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Documents for "Mathematics":
  • abacus in mathematics, simple device for performing arithmetic calculations. The type of abacus now best known is represented by a frame with sliding counters. An elementary abacus might have ten...
  • absolute value magnitude of a number or other mathematical expression disregarding its sign; thus, the absolute value is positive, whether the original expression is positive or negative. In symbols, if &124; a...
  • addition fundamental operation of arithmetic, denoted by +. In counting, a + b represents the number of items in the union of two collections having no common members (disjoint sets), having respectively a...
  • algebra branch of mathematics concerned with operations on sets of numbers or other elements that are often represented by symbols. Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as...
  • algebraic geometry branch of geometry , based on analytic geometry , that is concerned with geometric objects (loci) defined by algebraic relations among their coordinates (see Cartesian coordinates ). In plane geometry an algebraic curve is the locus of all points satisfying the polynomial equation f ( x,y )=0; in three dimensions the polynomial equation f ( x,y,z )=0 defines an algebraic surface. In general, points in n -space are defined by ordered sequences of numbers ( x1 , x2 , x3 , …  xn ), where each n -tuple specifies a unique point and x1 , x2 , x3 , …  xn are members of a given field (e.g., the complex numbers). An algebraic hypersurface is the locus of all such points satisfying the polynomial equation f ( x1 , x2 , x3 , …  xn )=0, whose coefficients are also chosen from the given field. The intersection of two or more algebraic hypersurfaces defines an algebraic set, or variety, a concept of particular importance in...
  • algorithm or algorism [for Al-Khowarizmi ], a clearly defined procedure for obtaining the solution to a general type of problem, often numerical. Much of ordinary arithmetic as traditionally taught consists of algorithms involving the...
  • analysis branch of mathematics that utilizes the concepts and methods of the calculus. It includes not only basic calculus, but also advanced calculus, in which such underlying concepts as that of a limit...
  • analytic geometry branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates , and in which the approach to geometric problems is primarily algebraic. Its most common application is in the representation of equations involving two or three variables as curves in two or three...
  • angle in mathematics, figure formed by the intersection of two straight lines; the lines are called the sides of the angle and their point of intersection the vertex of the angle. Angles are commonly...
  • arc in geometry, a curved line or any part of it; in particular, a portion of the circumference of a circle. The length s of an arc of a circle of radius r and subtending a central angle of θ radians...
  • area measure of the size of a surface region, usually expressed in units that are the square of linear units, e.g., square feet or square meters. In elementary geometry, formulas for the areas of the...
  • arithmetic branch of mathematics commonly considered a separate branch but in actuality a part of algebra. Conventionally the term has been most widely applied to simple teaching of the skills of dealing with Numbers for practical purposes, e.g., computation of areas, proportions, costs, and the like. The four fundamental operations of this study are addition, subtraction, multiplication, and division. In...
  • associative law in mathematics, law holding that for a given operation combining three quantities, two at a time, the initial pairing is arbitrary; e.g., using the operation of addition, the numbers 2, 3, and 4...
  • average number used to represent or characterize a group of numbers. The most common type of average is the arithmetic mean. See median ; mode.
  • axiom in mathematics and logic, general statement accepted without proof as the basis for logically deducing other statements (theorems). Examples of axioms used widely in mathematics are those related to equality (e.g., "Two things equal to the same thing are equal to each other" ; "If equals are added to equals, the sums are equal" ) and those related to operations (e.g., the associative law and the commutative law ). A postulate, like an axiom, is a statement that is accepted without proof; however, it deals with specific subject matter (e.g., properties of geometrical figures) and thus is not so general as...
  • binary system numeration system based on powers of 2, in contrast to the familiar decimal system , which is based on powers of 10. In the binary system, only the digits 0 and 1 are used. Thus, the first ten numbers in binary notation, corresponding to the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and...
  • binomial polynomial expression (see polynomial ) containing two terms, for example, x + y. The binomial theorem, or binomial formula, gives the expansion of the n th power of a binomial ( x + y ) for n=...
  • Boolean algebra an abstract mathematical system primarily used in computer science and in expressing the relationships between sets (groups of objects or concepts). The notational system was developed by the English mathematician George Boole c.1850 to permit an algebraic manipulation of logical statements. Such manipulation can demonstrate whether or not a statement is true and show how a complicated statement can be rephrased in a...
  • Bourbaki, Nicolas pseudonym under which a group of 20th cent. mathematicians has written a series of treatises on pure mathematics. The mathematicians have all been associated with the Ecole Normale Supérieure in Paris at some point in their careers; among them are the French scholars Claude Chevalley, André Weil, Henri Cartan, and Jean Dieudonné along with the American Samuel Eilenberg. The pseudonym...
  • calculus branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit —the notion of tending toward, or approaching, an ultimate value. The English physicist Isaac Newton and the German mathematician G. W. Leibniz , working independently, developed the calculus during the 17th cent. The calculus and its basic tools of differentiation and integration serve as the foundation for the larger branch of mathematics...
  • calculus of variations branch of mathematics concerned with finding maximum or minimum conditions for a relationship between two or more variables that depends not only on the variables themselves, as in the ordinary calculus , but also on an additional arbitrary relation, or constraint, between them. For example, the problem of finding the closed plane curve of given length that will enclose the greatest area is a type...
  • Cartesian coordinates [for René Descartes ], system for representing the relative positions of points in a plane or in space. In a plane, the point P is specified by the pair of numbers ( x,y ) representing the distances of the point from two intersecting straight lines, referred to as the x -axis and the y -axis. The point of intersection of these axes, which are called the coordinate axes, is known as the origin. In rectangular coordinates, the type most often used, the axes are taken to be...
  • chance in mathematics: see probability.
  • chaos theory in mathematics, physics, and other fields, a set of ideas that attempts to reveal structure in aperiodic, unpredictable dynamic systems such as cloud formation or the fluctuation of biological...
  • chord in geometry, straight line segment both end points of which lie on the circumference of a circle or other curve; it is a segment of a secant. A chord passing through the center of a circle is a diameter....
  • circle closed plane curve consisting of all points at a given distance from some fixed point, called the center. A circle is a conic section cut by a plane perpendicular to the axis of the cone. The term circle is also used to refer to the region enclosed by the curve, more properly called a circular region. The radius of a circle is any line segment connecting the center and a point on the curve; the...
  • combinatorics or combinatorial analysis , sometimes called the science of counting, the branch of mathematics concerned with the selection, arrangement, and operation of elements within sets. Combinatorial theory deals with existence (does a particular arrangement exist?), enumeration (how many such...
  • commutative law in mathematics, law holding that for a given binary operation (combining two quantities) the order of the quantities is arbitrary; e.g., in addition, the numbers 2 and 5 can be combined as 2+5=7 or...
  • complex variable analysis branch of mathematics that deals with the calculus of functions of a complex variable, i.e., a variable of the form z = x + iy, where x and y are real and i = -1 (see number ). A function w = f(z) of a complex variable z is separable into two parts, w = g1 ( x,y ) + ig2 ( x,y ), where g1 and g2 are real-valued functions of the real variables x and y. The theory of functions of a complex variable is concerned mainly with functions that have a derivative at every point of a given domain of values for z; such functions are called analytic, regular, or holomorphic. If a function is analytic in a given domain, then it also has continuous derivatives of higher order and can be expanded in an infinite series in terms of these derivatives (i.e., a Taylor's series). The function can also be expressed in the infinite series where z0 is a point in the domain. Also of interest in complex variable analysis are the points in a domain, called singular points, where a function fails to have a derivative. The theory of functions of a...
  • complexity in science, field of study devoted to the process of self-organization. The basic concept of complexity is that all things tend to organize themselves into patterns, e.g., ant colonies, immune...
  • cone or conical surface, in mathematics, surface generated by a moving line (the generator) that passes through a given fixed point (the vertex) and continually intersects a given fixed curve (the directrix). The generator...