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Pentated
In mathematics, pentation (or hyper-5) is the fifth hyperoperation. Pentation is defined to be repeated tetration, similarly to how tetration is repeated exponentiation, exponentiation is repeated multiplication, and multiplication is repeated addition. The concept of "pentation" was named by English mathematician Reuben Goodstein in 1947, when he came up with the naming scheme for hyperoperations. The number ''a'' pentated to the number ''b'' is defined as ''a'' tetrated to itself ''b - 1'' times. This may variously be denoted as a , a\uparrow\uparrow\uparrow b, a\uparrow^3 b, a\to b\to 3, or , depending on one's choice of notation. For example, 2 pentated to 2 is 2 tetrated to 2, or 2 raised to the power of 2, which is 2^2 = 4. As another example, 2 pentated to 3 is 2 tetrated to the result of 2 tetrated to 2. Since 2 tetrated to 2 is 4, 2 pentated to 3 is 2 tetrated to 4, which is 2^ = 65536. Based on this definition, pentation is only defined when ''a'' and ''b'' are both po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterated Function
In mathematics, an iterated function is a function that is obtained by composing another function with itself two or several times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial object, the result of applying a given function is fed again into the function as input, and this process is repeated. For example, on the image on the right: : Iterated functions are studied in computer science, fractals, dynamical systems, mathematics and renormalization group physics. Definition The formal definition of an iterated function on a set ''X'' follows. Let be a set and be a function. Defining as the ''n''-th iterate of , where ''n'' is a non-negative integer, by: f^0 ~ \stackrel ~ \operatorname_X and f^ ~ \stackrel ~ f \circ f^, where is the identity function on and denotes function composition. This notation has been traced to and John Frederick William Herschel in 1813. Herschel credited ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of Large Numbers
Different cultures used different traditional numeral systems for naming large numbers. The extent of large numbers used varied in each culture. Two interesting points in using large numbers are the confusion on the term billion and milliard in many countries, and the use of ''zillion'' to denote a very large number where precision is not required. Indian mathematics The Shukla Yajurveda has a list of names for powers of ten up to 1012. The list given in the Yajurveda text is: :''eka'' (1), ''daśa'' (10), ''mesochi'' (100), ''sahasra'' (1,000), ''ayuta'' (10,000), ''niyuta'' (100,000), ''prayuta'' (1,000,000), ''arbuda'' (10,000,000), ''nyarbuda'' (100,000,000), ''saguran'' (1,000,000,000), ''madhya'' (10,000,000,000), ''anta'' (100,000,000,000), ''parârdha'' (1,000,000,000,000). Later Hindu and Buddhist texts have extended this list, but these lists are no longer mutually consistent and names of numbers larger than 108 differ between texts. For example, the Panchavimsha Bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graham's Number
Graham's number is an Large numbers, immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. It is much larger than many other large numbers such as Skewes's number and Moser's number, both of which are in turn much larger than a googolplex. As with these, it is so large that the observable universe is far too small to contain an ordinary Numerical digit, digital representation of Graham's number, assuming that each digit occupies one Planck volume, possibly the smallest measurable space. But even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of ''that'' number—and so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe. Thus, Graham's number cannot be expressed even by physical universe-scale Tetrat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large Numbers
Large numbers, far beyond those encountered in everyday life—such as simple counting or financial transactions—play a crucial role in various domains. These expansive quantities appear prominently in mathematics, cosmology, cryptography, and statistical mechanics. While they often manifest as large positive integers, they can also take other forms in different contexts (such as P-adic number). Googology delves into the naming conventions and properties of these immense numerical entities. Since the customary, traditional (non-technical) decimal format of large numbers can be lengthy, other systems have been devised that allows for shorter representation. For example, a billion is represented as 13 characters (1,000,000,000) in decimal format, but is only 3 characters (109) when expressed in exponential format. A trillion is 17 characters in decimal, but only 4 (1012) in exponential. Values that vary dramatically can be represented and compared graphically via logarithmic sca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Tower (mathematics)
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though Knuth's up arrow notation \uparrow \uparrow and the left-exponent ^b are common. Under the definition as repeated exponentiation, means , where ' copies of ' are iterated via exponentiation, right-to-left, i.e. the application of exponentiation n-1 times. ' is called the "height" of the function, while ' is called the "base," analogous to exponentiation. It would be read as "the th tetration of ". For example, 2 tetrated to 4 (or the fourth tetration of 2) is =2^=2^=2^=65536. It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration. Tetration is also defined recursively as : := \begin 1 &\textn=0, \\ a^ &\textn>0, \end allowing for the holomorphic extension of tetration to non-natural numbers such as real, complex, and ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triviality (mathematics)
In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or a particularly simple object possessing a given structure (e.g., group (mathematics), group, topological space). The noun triviality usually refers to a simple technical aspect of some proof or definition. The origin of the term in mathematical language comes from the medieval Trivium (education), trivium curriculum, which distinguishes from the more difficult quadrivium curriculum. The opposite of trivial is nontrivial, which is commonly used to indicate that an example or a solution is not simple, or that a statement or a theorem is not easy to prove. Triviality does not have a rigorous definition in mathematics. It is Subjectivity and objectivity (philosophy), subjective, and often determined in a given situation by the knowledge and experience of those considering the case. Trivial and nontrivial solutions In mathematics, the term "trivial" is ofte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Für Die Reine Und Angewandte Mathematik
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by August Leopold Crelle (Berlin) in 1826 and edited by him until his death in 1855. It was one of the first major mathematical journals that was not a proceedings of an academy. It has published many notable papers, including works of Niels Henrik Abel, Georg Cantor, Gotthold Eisenstein, Carl Friedrich Gauss and Otto Hesse. It was edited by Carl Wilhelm Borchardt from 1856 to 1880, during which time it was known as ''Borchardt's Journal''. The current editor-in-chief is Daniel Huybrechts (Rheinische Friedrich-Wilhelms-Universität Bonn). Past editors * 1826–1856: August Leopold Crelle * 1856–1880: Carl Wilhelm Borchardt * 1881–1888: Leopold Kronecker, Karl Weierstrass Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 � ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hellmuth Kneser
Hellmuth Kneser (16 April 1898 – 23 August 1973) was a German mathematician who made notable contributions to group theory and topology. His most famous result may be his theorem on the existence of a prime decomposition for 3-manifolds. His proof originated the concept of normal surface, a fundamental cornerstone of the theory of 3-manifolds. He was born in Dorpat, Russian Empire (now Tartu, Estonia) and died in Tübingen, Germany. He was the son of the mathematician Adolf Kneser and the father of the mathematician Martin Kneser. He assisted Wilhelm Süss in the founding of the Mathematical Research Institute of Oberwolfach and served as the director of the institute from 1958 to 1959. He was an editor of Mathematische Zeitschrift, Archiv der Mathematik and Aequationes Mathematicae. Kneser formulated the problem of non-integer iteration of functions and proved the existence of the entire Abel function of the exponential; on the base of this Abel function, he construct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ackermann Function
In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total function, total computable function that is not Primitive recursive function, primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total computable functions are primitive recursive. After Ackermann's publication of his function (which had three non-negative integer arguments), many authors modified it to suit various purposes, so that today "the Ackermann function" may refer to any of numerous variants of the original function. One common version is the two-argument Ackermann–Péter function developed by Rózsa Péter and Raphael Robinson. This function is defined from the recurrence relation \operatorname(m+1, n+1) = \operatorname(m, \operatorname(m+1, n)) with appropriate Base case (recursion), base cases. Its value grows very rapidly; for example, \o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conway Chained Arrow Notation
Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers. It is simply a finite sequence of positive integers separated by rightward arrows, e.g. 2\to3\to4\to5\to6. As with most combinatorial notations, the definition is recursive. In this case the notation eventually resolves to being the leftmost number raised to some (usually enormous) integer power. Definition and overview A "Conway chain" is defined as follows: * Any positive integer is a chain of length 1. * A chain of length ''n'', followed by a right-arrow → and a positive integer, together form a chain of length n+1. Any chain represents an integer, according to the six rules below. Two chains are said to be equivalent if they represent the same integer. Let a, b, c denote positive integers and let \# denote the unchanged remainder of the chain. Then: #An empty chain (or a chain of length 0) is equal to 1. #The chain a represents the number a. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knuth's Up-arrow Notation
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called ''hyperoperations''. Goodstein also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation. The sequence starts with a unary operation (the successor function with ''n'' = 0), and continues with the binary operations of addition (''n'' = 1), multiplication (''n'' = 2), exponentiation (''n'' = 3), tetration (''n'' = 4), pentation (''n'' = 5), etc. Various notations have been used to represent hyperoperations. One such notation is H_n(a,b). Knuth's up-arrow notation \uparrow is another. For example: * the single arrow \uparrow represents exponentiation (iterated multiplication) 2 \uparrow 4 = H_3(2,4) = 2\times(2\times(2\times 2)) = 2^4 = 16 * the double arrow \uparrow\uparrow represents tetration (iterated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
