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Riemann sum – semigroup

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
1. the set is an Abelian group under the addition operation, and
2. 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

Roth’s Theorem   (Klaus Friedrich Roth, 1955) Given a real algebraic number a, consider the least upper bound m(a) of all numbers m for which there are infinitely many rational numbers p/q such that

Then for all a, m(a) = 2. This result improves on earlier theorems of Joseph Liouville, Axel Thue, and Carl Ludwig Siegel regarding the approximation of irrational numbers by rational numbers.

 Bertrand Russell
Russell, Bertrand
(born 1872)   English philosopher and logician. Wrote Principia Mathematica (1913) with Alfred North Whitehead, an attempt to reduce all of mathematics to symbolic logic. This was perhaps the most important effort in the logicist program. Russell introduced type theory into the theory of sets and classes in an effort to avoid the kind of antinomy in the foundations of mathematics as was exemplified in the so-called Russell paradox. The goal of Russell’s system was to show that any true mathematical proposition can be established by logic alone, a goal which was severely compromised by Kurt Gödel’s proof of the Gödel Incompleteness Theorem in 1931.

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.

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).

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.

sec   See secant.

secant   Geometry: A line connecting two points on a curve.

Trigonometry: A periodic trigonometric function defined on angles, usually abbreviated “sec.” It is the multiplicative inverse of the cosine function.

See the related article for a complete exposition.

Related article: Trig Functions and Identities

semi-finite measure   See measure.

semigroup   See group.

Riemann sum – semigroup

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