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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 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[5]b, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reuben Goodstein
Reuben Louis Goodstein (15 December 1912 – 8 March 1985) was an English mathematician with an interest in the philosophy and teaching of mathematics. Education Goodstein was educated at St Paul's School in London. He received his Master's degree from Magdalene College, Cambridge. After this, he worked at the University of Reading but ultimately spent most of his academic career at the University of Leicester. He earned his PhD from the University of London in 1946 while still working in Reading. Goodstein also studied under Ludwig Wittgenstein. Research He published many works on finitism and the reconstruction of analysis from a finitistic viewpoint, for example "Constructive Formalism. Essays on the foundations of mathematics." Goodstein's theorem was among the earliest examples of theorems found to be unprovable in Peano arithmetic but provable in stronger logical systems (such as second-order arithmetic). He also introduced a variant of the Ackermann function that i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skewes's Number
In number theory, Skewes's number is the smallest natural number x for which the prime-counting function \pi(x) exceeds the logarithmic integral function \operatorname(x). It is named for the South African mathematician Stanley Skewes who first computed an upper bound on its value. The exact value of Skewes's number is still not known, but it is known that there is a crossing between \pi(x) \operatorname(x) near e^ \operatorname(x), Skewes's research supervisor J.E. Littlewood had proved in that there is such a number (and so, a first such number); and indeed found that the sign of the difference \pi(x) - \operatorname(x) changes infinitely many times. Littlewood's proof did not, however, exhibit a concrete such number x, nor did it even give any bounds on the value. Skewes's task was to make Littlewood's existence proof effective: exhibit some concrete upper bound for the first sign change. According to Georg Kreisel, this was not considered obvious even in principle at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplication
Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathematics), product''. Multiplication is often denoted by the cross symbol, , by the mid-line dot operator, , by juxtaposition, or, in programming languages, by an asterisk, . 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''. This is to be distinguished from term (arithmetic), ''terms'', which are added. :a\times b = \underbrace_ . Whether the first factor is the multiplier or the multiplicand may be ambiguous or depend upon context. For example, the expression 3 \times 4 , can be phrased as "3 ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol, +) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication, and Division (mathematics), division. The addition of two Natural number, whole numbers results in the total or ''summation, sum'' of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as , which is read as "three plus two Equality (mathematics), equals five". Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers, and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as Euclidean vector, vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
0⁰
Zero to the power of zero, denoted as , is a mathematical expression with different interpretations depending on the context. In certain areas of mathematics, such as combinatorics and algebra, is conventionally defined as 1 because this assignment simplifies many formulas and ensures consistency in operations involving exponents. For instance, in combinatorics, defining aligns with the interpretation of choosing 0 elements from a set and simplifies polynomial and binomial expansions. However, in other contexts, particularly in mathematical analysis, is often considered an indeterminate form. This is because the value of as both and approach zero can lead to different results based on the limiting process. The expression arises in limit problems and may result in a range of values or diverge to infinity, making it difficult to assign a single consistent value in these cases. The treatment of also varies across different computer programming languages and software. While m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduction Strategy
In rewriting, a reduction strategy or rewriting strategy is a relation specifying a rewrite for each object or term, compatible with a given reduction relation. Some authors use the term to refer to an evaluation strategy. Definitions Formally, for an abstract rewriting system (A, \to), a reduction strategy \to_S is a binary relation on A with \to_S \subseteq \overset , where \overset is the transitive closure of \to (but not the reflexive closure). In addition the normal forms of the strategy must be the same as the normal forms of the original rewriting system, i.e. for all a, there exists a b with a\to b iff \exists b'. a\to_S b'. A ''one step'' reduction strategy is one where \to_S \subseteq \to. Otherwise it is a ''many step'' strategy. A ''deterministic'' strategy is one where \to_S is a partial function, i.e. for each a\in A there is at most one b such that a \to_S b. Otherwise it is a ''nondeterministic'' strategy. Term rewriting In a term rewriting system a rewr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stack (abstract Data Type)
In computer science, a stack is an abstract data type that serves as a collection (abstract data type), collection of elements with two main operations: * Push, which adds an element to the collection, and * Pop, which removes the most recently added element. Additionally, a peek (data type operation), peek operation can, without modifying the stack, return the value of the last element added. The name ''stack'' is an analogy to a set of physical items stacked one atop another, such as a stack of plates. The order in which an element added to or removed from a stack is described as last in, first out, referred to by the acronym LIFO. As with a stack of physical objects, this structure makes it easy to take an item off the top of the stack, but accessing a Data, datum deeper in the stack may require removing multiple other items first. Considered a sequential collection, a stack has one end which is the only position at which the push and pop operations may occur, the ''top'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rewriting
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduction systems). In their most basic form, they consist of a set of objects, plus relations on how to transform those objects. Rewriting can be non-deterministic. One rule to rewrite a term could be applied in many different ways to that term, or more than one rule could be applicable. Rewriting systems then do not provide an algorithm for changing one term to another, but a set of possible rule applications. When combined with an appropriate algorithm, however, rewrite systems can be viewed as computer programs, and several theorem provers and declarative programming languages are based on term rewriting. Example cases Logic In logic, the procedure for obtaining the conjunctive normal form (CNF) of a formula can be implemented as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: \begin (2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\ 2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 . \end Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real numbers, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Successor Function
In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor function is one of the basic components used to build a primitive recursive function. Successor operations are also known as zeration in the context of a zeroth hyperoperation: H0(''a'', ''b'') = 1 + ''b''. In this context, the extension of zeration is addition, which is defined as repeated succession. Overview The successor function is part of the formal language used to state the Peano axioms, which formalise the structure of the natural numbers. In this formalisation, the successor function is a primitive operation on the natural numbers, in terms of which the standard natural numbers and addition are defined. For example, 1 is defined to be ''S''(0), and addition on natural numbers is defined recursively by: : This can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
