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complex number – cross product
complex number
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.
concave
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°.
Cf. convex.
concave function
A function is concave if the chord connecting any two points of its graph lies entirely below the graph.
Cf. convex function.
conditional statement
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.
cone
ARTICLE
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.
congruence relation
Given a structure X (i.e., X is a space, algebra, etc.), then an equivalence relation ~ on X is called a congruence relation if it is preserved by every defined operation on X. That is, for any x, y in X, if x ~ y and f is defined on X, then f(x) ~ f(y).
conic section
ARTICLE
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.
connected
A topological space X is connected if there are no two open sets of X whose union is X and whose intersection is empty.
constant
An unvarying quantity, usually represented notationally by an alphabetic letter such as k, c, etc.
constant function
A constant function f is one whose value is the same at all points of its domain.
continuous
Analysis: A function f is continuous at a point x of its domain if, whenever we are given a number e greater than 0, we may find a d greater than 0 so that whenever y is within a d-neighborhood of x, then f(y) is within an e-neighborhood of f(x).
Topology: A transformation of one topological space into another is continuous if the inverse image of every open set is open, or equivalently if the inverse image of every closed set is closed.
Cf. uniformly continuous, equicontinuous, absolutely continuous.
continuum
In mathematics, the real numbers or real number line.
Related article: Gödel's Theorems
Related MiniText: Infinity -- You Can't Get There From Here...
contradiction
An assertion that some statement is simultaneously true and false.
When a contradiction arises in mathematics, it is an indication that a mistaken assumption has been made, and this is often used to write proofs by contradiction, a strategy in which the negation of the statement to be proved is assumed, and a contradiction then derived. The contradiction is then taken as proving the original (non-negated) statement. (This strategy relies on the supposition that the mathematical theory under discussion is itself consistent, that is, free of contradiction.)
It sometimes happens that a contradiction arises, and yet none of the assumptions leading to the contradiction can be sacrificed. This is known as a paradox. If the assumptions are formal axioms, the result is called an antinomy.
convergent sequence
See sequence.
convergent series
See series.
Related article: Series
converse statement
Given a conditional, i.e., a statement of the form if A then B, or A implies B, its converse is B implies A. A conditional neither implies nor is implied by its converse. However, the converse of a conditional and its inverse are logically equivalent, since they are contrapositives of each other.
convex
Naively, a region of space is convex if the line segement joining any two points of the region lies wholly within it. Thus, a polygon is convex if every line segment joining any two points on its sides lies entirely within the polygon. (This is equivalent to the condition that all its interior angles be less than 180°.)
More generally, a region in a real vector space is convex if whenever two points x and y are in the region then so is any point tx + (1 - t)y, where t lies in the interval [0, 1]. See the immediately following entries for additional uses of the descriptor convex.
Cf. concave.
convex function
A function is convex if the chord joining any two points of its graph lies entirely above the graph.
Cf. concave function.
countable
A set is countable if it is finite, or if it is infinite and bijective to the set of natural numbers (finite ordinals), i.e., if there exists a complete one-to-one mapping of the set in question onto the set N. Sets that are both countable and infinite are sometimes called denumerable. Georg Cantor proved that sets may be uncountably infinite, for example the set of real numbers.
Related MiniText: Infinity -- You Can't Get There From Here...
countably infinite
See countable
cross product
See vector product.
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