BROWSE ALPHABETICALLY LEVEL:    Elementary    Advanced    Both INCLUDE TOPICS:    Basic Math    Algebra    Analysis    Biography    Calculus    Comp Sci    Discrete    Economics    Foundations    Geometry    Graph Thry    History    Number Thry    Phys Sci    Statistics    Topology    Trigonometry metric – nowhere dense metric   Given a set X, a metric on X is a function d with domain X and range in R* (the set of non-negative real numbers) satisfying, for all a, b, and c in X:d(a,b) = 0 if and only if a = b;d(a,b) = d(b,a); andd(a,b) + d(b,c) >= d(a,c (where >= means ‘greater than or equal to’).Cf. metric space. metric space   A set with a metric defined on its elements. monotone function   Also called monotonic function. See order-preserving function. multiplication   A binary operation on numbers or quantities resulting in a product, usually but not always amounting to repeated addition. On the natural numbers multiplication is defined recursively by the Peano axioms, such that the product of two numbers n and m, denoted by n × m, is found by adding up m copies of n (or n copies of m).Multiplication of most kinds of numbers is associative and commutative, but these properties sometimes fail, for example in the case of matrix multiplication.When a product of more than two numbers or quantities is taken, the general product may be denoted by the capital Greek letter Pi, i.e., Pai denotes the product a1 × a2 × . . . × an. multiply   To find the product of two numbers or quantities by multiplication. mutually prime   Two integers are mutually prime if they have no common factors larger than 1 or -1. mutually singular   Given a measurable space (X, M), two signed measures m and n on X are called mutually singular, denoted byif there exist sets E and F in M such that their union is X, their intersection is empty, and m(E) = 0 and n(F) = 0. nth-term test   A test for the divergence of a series. See the related article for a complete description. Related article: Series natural base   See Euler number. natural logarithm   A logarithm with base e, the Euler number. Often written “ln” rather than “log” to distinguish it from logarithms using other bases. natural number   An element of the set N = {1, 2, 3, ...} consisting of all the “counting numbers.” When the number 0 is included, this set is sometimes called the whole numbers. In set theory, the natural numbers (incuding 0) are identified with the set w of finite ordinals. The natural numbers are a well-founded linear order with no largest member, and are countably infinite.Cf. Peano axioms, rational number, real number. Related MiniText: Number -- What Is How Many? negation   If j is a statement, sentence, or formula of logic, then the negation of j, denoted by j, is that formula which is true whenever j is false, and false whenever j is true. negative   The negative of a number or quantity x is the number, denoted -x, which when added to x yields 0. That is, the negative of a number is its additive inverse. negative set   Given a signed measure m on a measure space X, a measurable set A in X is called a negative set if the measure of all measurable subsets of A is less than or equal to zero.Cf. positive set, null set. neighborhood   A neighborhood of a point x of a topological space is an open set of the space containing x. In a metric space, a d-neighborhood of x is the collection of all points of the space whose distance from x is less than d. non-denumerable   norm   Analysis: A non-negative real-valued function “|| x ||” defined on a vector space, satisfying|| –x || = || x ||,|| cx || = || c || × || x || for all scalars c, and|| x + y || <= || x || + || y || (triangle inequality)Statistics: Another term for the mode of a frequency distribution. normal   A line intersecting a curve (or surface) perpendicular to the tangent line (or tangent plane) at the point of intersection. The normal to a surface expressed as a function of several variables xi is given by the gradient. normalized bounded variation   See: bounded variation. normed space   A vector space with a norm defined on it. nowhere dense   Given a space X and a subset A of X, we say that A is nowhere dense if every open set of X contains an open subset that is disjoint from A. This is equivalent to saying that the complement of A is dense, or that A has empty interior. metric – nowhere dense
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