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 real number – scientific notation 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: 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 large   A set A of natural numbers is called relatively large if the number of elements of A is greater than the least element of A. 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. ring of sets   Given a set X, a ring on X is a collection of subsets of X which is closed under finite unions and set differences. If the ring includes X itself then it is an algebra of sets. If the ring is closed under countable unions, then it is called a s-ring. 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. Russell Paradox   (Bertrand Russell, 1901) A paradox of set theory which necessitated a more careful axiomatization of set theory in the 1920’s and 1930’s: Naively, some sets are members of themselves and some are not. For instance, the set of all apples is not itself an apple, but the set of all sets does seem to be a set. So consider the set X of all sets that are not members of themselves. We may ask, is X a member of itself? If it is then it cannot be, because of the way in which X itself was defined, but if it isn’t then it must be, by the same reasoning. Contradiction. The Russell paradox is resolved in modern set theory by a foundation axiom or axiom of regularity, and by limiting the “size” of objects we call sets. For example, the “set of all sets” is considered not to be a set but a proper class. scalar   A quantity having only magnitude, not direction (typically an element of a field, such as the real numbers or complex numbers).Cf. vector. scalene   A triangle is called scalene if all of its sides are unequal (equivalently, if all of its angles are unequal). Schroeder-Bernstein Theorem   If there exists an injection from a set X into a set Y, and also an injection from Y into X, then there exists a bijection from X to Y, and hence X and Y have the same cardinality. scientific notation   A number is written in scientific notation when it is written as the product of a real number between 1 and 10 and a power of 10. E.g., 320 is written in scientific notation as 3.2 × 102. On some calculators and in some textbooks, this may be written as 3.2E2. Scientific notation is a convenient way to represent very large and very small numbers. real number – scientific notation
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