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 Pythagorean theorem – regular measure Pythagorean theorem   In Euclidean geometry, the sum of the areas of the squares on the legs of any right triangle is equal to the area of the square on the hypotenuse. This is arguably the most important theorem of classical mathematics, and perhaps of all time. Pythagorean triple   An ordered triple (a,b,c) of natural numbers satisfying a2 + b2 = c2. The triples (3,4,5) and (5,12,13) are the first of infinitely many examples. quadratic formula   Given a quadratic function, i.e., a polynomial function of second degree y = ax 2 + bx + c, the zeros of the function are given byThe expression under the radical is called the determinant. If the determinant is positive, both solutions are real; if negative, both solutions are complex; and if zero, there is a single solution of multiplicity two. quadrilateral   A closed, plane figure with four straight sides.Cf. polygon. quantifier   quasi-disjoint family   Set Theory: See D-system. Quine atom   In set theory, a set whose only member is itself, i.e., x = { x }. More generally, some phrases may be “Quined” to form meaningful sentences, e.g., “is a five-word phrase” is a five-word phrase. quotient   The number that results from dividing one number by another.Cf. division algorithm. radian   A dimensionless unit of measure of angles. An angle of one radian is given by the central angle of a circle subtending an arc of length equal to the radius of the circle. Consequently, 360 degrees is the same as 2p radians. See the related article for a more extensive exposition. Related article: Trig Functions and Identities Ramsey cardinal   A cardinal k is called a Ramsey cardinal ifCf. Ramsey’s Theorem. Ramsey number   See Ramsey’s Theorem. Ramsey’s Theorem   Graph theory: given any two positive integers m and n, there exists a number, called the Ramsey number of m and n, and denoted R(m, n), such that any simple graph with R(m, n) vertices either contains a clique with m vertices (all vertices adjacent) or an independent set with n vertices (no vertices adjacent).The question of how to determine the Ramsey number for arbitrary m and n is unsolved and believed to be very difficult, but various lower and upper bounds on it have been proven over the years. In particular, it is known that R(m, m) 2m/2, and that R(m, n) (m + n - 2)C(m - 1).Also, certain specific Ramsey numbers are known, e.g., R(3, 3) = 6. Thus, in any group of six people, either three are mutual acquaintances or three are mutual strangers.Set theory: Let n be a positive integer, k and l be (finite or transfinite) cardinals, and denote by kln the assertion that the complete graph of size k in which each edge has been “colored” by one of n colors contains a complete subgraph of size l all of whose edges are the same color. More generally still, denote by the assertion that if the subsets of k of size b have each been colored with one of a-many colors, then there is a subset of X of size l all of whose b-sized subsets are the same color (homogeneous). Ramsey’s Theorem (infinite case) may now be stated as: For all positive integers k and l, we havei.e., for every coloring of the k-sized subsets of the natural numbers (w) into l colors, there is an infinite homogeneous set (an infinite collection of k-sized subsets all the same color).This more general notion gives rise to a field of set theory known as Ramsey Theory, in which far more is known about infinite sets than finite ones. range   The set of elements to which a function maps the elements of its domain set. rational exponent   An exponent of the form p/q, with p and q integers and q not zero. Evaluated as the qth root of the base, raised to the pth power, or equivalently, as the qth root of the pth power of the base. For a negative base, this operation is not defined except when q is odd. Irrational roots may be considered as limits of sequences of rational roots.Cf. laws of exponents. rational number   An element of the set Q consisting of ordered pairs (p, q) of integers, with q not 0, and with the order relation (p, q) < (r, s) if and only if ps < rq as integers. (The ordered pairs are usually written p/q, i.e., as a fraction (ratio) with integer numerator and denominator.) The rational numbers are countably infinite.Cf. natural number, real number. Related MiniText: Number -- What Is How Many? ratio test    ARTICLE   A test for the convergence of a series. See the related article for a complete description. real number   An element of the set R consisting of all of the rational numbers together with all of the irrational numbers. Sometimes called the continuum. Usually defined formally by a Dedekind cut of rational numbers. The real numbers form (uniquely) a complete ordered field, but are not algebraically complete.It is a famous theorem of Georg Cantor that the real numbers are not countable.Cf. complex number. Related MiniText: Number -- What Is How Many? real number line   A geometrical line graphically representing the set of real numbers, in which every real number corresponds to a unique point on the line, and every point on the line corresponds to a unique real number. reflexive relation   A relation “ ~ ” on a set X is reflexive if for every element x in X we have x ~ x. The relation “ ~ ” is called irreflexive if for every x we have that x ~ x is false. Note that a relation may be neither reflexive nor irreflexive.Cf. symmetric relation, transitive relation. regular   Set Theory: An ordinal a is called regular if and only if it is a limit ordinal and the cofinality of a is a. All regular ordinals are cardinals. All successor cardinals are regular, but limit cardinals can fail to be regular. A cardinal which is not regular is called singular. regular measure   A Borel measure m on a finite-dimensional real space is called regular if for all compact measurable sets K and Borel sets E we have: Pythagorean theorem – regular measure
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