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Euclidean Line
Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. Geometry *Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry as well as their higher dimensional generalizations *Euclidean geometry, the study of the properties of Euclidean spaces *Non-Euclidean geometry, systems of points, lines, and planes analogous to Euclidean geometry but without uniquely determined parallel lines *Euclidean distance, the distance between pairs of points in Euclidean spaces * Euclidean ball, the set of points within some fixed distance from a center point Number theory *Euclidean division, the division which produces a quotient and a remainder *Euclidean algorithm, a method for finding greatest common divisors *Extended Euclidean algorithm, a method for solving the Diophantine equation ''ax'' + ''by'' = ''d'' where ''d'' is the greatest common divisor of ''a'' and ''b'' *Euclid's lemma: if a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid
Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics. Very little is known of Euclid's life, and most information comes from the scholars Proclus and Pappus of Alexandria many centuries later. Medieval Islamic mathematicians invented a fanciful biography, and medieval Byzantine and early Renaissance scholars mistook him for the earlier philo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of Euclidean division of integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them ( Bézout's identity). In particular, the existence of efficient algorithms for Euclidean division of integers and of polynomials in one variable over a field is of basic importance in computer algebra. It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs). An arbitrary PID has much the same "structural proper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Elements
The ''Elements'' ( ) is a mathematics, mathematical treatise written 300 BC by the Ancient Greek mathematics, Ancient Greek mathematician Euclid. ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. Drawing on the works of earlier mathematicians such as Hippocrates of Chios, Eudoxus of Cnidus and Theaetetus (mathematician), Theaetetus, the ''Elements'' is a collection in 13 books of definitions, postulates, propositions and mathematical proofs that covers plane and solid Euclidean geometry, elementary number theory, and Commensurability (mathematics), incommensurable lines. These include Pythagorean theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many prime numbers, and the Compass-and-straightedge construction, construction of regular polygons and Regular polyhedra, polyhedra. Often referred to as the most successful textbook ever written, the ''Elements'' has continued to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid (other)
Euclid generally refers to the ancient Greek mathematician Euclid of Alexandria (3rd century BC), who wrote a work on geometry called the ''Elements''. Euclid, Euclides, or Eucleides may also refer to: People * Euclid of Megara (c. 435 BC–c. 365 BC), ancient Greek philosopher * Eucleides, archon of Athens (5th century BC) * Euclid Bertrand (born 1974), Dominican former footballer * Euclides da Cunha (1866–1909), Brazilian sociologist * Euclid James Sherwood (1942–2011), American musician * Euclid Kyurdzidis (born 1968), Russian actor * Euclid Tsakalotos (born 1960), Greek economist and Minister of Finance * Nicholas Euclid (1932–2007), Australian rugby league player, coach, and official Mathematics, science, and technology * Euclid (computer program) * Euclid (programming language) * ''Euclid'', a space telescope built by ESA, launched in 2023 * Euclid, a computer system used by Euroclear * Euclid Contest, a maths competition held by the Centre for Education i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intermediate Math League Of Eastern Massachusetts
The Intermediate Math League of Eastern Massachusetts (or IMLEM) is a math league for middle schools across Eastern Massachusetts. A brief history of IMLEM is given in its By-Laws: Schools As of 2017, 86 different schools attend the competition. Each school is allowed to send more than 1 team and each team can consist of at most 10 people. Alternates, people who are not officially part of team, can be taken too. There are a total of 15 different geographic clusters of schools and there is even a cluster of schools from Pennsylvania. The schools are then separated into different divisions with the schools in each division be approximately the same level. Schools can then make their way up through divisions to try to get into the top division, which is the Lexington Division. In total there are 13 divisions. Schools may send more than one team, however no student can compete on more than one team in a year. Also, a school may send alternates to gain the experience of a meet. Meet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Zoning
Zoning is a law that divides a jurisdiction's land into districts, or zones, and limits how land in each district can be used. In the United States, zoning includes various land use laws enforced through the police power rights of state governments and local governments to exercise authority over privately owned real property. Zoning laws in major cities originated with the New York City 1916 Zoning Resolution. Before zoning, some cities had local ordinances like those in Los Angeles in 1904 limiting "wash houses" (laundries) from operating in a residential area. These early city ordinances were in some cases motivated by racism and classism. After the Supreme Court declared racial ordinances unconstitutional in 1917, many localities discovered zoning and began setting down citywide restrictions. In suburban localities, zoning often mandates single-family housing. Zoning ordinances did not allow African-Americans moving into or using residences that were occupied by majority whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Rhythm
The Euclidean rhythm in music was discovered by Godfried Toussaint in 2004 and is described in a 2005 paper "The Euclidean Algorithm Generates Traditional Musical Rhythms".The Euclidean algorithm generates traditional musical rhythms by G. T. Toussaint, ''Proceedings of BRIDGES: Mathematical Connections in Art, Music, and Science'', Banff, Alberta, Canada, July 31 to August 3, 2005, pp. 47–56. The of two numbers is used ically giving the number of [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Distance Map
A distance transform, also known as distance map or distance field, is a derived representation of a digital image. The choice of the term depends on the point of view on the object in question: whether the initial image is transformed into another representation, or it is simply endowed with an additional map or field. Distance fields can also be signed, in the case where it is important to distinguish whether the point is inside or outside of the shape. The map labels each pixel of the image with the distance to the nearest ''obstacle pixel''. A most common type of obstacle pixel is a ''boundary pixel'' in a binary image. See the image for an example of a Chebyshev distance transform on a binary image. Usually the transform/map is qualified with the chosen metric. For example, one may speak of Manhattan distance transform, if the underlying metric is Manhattan distance. Common metrics are: * Euclidean distance * Taxicab geometry, also known as ''City block distance'' or '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Relation
In mathematics, Euclidean relations are a class of binary relations that formalize ":wikisource:Page:First six books of the elements of Euclid 1847 Byrne.djvu/26, Axiom 1" in Euclid's Elements, Euclid's ''Elements'': "Magnitudes which are equal to the same are equal to each other." Definition A binary relation ''R'' on a set (mathematics), set ''X'' is Euclidean (sometimes called right Euclidean) if it satisfies the following: for every ''a'', ''b'', ''c'' in ''X'', if ''a'' is related to ''b'' and ''c'', then ''b'' is related to ''c''.. To write this in predicate logic: :\forall a, b, c\in X\,(a\,R\, b \land a \,R\, c \to b \,R\, c). Dually, a relation ''R'' on ''X'' is left Euclidean if for every ''a'', ''b'', ''c'' in ''X'', if ''b'' is related to ''a'' and ''c'' is related to ''a'', then ''b'' is related to ''c'': :\forall a, b, c\in X\,(b\,R\, a \land c \,R\, a \to b \,R\, c). Properties # Due to the commutativity of ∧ in the definition's antecedent, ''aRb'' ∧ ''aRc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Lemma
In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers: For example, if , , , then , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well. In fact, . The lemma first appeared in Euclid's '' Elements'', and is a fundamental result in elementary number theory. If the premise of the lemma does not hold, that is, if is a composite number, its consequent may be either true or false. For example, in the case of , , , composite number 10 divides , but 10 divides neither 4 nor 15. This property is the key in the proof of the fundamental theorem of arithmetic. It is used to define prime elements, a generalization of prime numbers to arbitrary commutative rings. Euclid's lemma shows that in the integers irreducible elements are also prime elements. The proof uses induction so it does not apply to all integral domains. Formulations Euclid's lemma is commonly used in the following e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Euclidean Algorithm
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers ''a'' and ''b'', also the coefficients of Bézout's identity, which are integers ''x'' and ''y'' such that : ax + by = \gcd(a, b). This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs. It allows one to compute also, with almost no extra cost, the quotients of ''a'' and ''b'' by their greatest common divisor. also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials. The extended Euclidean algorithm is particularly useful when ''a'' and ''b'' are coprime. With that provision, ''x'' is the modular multiplicative inverse of ''a'' modulo ''b'', and ''y'' is the modular multiplicative inverse of ''b'' mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
