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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 an alternative notation. It is obtained by replacing /math> in the square bracket notation by n-2 arrows. For example: * the single arrow \uparrow represents exponentiation (iterated multiplication) 2 \uparrow 4 = H_3(2,4) = 2\times ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a ''product''. The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the ''multiplicand'', as the quantity of the other one, the ''multiplier''. Both numbers can be referred to as ''factors''. :a\times b = \underbrace_ For example, 4 multiplied by 3, often written as 3 \times 4 and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together: :3 \times 4 = 4 + 4 + 4 = 12 Here, 3 (the ''multiplier'') and 4 (the ''multiplicand'') are the ''factors'', and 12 is the ''product''. One of the main properties of multiplication is ... [...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 as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least 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 that is greater than every natural number, along with ordinal numbers \omega + 1, \omega + 2, etc., which are even greater than \omega. A linear order such that every subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to ... [...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 p represents the number p. # ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyper Operator
In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called ''hyperoperations'' in this context) that starts with a unary operation (the successor function with ''n'' = 0). The sequence continues with the binary operations of addition (''n'' = 1), multiplication (''n'' = 2), and exponentiation (''n'' = 3). After that, the sequence proceeds with further binary operations extending beyond exponentiation, using Operator associativity, right-associativity. For the operations beyond exponentiation, the ''n''th member of this sequence is named by Reuben Goodstein after the Numerical prefix, Greek prefix of ''n'' suffixed with ''-ation'' (such as tetration (''n'' = 4), pentation (''n'' = 5), hexation (''n'' = 6), etc.) and can be written as using ''n'' − 2 arrows in Knuth's up-arrow notation. Each hyperoperation may be understood Recursion (computer science), recursively in terms of the previous one by: :a[n]b = \underbrace_,\quad n \ge 2 I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Caret
Caret is the name used familiarly for the character , provided on most QWERTY keyboards by typing . The symbol has a variety of uses in programming and mathematics. The name "caret" arose from its visual similarity to the original proofreader's caret, a mark used in proofreading to indicate where a punctuation mark, word, or phrase should be inserted into a document. The formal ASCII standard (X3.64.1977) calls it a "circumflex". History Typewriters On typewriters designed for languages that routinely use diacritics (accent marks), there are two possible ways to type these. Keys can be dedicated to precomposed characters (with the diacritic included) or alternatively a dead key mechanism can be provided. With the latter, a mark is made when a dead key is typed but, unlike normal keys, the paper carriage does not move on and thus the next letter to be typed is printed under the accent. The symbol was originally provided in typewriters and computer printers so that circumfl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Character Set
Character encoding is the process of assigning numbers to graphical characters, especially the written characters of human language, allowing them to be stored, transmitted, and transformed using digital computers. The numerical values that make up a character encoding are known as "code points" and collectively comprise a "code space", a " code page", or a "character map". Early character codes associated with the optical or electrical telegraph could only represent a subset of the characters used in written languages, sometimes restricted to upper case letters, numerals and some punctuation only. The low cost of digital representation of data in modern computer systems allows more elaborate character codes (such as Unicode) which represent most of the characters used in many written languages. Character encoding using internationally accepted standards permits worldwide interchange of text in electronic form. History The history of character codes illustrates the evol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superscript
A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, while superscripts are above. Subscripts and superscripts are perhaps most often used in formulas, mathematical expressions, and specifications of chemical compounds and isotopes, but have many other uses as well. In professional typography, subscript and superscript characters are not simply ordinary characters reduced in size; to keep them visually consistent with the rest of the font, typeface designers make them slightly heavier (i.e. medium or bold typography) than a reduced-size character would be. The vertical distance that sub- or superscripted text is moved from the original baseline varies by typeface and by use. In typesetting, such types are traditionally called "superior" and "inferior" letters, figures, etc., or just "superior ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E-mail
Electronic mail (email or e-mail) is a method of exchanging messages ("mail") between people using electronic devices. Email was thus conceived as the electronic ( digital) version of, or counterpart to, mail, at a time when "mail" meant only physical mail (hence '' e- + mail''). Email later became a ubiquitous (very widely used) communication medium, to the point that in current use, an email address is often treated as a basic and necessary part of many processes in business, commerce, government, education, entertainment, and other spheres of daily life in most countries. ''Email'' is the medium, and each message sent therewith is also called an ''email.'' The term is a mass noun. Email operates across computer networks, primarily the Internet, and also local area networks. Today's email systems are based on a store-and-forward model. Email servers accept, forward, deliver, and store messages. Neither the users nor their computers are required to be online simult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Programming Language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming language is usually split into the two components of syntax (form) and semantics (meaning), which are usually defined by a formal language. Some languages are defined by a specification document (for example, the C programming language is specified by an ISO Standard) while other languages (such as Perl) have a dominant implementation that is treated as a reference. Some languages have both, with the basic language defined by a standard and extensions taken from the dominant implementation being common. Programming language theory is the subfield of computer science that studies the design, implementation, analysis, characterization, and classification of programming languages. Definitions There are many considerations when defini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexation
In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called ''hyperoperations'' in this context) that starts with a unary operation (the successor function with ''n'' = 0). The sequence continues with the binary operations of addition (''n'' = 1), multiplication (''n'' = 2), and exponentiation (''n'' = 3). After that, the sequence proceeds with further binary operations extending beyond exponentiation, using right-associativity. For the operations beyond exponentiation, the ''n''th member of this sequence is named by Reuben Goodstein after the Greek prefix of ''n'' suffixed with ''-ation'' (such as tetration (''n'' = 4), pentation (''n'' = 5), hexation (''n'' = 6), etc.) and can be written as using ''n'' − 2 arrows in Knuth's up-arrow notation. Each hyperoperation may be understood recursively in terms of the previous one by: :a = \underbrace_,\quad n \ge 2 It may also be defined according to the recursion rule part of the definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentation
In mathematics, pentation (or hyper-5) is the next hyperoperation after tetration and before hexation. It is defined as iterated (repeated) tetration, just as tetration is iterated exponentiation. It is a binary operation defined with two numbers ''a'' and ''b'', where ''a'' is tetrated to itself ''b-1'' times. For instance, using hyperoperation notation for pentation and tetration, 2 means 2 to itself 2 times, or 2 2 ). This can then be reduced to 2 2^2)=2 =2^=2^=2^=65536. Etymology The word "pentation" was coined by Reuben Goodstein in 1947 from the roots penta- (five) and iteration. It is part of his general naming scheme for hyperoperations. Notation There is little consensus on the notation for pentation; as such, there are many different ways to write the operation. However, some are more used than others, and some have clear advantages or disadvantages compared to others. *Pentation can be written as a hyperoperation as a . In this format, a may be interpreted as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
