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real number – root test
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 sum
is 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
A set together with two binary operations (called addition and multiplication) defined on its elements, and satisfying - the set is an Abelian group under the addition operation, and
- multiplication is distributive with respect to addition, i.e., for all elements a, b, and c in the ring, we have a(b + c) = ab + ac and (b + c)a = ba + ca.
If in addition multiplication is commutative, the ring is called a commutative ring. If there is a multiplicative identity element, i.e., an element 1 such that for every element a in the ring we have 1a = a1 = a, then it is called a ring with unity. A commutative ring with unity is called an integral domain if no product of nonzero elements is zero. If everly element of the ring except the additive identity has a multiplicative inverse, it is called a division ring (equivalently, if the nonzero elements form a group under multiplication). A commutative division ring is called a field.
Cf. ideal.
ring automorphism
A ring isomorphism from a ring to itself. That is, a bijective function from a ring to itself that preserves addition (the group operation) and multiplication: f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b). If the domain is a ring with unity, we also require that f preserve unity, i.e. f(1) = 1.
Cf. ring homomorphism, ring isomorphism.
ring homomorphism
A function from one ring to another that preserves addition (the group operation) and multiplication: f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b). If the domain is a ring with unity, we also require that f preserve unity, i.e. f(1) = 1.
Cf. ring automorphism, ring isomorphism.
ring isomorphism
A ring homomorphism that is both one-to-one and onto. That is, a bijective function from one ring to another that preserves addition and multiplication: f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b). If the domain is a ring with unity, we also require that f preserve unity, i.e. f(1) = 1.
Cf. ring automorphism, ring homomorphism.
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.
root test
A test for the convergence of a series. See the related article for a complete description.
Related article: Series
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