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Surreal Number
In mathematics, the surreal number system is a total order, totally ordered proper class containing not only the real numbers but also Infinity, infinite and infinitesimal, infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book ''Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness''. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surreal Number Tree
Surreal may refer to: *Anything related to or characteristic of Surrealism, a movement in philosophy and art *Surreal (song), "Surreal" (song), a 2000 song by Ayumi Hamasaki *"Surreal", a 2023 song by Luísa Sonza and Baco Exu do Blues *Surreal (album), ''Surreal'' (album), an album by Man Raze *Surreal humour, a common aspect of humor *Surreal numbers, a superset of the real numbers in mathematics *Surreal Software, an American video game studio See also *Surrealist automatism *Surrealist Manifesto *Surrealist techniques *Surrealist music {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epsilon Numbers (mathematics)
In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication. The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ''ε'' that satisfy the equation :\varepsilon = \omega^\varepsilon, \, in which ω is the smallest infinite ordinal. The least such ordinal is ''ε''0 (pronounced epsilon nought (chiefly British), epsilon naught (chiefly American), or epsilon zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals: :\varepsilon_0 = \omega^ = \sup \left\\,, where is the supremum, which is equivalent to set union in the case of the von Neumann representation of ordinals. Larger ordinal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Power Series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, of the form \sum_^\infty a_nx^n=a_0+a_1x+ a_2x^2+\cdots, where the a_n, called ''coefficients'', are numbers or, more generally, elements of some ring, and the x^n are formal powers of the symbol x that is called an indeterminate or, commonly, a variable. Hence, power series can be viewed as a generalization of polynomials where the number of terms is allowed to be infinite, and differ from usual power series by the absence of convergence requirements, which implies that a power series may not represent a function of its variable. Formal power series are in one to one correspondence with their sequences of coefficients, but the two concepts must not be confused, sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hahn Series
In mathematics, Hahn series (sometimes also known as Hahn–Mal'cev–Neumann series) are a type of formal series, formal infinite series. They are a generalization of Puiseux series (themselves a generalization of formal power series) and were first introduced by Hans Hahn (mathematician), Hans Hahn in 1907 (and then further generalized by Anatoly Maltsev and Bernhard Neumann to a non-commutative setting). They allow for arbitrary exponents of the Indeterminate (variable), indeterminate so long as the set supporting them forms a well-ordered subset of the Valuation (algebra), value group (typically \mathbb or \mathbb). Hahn series were first introduced, as groups, in the course of the mathematical proof, proof of the Hahn embedding theorem and then studied by him in relation to Hilbert's second problem. Formulation The field (mathematics), field of Hahn series K\left[\left[T^\Gamma\right]\right] (in the indeterminate T) over a field K and with value group \Gamma (an ordered group) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hans Hahn (mathematician)
Hans Hahn (; ; 27 September 1879 – 24 July 1934) was an Austrian mathematician and philosopher who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory. In philosophy he was among the main logical positivists of the Vienna Circle. Biography Born in Vienna as the son of a higher government official of the K.K. Telegraphen-Korrespondenz Bureau (since 1946 named "Austria Presse Agentur"), in 1898 Hahn became a student at the Universität Wien starting with a study of law. In 1899 he switched over to mathematics and spent some time at the universities of Strasbourg, Munich and Göttingen. In 1902 he took his Ph.D. in Vienna, on the subject "Zur Theorie der zweiten Variation einfacher Integrale". He was a student of Gustav von Escherich. He was appointed to the teaching staff ( Habilitation) in Vienna in 1905. After 1905/1906 as a stand-in for Otto Stolz at the University of Innsbruck and some further year ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
On Numbers And Games
''On Numbers and Games'' is a mathematics book by John Horton Conway first published in 1976. The book is written by a pre-eminent mathematician, and is directed at other mathematicians. The material is, however, developed in a playful and unpretentious manner and many chapters are accessible to non-mathematicians. Martin Gardner discussed the book at length, particularly Conway's construction of surreal numbers, in his Mathematical Games column in ''Scientific American'' in September 1976. The book is roughly divided into two sections: the first half (or ''Zeroth Part''), on numbers, the second half (or ''First Part''), on games. In the ''Zeroth Part'', Conway provides axioms for arithmetic: addition, subtraction, multiplication, division and inequality. This allows an axiomatic construction of numbers and ordinal arithmetic, namely, the integers, reals, the countable infinity, and entire towers of infinite ordinals. The object to which these axioms apply takes the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Go Strategy And Tactics
The game of Go has simple rules that can be learned very quickly but, as with chess and similar board games, complex strategies may be employed by experienced players. Go opening theory The whole board opening is called fuseki. An important principle to follow in early play is "corner, side, center." In other words, the corners are the easiest places to take territory, because two sides of the board can be used as boundaries. Once the corners are occupied, the next most valuable points are along the sides, aiming to use the edge as a territorial boundary. Capturing territory in the middle, where it must be surrounded on all four sides, is extremely difficult. The same is true for founding a living group: Easiest in the corner, most difficult in the center. The first moves are usually played on or near the 4-4 star points in the corners, because in those places it is easiest to gain territory or influence. In order to be totally secure alone, a corner stone must be placed on t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Arithmetic
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations. Addition The sum of two well-ordered sets and is the ordinal representing the variant of lexicographical order with least significant position first, on the union of the Cartesian products and . This way, every element of is smaller than every element of , comparisons within keep the order they already have, and likewise for comparisons within . The definition of addition can also be given by transfinite recursion on . When the right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally using linearly ordered greek letter variables that include the natural numbers and have the property that every set of ordinals has a least or "smallest" element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number \omega (omega) to be the least element that is greater than every natural number, along with ordinal numbers , , etc., which are even greater than . A linear order such that every non-empty subset has a least element is called a well-order. The axiom of choice implies that every set can be well-orde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transfinite Number
In mathematics, transfinite numbers or infinite numbers are numbers that are " infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term ''transfinite'' was coined in 1895 by Georg Cantor, who wished to avoid some of the implications of the word ''infinite'' in connection with these objects, which were, nevertheless, not ''finite''. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as ''infinite numbers''. Nevertheless, the term ''transfinite'' also remains in use. Notable work on transfinite numbers was done by Wacław Sierpiński: ''Leçons sur les nombres transfinis'' (1928 book) much expanded into '' Cardinal and Ordinal Numbers'' (1958, 2nd ed. 1965). Definition Any finite natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperreal Number
In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
