Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians,[1] Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system.[2] The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.[3]
250: 名無しさん：14/06/15 14:06 ID:Ues
For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only where the gravitational field is weak.[4]
251: 名無しさん：14/06/15 14:07 ID:Ues
Euclidean Geometry is constructive. Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge.[9] In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory.
252: 名無しさん：14/06/15 14:07 ID:Ues
Euclidean geometry has two fundamental types of measurements: angle and distance. The angle scale is absolute, and Euclid uses the right angle as his basic unit, so that, e.g., a 45-degree angle would be referred to as half of a right angle. The distance scale is relative; one arbitrarily picks a line segment with a certain nonzero length as the unit, and other distances are expressed in relation to it. Addition of distances is represented by a construction in which one line segment is copied onto the end of another line segment to extend its length, and similarly for subtraction.
253: 名無しさん：14/06/15 14:08 ID:Ues
Measurements of area and volume are derived from distances. For example, a rectangle with a width of 3 and a length of 4 has an area that represents the product, 12. Because this geometrical interpretation of multiplication was limited to three dimensions, there was no direct way of interpreting the product of four or more numbers, and Euclid avoided such products, although they are implied, e.g., in the proof of book IX, proposition 20.
254: 名無しさん：14/06/15 14:09 ID:Ues
De euclidische meetkunde is een wiskundig systeem dat wordt toegeschreven aan de Griekse wiskundige Euclides van Alexandri？. Zijn werk, de Elementen, is de vroegst bekende systematische bespreking van de meetkunde. De Elementen is een van de meest invloedrijke boeken uit de geschiedenis, niet alleen om de wiskundige inhoud, maar vooral vanwege de gehanteerde methode. Deze methode bestaat eruit om uitgaande van een kleine verzameling van intu？tief aansprekende axioma's, vervolgens vele andere proposities, lemma's en stellingen te bewijzen.
255: 名無しさん：14/06/15 14:09 ID:Ues
De euclidische meetkunde is de meetkunde van ruimte die niet gekromd is. Eerste voorbeeld van een ruimte die wel gekromd is, is het oppervlak van een bol. Belangrijke begrippen in de euclidische meetkunde zijn onder andere de punt, lijn, lijnstuk, kant van de lijn, cirkel met straal en middelpunt, rechte hoek en congruentie. Deze begrippen kennen we, het zijn de begrippen waar het onderwijs in de wiskunde mee begint. We hebben ook een intu？tief beeld van de euclidische meetkunde, maar voor een exacte beschrijving ervan zijn de vijf postulaten van Euclides nodig.
256: 名無しさん：14/06/15 14:10 ID:Ues
Als eerste axiomatisch systeem begint de Elementen met de meetkunde op een vlak en gebruikt daarbij bovengenoemde begrippen. Hier vindt men ook de eerste voorbeelden van formele bewijzen. De Elementen gaat vervolgens verder met meetkunde van de drie-dimensionale ruimte, de stereometrie. Vooral in de 19e eeuw is de euclidische meetkunde uitgebreid naar elk eindig aantal dimensies. Vooral de leerboeken van de planimetrie en de stereometrie liggen ten grondslag aan de elementaire mechanica en natuurkunde.
257: 名無しさん：14/06/15 14:11 ID:Ues
Tegenwoordig wordt het niet langer vanzelfsprekend beschouwd dat de euclidische meetkunde de natuurkundige ruimte, het heelal, beschrijft. Een implicatie van Einsteins algemene relativiteitstheorie is dat de euclidische meetkunde alleen een goede benadering van de eigenschappen van het heelal vormt als het zwaartekrachtsveld niet te sterk is.
L'algorithme d'Euclide est un algorithme permettant de d？terminer le plus grand commun diviseur (P.G.C.D.) de deux entiers dont on ne conna？t pas la factorisation. Il est d？j？ d？rit dans le livre VII des ？l？ments d'Euclide.
263: 名無しさん：14/06/16 04:46 ID:kig
Dans la tradition grecque, en comprenant un nombre entier comme une longueur, un couple d'entiers comme un rectangle, leur PGCD est la longueur du c？t？ du plus grand carr？ permettant de carreler enti？rement ce rectangle. L'algorithme d？ompose ce rectangle en carr？s, de plus en plus petits, par divisions euclidiennes successives, de la longueur par la largeur, puis de la largeur par le reste, jusqu'？ un reste nul.
n mathematics, the Euclidean algorithm[a], or Euclid's algorithm, is a method for computing the greatest common divisor (GCD) of two (usually positive) integers, also known as the greatest common factor (GCF) or highest common factor (HCF). It is named after the Greek mathematician Euclid, who described it in Books VII and X of his Elements.[1]
266: 名無しさん：14/06/16 04:53 ID:kig
The main principle is that the GCD does not change if the smaller number is subtracted from the larger number. For example, the GCD of 252 and 105 is exactly the GCD of 147 (= 252 − 105) and 105. Since the larger of the two numbers is reduced, repeating this process gives successively smaller numbers, so this repetition will necessarily stop sooner or later ― when the numbers are equal (if the process is attempted once more, one of the numbers will become 0).
267: 名無しさん：14/06/16 04:53 ID:kig
The earliest surviving description of the Euclidean algorithm is in Euclid's Elements (c. 300 BC), making it one of the oldest numerical algorithms still in common use. The original algorithm was described only for natural numbers and geometric lengths (real numbers), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials in one variable. This led to modern abstract algebraic notions, such as Euclidean domains. The Euclidean algorithm has been generalized further to other mathematical structures, such as knots and multivariate polynomials.
268: 名無しさん：14/06/16 04:54 ID:kig
If implemented using remainders of Euclidean division rather than subtractions, Euclid's algorithm computes the GCD of large numbers efficiently: it never requires more division steps than five times the number of digits (in base 10) of the smaller integer. This was proved by Gabriel Lam？ in 1844, and marks the beginning of computational complexity theory. Methods for improving the algorithm's efficiency were developed in the 20th century.
269: 名無しさん：14/06/16 04:55 ID:kig
The Euclidean algorithm calculates the greatest common divisor (GCD) of two natural numbers a and b. The greatest common divisor g is the largest natural number that divides both a and b without leaving a remainder. Synonyms for the GCD include the greatest common factor (GCF), the highest common factor (HCF), and the greatest common measure (GCM). The greatest common divisor is often written as gcd(a, b) or, more simply, as (a, b),[3] although the latter notation is also used for other mathematical concepts, such as two-dimensional vectors.
270: 名無しさん：14/06/16 04:55 ID:kig
If gcd(a, b) = 1, then a and b are said to be coprime (or relatively prime).[4] This property does not imply that a or b are themselves prime numbers.[5] For example, neither 6 nor 35 is a prime number, since they both have two prime factors: 6 = 2 × 3 and 35 = 5 × 7. Nevertheless, 6 and 35 are coprime. No natural number other than 1 divides both 6 and 35, since they have no prime factors in common.
271: 名無しさん：14/06/16 04:56 ID:kig
Let g = gcd(a, b). Since a and b are both multiples of g, they can be written a = mg and b = ng, and there is no larger number G > g for which this is true. The natural numbers m and n must be coprime, since any common factor could be factored out of m and n to make g greater. Thus, any other number c that divides both a and b must also divide g. The greatest common divisor g of a and b is the unique (positive) common divisor of a and b that is divisible by any other common divisor c.[6]
272: 名無しさん：14/06/16 05:18 ID:RTc
またもや荒らし。
273: 名無しさん：14/06/16 12:25 ID:kig
Error: 502
274: 名無しさん：14/06/17 03:46 ID:6pI
Error: 503a
275: 名無しさん：14/06/17 03:47 ID:6pI
An ASI signal can carry one or multiple SD, HD or audio programs that are already compressed, not like an uncompressed SD-SDI (270 Mbit/s) or HD-SDI (1.485 Gbit/s). An ASI signal can be at varying transmission speeds and is completely dependent on the user's engineering requirements. For example, an ATSC (US digital standard for broadcasting) has a maximum bandwidth of 19.392658 Mbit/s. Generally, the ASI signal is the final product of video compression, either MPEG2 or MPEG4, ready for transmission to a transmitter or microwave system or other device.
276: 名無しさん：14/06/17 03:48 ID:6pI
In coding theory, Reed？Solomon (RS) codes are non-binary cyclic error-correcting codes invented by Irving S. Reed and Gustave Solomon. They described a systematic way of building codes that could detect and correct multiple random symbol errors. By adding t check symbols to the data, an RS code can detect any combination of up to t erroneous symbols, or correct up to ？t/2？ symbols. As an erasure code, it can correct up to t known erasures, or it can detect and correct combinations of errors and erasures. Furthermore, RS codes are suitable as multiple-burst bit-error correcting codes, since a sequence of b + 1 consecutive bit errors can affect at most two symbols of size b. The choice of t is up to the designer of the code, and may be selected within wide limits.
277: 名無しさん：14/06/17 03:48 ID:6pI
Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry.
278: 名無しさん：14/06/17 03:49 ID:6pI
Conventions regarding the definition of an algebraic variety differ slightly. For example, some authors require that an "algebraic variety" is, by definition, irreducible (which means that it is not the union of two smaller sets that are closed in the Zariski topology), while others do not. When the former convention is used, non-irreducible algebraic varieties are called algebraic sets.
279: 名無しさん：14/06/17 03:49 ID:6pI
Proven around the year 1800, the fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial in one variable with complex coefficients (an algebraic object) is determined by the set of its roots (a geometric object). Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specifity of algebraic geometry among the other subareas of geometry.
280: 名無しさん：14/06/17 03:50 ID:6pI
Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales (6th Century BC). By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment―Euclidean geometry―set a standard for many centuries to follow.[1] Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. In the classical world, both geometry and astronomy were considered to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.
281: 名無しさん：14/06/17 03:51 ID:6pI
Rooted in the basic curriculum - the enkuklios paideia or "education in a circle" - of late Classical and Hellenistic Greece, the "liberal arts" or "liberal pursuits" (Latin liberalia studia) were already so called in formal education during the Roman Empire: thus Seneca the Younger discusses liberal arts in education from a critical Stoic point of view in Moral Epistle 88.[5] Contrary to popular belief, freeborn girls were as likely to receive formal education as boys, especially during the Roman Empire―unlike the lack of education, or purely manual/technical skills, proper to a slave.[6] The exact classification of the liberal arts varied however in Roman times,[7] and it was only after Martianus Capella in the 5th century AD influentially brought the seven liberal arts as bridesmaids to the Marriage of Mercury and Philology,[8] that they took on canonical form.
282: 名無しさん：14/06/17 04:56 ID:pEU
せっかくのかなえちゃんコーナーが荒らし放題になってる。
283: 名無しさん：14/06/17 08:19 ID:6pI
Error: 208
284: 名無しさん：14/06/17 11:32 ID:fy6
In 3 dimensions, the volume inside a sphere (that is, the volume of a ball) is derived to be \!V = \frac{4}{3}\pi r^3 where r is the radius of the sphere and π is the constant pi. Archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. (This assertion follows from Cavalieri's principle.) In modern mathematics, this formula can be derived using integral calculus, i.e. disk integration to sum the volumes of an infinite number of circular disks of infinitesimally small thickness stacked centered side by side along the x axis from x = 0 where the disk has radius r (i.e. y = r) to x = r where the disk has radius 0 (i.e. y = 0).
At any given x, the incremental volume (δV) equals the product of the cross-sectional area of the disk at x and its thickness (δx):
285: 名無しさん：14/06/17 11:33 ID:fy6
Pairs of points on a sphere that lie on a straight line through the sphere's center are called antipodal points. A great circle is a circle on the sphere that has the same center and radius as the sphere and consequently divides it into two equal parts. The shortest distance along the surface between two distinct non-antipodal points on the surface is on the unique great circle that includes the two points. Equipped with the great-circle distance, a great circle becomes the Riemannian circle.
If a particular point on a sphere is (arbitrarily) designated as its north pole, then the corresponding antipodal point is called the south pole, and the equator is the great circle that is equidistant to them. Great circles through the two poles are called lines (or meridians) of longitude, and the line connecting the two poles is called the axis of rotation. Circles on the sphere that are parallel to the equator are lines of latitude.
286: 名無しさん：14/06/17 11:33 ID:fy6
Any plane that includes the center of a sphere divides it into two equal hemispheres. Any two intersecting planes that include the center of a sphere subdivide the sphere into four lunes or biangles, the vertices of which all coincide with the antipodal points lying on the line of intersection of the planes.
The antipodal quotient of the sphere is the surface called the real projective plane, which can also be thought of as the northern hemisphere with antipodal points of the equator identified.
The round hemisphere is conjectured to be the optimal (least area) filling of the Riemannian circle.
The circles of intersection of any plane not intersecting the sphere's center and the sphere's surface are called spheric sections.[
287: 名無しさん：14/06/17 11:35 ID:fy6
This 'paradox' was discovered by Stephen Smale (1958). It is difficult to visualize a particular example of such a turning, although some digital animations have been produced that make it somewhat easier. The first example was exhibited through the efforts of several mathematicians, including Arnold S. Shapiro and Bernard Morin who was blind. On the other hand, it is much easier to prove that such a "turning" exists and that is what Smale did.
Smale's graduate adviser Raoul Bott at first told Smale that the result was obviously wrong (Levy 1995). His reasoning was that the degree of the Gauss map must be preserved in such "turning"？in particular it follows that there is no such turning of S1in R2. But the degree of the Gauss map for the embeddings f, ？ in R3 are both equal to 1, and do not have opposite sign as one might incorrectly guess. The degree of the Gauss map of all immersions of a 2-sphere in R3 is 1;
288: 名無しさん：14/06/17 11:35 ID:fy6
Aitchison's 'holiverse' (2010): this uses a combination of topological and geometric methods, and is specific to the actual regular homotopy between a standardly embedded 2-sphere, and the embedding with reversed orientation. This provides conceptual understanding for the process, revealed as arising from the concrete structure of the 3-dimensional projective plane and the underlying geometry of the Hopf fibration. Understanding details of these mathematical concepts is not required to conceptually appreciate the concrete eversion that arises, which in essence only requires understanding a specific embedded circle drawn on a torus in 3-space.
In forma simplicissima, algorithmus Euclidis pari integrorum incipit, tunc parem novum fingit ex numere minore et differentia inter numeros maiorem et minorem consistentem. Processus iteratur dum numeri sint aequi. Hic numerus igitur est paris principalis divisor communis maximus.
292: 名無しさん：14/06/17 23:17 ID:6pI
Unus est ex veterrimis algorithmis numericis adhuc in uso commune. Algorithmus principalis in Elementis (c. 300 a.C.n.) descriptus solos numeros naturales et longitudines geometricas (numeros reales) tractavit, sed saeculo 19 definitio amplificata est ad alia genera numerorum comprehenda, qualia sunt numeri integri Gaussiani et polynomia unius variabilis. Hoc notiones hodiernas algebrae abstractae attulit, e.g. anulos Euclidianos. Nunc algorithmus Euclidis latiore definitur ut et alias structuras mathematicas comprehendat, e.g. nodos et polynomia multiplarum variabilium.
293: 名無しさん：14/06/17 23:18 ID:6pI
Algorithmus multes applicationes theoreticas et practicas habet. Fere ut omnes rhythmos musicales traditionales gentium variarum generat usurpari potest.[2] Pars est praecipua algorithmi RSA, methodi incryptionis clavi publica in commercio electronico pervagati. Ut aequationes Diaphonteas solvat adhibet, e.g., ut numeros qui multiplas congruentias satisfaciunt (vide theorema Sericum de residuis) vel inversos multiplicativos corporis finiti inveniat. Etiam usurpatur ut fractiones continuas construat, in methoda catenarum Sturm ut radices reales polynomii inveniat, et in pluribus algorithmis recentibus factorizationis numerorum integrorum. Denique instrumentum est elementarium ad theoremata demonstranda in theoria numerorum hodierna, talia quales sunt theorema quatuor quadratorum Lagrange et theorema fundamentale arithmeticae.
294: 名無しさん：14/06/17 23:19 ID:6pI
Divisor communis maximus duorum numerorum est maximus numerus qui ambos ita dividit ut residuum non relinquat. Algorithmus Euclidis in elemento nititur, quod DCM duorum numerorum non mutatur si numerus maior ex minore subtrahitur. Nam si k, m, n sunt integri, et k est factor communis duorum integrorum A et B, ergo A = nk et B = mk significat A − B = (n − m)k, ergo k est etiam factor communis differentiae. Quod k etiam divisorem communem maximum potest representare infra demonstratur. Exempli gratia, 21 est DCM 252 et 105 (252 = 12 × 21; 105 = 5 × 21); quia 252 − 105 = (12 − 5) × 21 = 147, DCM 147 et 105 est etiam 21.
295: 名無しさん：14/06/17 23:19 ID:6pI
Questi algoritmi possono essere usati, oltre che con i numeri interi, in ogni contesto in cui ？ possibile eseguire la divisione col resto. Ad esempio, l'algoritmo funziona per i polinomi ad una indeterminata su un campo K, o polinomi omogenei a due indeterminate su un campo, o gli interi gaussiani. Un oggetto algebrico in cui ？ possibile eseguire la divisione col resto ？ chiamato anello euclideo.
296: 名無しさん：14/06/17 23:20 ID:6pI
Siano a e b interi positivi assegnati, e sia d il loro MCD. Definiamo la successione di ricorrenza corrispondente ai passi dell'algoritmo di Euclide: a0 = a, b0 = b, an+1=bn, e bn+1 ？ il resto della divisione di an per bn, cio？ an = qnbn + bn+1. Per definizione di resto nella divisione, bn+1 < bn per ogni n, quindi la successione dei bn ？ strettamente decrescente, e quindi esiste un N tale che bN = 0. Vogliamo dimostrare che d = aN. Infatti, per induzione si ha che per ogni n, d|an = bn-1 = an-2 - qn-2bn-2. Inoltre, sempre per induzione, aN divide an per ogni n？N, quindi divide anche bn per ogni n<N, quindi aN = d.
297: 名無しさん：14/06/17 23:21 ID:6pI
Quando si analizza il tempo di calcolo dell'algoritmo di Euclide, si trova che i valori di input che richiedono il maggior numero di divisioni sono due successivi numeri di Fibonacci, e il caso peggiore richiede O(n) divisioni, dove n ？ il numero di cifre nell'input. Occorre anche notare che le divisioni non sono operazioni atomiche (se i numeri sono pi？ grandi della dimensione naturale delle operazioni aritmetiche del computer), visto che la dimensione degli operandi pu？ essere di n cifre. Allora il tempo di calcolo reale ？ quindi O(n？).
298: 名無しさん：14/06/17 23:21 ID:6pI
L'algoritmo di Euclide ？ ampiamente usato nella pratica, specialmente per numeri piccoli, grazie alla sua semplicit？. Un algoritmo alternativo, l'algoritmo del MCD binario, utilizza la rappresentazione binaria dei computer per evitare le divisioni e quindi aumentare l'efficienza, sebbene anch'esso sia dell'ordine di O(n？): infatti su molte macchine reali permette di diminuire le costanti nascoste nella notazione "O grande".