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 range – scalene 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 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: regular polygon   A polygon all of whose sides are equal in length and all of whose interior angles are equal. regular solid   A polyhedron having congruent faces, which are themselves regular polygons. Also called Platonic solid. Related article: Platonic Solids relation   An n-place relation is defined on a Cartesian product of n sets, and is represented by a set of ordered n-tuples. For example, the less-than (“<”) relation is a binary relation on numbers, and the membership relation (“e”) is a binary relation on sets. The property of forming a Pythagorean triple is a ternary relation on natural numbers, of which for example (3,4,5) is a member since 32 + 42 = 52.In a binary (two-place) relation, the set from which the abscissae are taken is called the domain, and the set providing the ordinates is called the range. Binary relations are classified according to whether they are reflexive, transitive, and/or symmetric.Cf. function, partial order, lattice. relatively prime   Two natural numbers a and b are relatively prime if their greatest common divisor is 1. Riemann Hypothesis   The conjecture that the zeta function has no non-trivial zeros off of the line Re(z) = 1/2. Riemann integral   See integral. Riemann sum   Let f be a real-valued function defined on the closed interval [a, b], and let D be a partition of [a, b], i.e., a = x0 < x1 < ... < xn = b, and where Dxi is the width of the i th subinterval. If c i is any point in the i th subinterval, then the sumis called the Riemann sum of f for the partition D. right angle   An angle of 90 degrees (p/2 radians). Equivalently, it can be said that two right angles are supplemental angles, i.e., they add up to a straight line (180 degrees or p radians).Cf. complementary angles, acute, obtuse. root   An nth root of a real or complex number x is a number which when multiplied by itself n times yields x.Of a polynomial p: A number x such that p(x) = 0. root test   A test for the convergence of a series. See the related article for a complete description. Related article: Series scalar   A quantity having only magnitude, not direction (typically an element of a field, such as the real numbers or complex numbers).Cf. vector. scalar product   The scalar product, also called dot product, of two vectors is the sum of the products of the corresponding components of the two vectors. I.e., given two vectors x = (x1, x2, ..., xn) and y = (y1, y2, ..., yn), their scalar product is the scalar x1y1 + x2y2 + ... + xnyn.Cf. vector product. scalene   A triangle is called scalene if all of its sides are unequal (equivalently, if all of its angles are unequal). range – scalene
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