Cartesian plane constant
The Cartesian product R2, represented graphically by two real number lines at right angles to one another, with the point (0,0) at the intersection.
Used for graphing functions from the set of real numbers to itself. The quadrants, numbered I - IV as shown, indicate the regions of the plane where the x and y axes are positive and negative. Compare: Argand plane.
A straight line segment connecting two points on a curve or surface.
In a plane, the locus of all points equidistant from a given point, called the center. The general equation for a circle in the Cartesian plane is given by (x - h) 2 + (y - k) 2 = r 2, where r is the radius of the circle (distance from the center to the locus of points), and (h, k) are the coordinates of the center.
The interior of a circle is referred to as an open disk.
A circle is also a conic section; a special case of an ellipse in which the foci coincide.
Geometry: The distance around a circle in the plane, or around a great circle of a sphere.
Graph Theory: The circumference of a graph G is defined as the length of the longest cycle of G. The circumference is ususally denoted by c(G), and is undefined if G has no cycles.
General: A set is closed under an operation if applying the operation to its elements returns only elements in the set. For example, the set of integers is closed under addition, since adding two integers always gives another integer, but it is not closed under division, since dividing two integers may result in a non-integer.
Geometry: A plane figure is closed if it consists of lines and/or curves that entirely enclose an area. Similarly, a figure in 3-dimensional space is called closed if it entirely encloses a volume.
See the following listings for other uses of the word closed in mathematics.
Topology: A set is topologically closed if it is not open.
An interval of the real number line (or any other totally ordered set) which includes its endpoints. An interval containing only one of its endpoints is called half-open.
Cf. open interval.
A subselection of a set of r elements from a set of n elements. The number of such combinations, i.e., the number of ways in which r elements may be chosen from a set of n elements, is given by the formula
This operation is also sometimes denoted by nC r, and is read n choose r.
Cf. permutation, factorial.
A logarithm with base 10.
An operation · on elements of a set A is commutative if for all elements a, b in A, a · b = b · a.
Cf. associative, distributive property.
A property of numbers which states that the operations of addition and multiplication are commutative. Cf. distributive property, associative.
A test for the convergence of a series. See the article for a complete description.
The complement of a set A is the set of all elements that are not elements of A.
Graph Theory: The complement of a simple graph G with vertex set V is the simple graph Gc, which also has vertex set V, and in which two vertices are adjacent if and only if they are not adjacent in G.
Two angles are complementary if they add up to a right angle.
An element of the set C of numbers of the form a + bi, where a and b are real numbers and i 2 = -1. The number i is called the imaginary number.
A region of space is concave if there are two points of the region such that a line joining the two points is not entirely contained within the region. In particular, a polygon is concave if any of its interior angles is greater than 180°.
A function is concave if the chord connecting any two points of its graph lies entirely below the graph.
Cf. convex function.
A statement of the form if A then B, or A implies B. A conditional is equivalent to its contrapositive, not B implies not A. See also: inverse statement and converse statement.
The infinite surface of revolution generated as shown
The term also refers to the solid bounded by one of the nappes and a flat elliptical base. If in this case the base is circular (at right angles to the axis), the cone is called a right circular cone. The surface area S (excluding the base) and volume V of a right circular cone are given by
Cf. conic section.
A plane curve, either the ellipse, parabola, or hyperbola, which results from the intersection of a plane with a cone. See the article for a full exposition
Cf. Dandelin's Spheres.
An unvarying quantity, usually represented notationally by an alphabetic letter such as k, c, etc.