x,y), the…

A scenario for the transition of a natural process from regular motion to chaos. Various natural processes develop in time in a way that depends upon prevailing environmental details. A quantity that specifies the particular state of the environment…