|
Steinhaus–Moser Notation
In mathematics, Steinhaus–Moser notation is a notation for expressing certain large numbers. It is an extension (devised by Leo Moser) of Hugo Steinhaus's polygon notation. Definitions : 20px, n in a triangle a number in a triangle means nn. : 20px, n in a square a number in a square is equivalent to "the number inside triangles, which are all nested." : 20px, n in a pentagon a number in a pentagon is equivalent with "the number inside squares, which are all nested." etc.: written in an ()-sided polygon is equivalent with "the number inside nested -sided polygons". In a series of nested polygons, they are associated inward. The number inside two triangles is equivalent to nn inside one triangle, which is equivalent to nn raised to the power of nn. Steinhaus defined only the triangle, the square, and the circle 20px, n in a circle, which is equivalent to the pentagon defined above. Special values Steinhaus defined: *mega is the number equivalent to 2 in a c ... [...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] |
Megagon
A megagon or 1,000,000-gon is a polygon with one million sides (mega-, from the Greek μέγας, meaning "great", being a unit prefix denoting a factor of one million). Regular megagon A regular megagon is represented by the Schläfli symbol and can be constructed as a truncated 500,000-gon, t, a twice-truncated 250,000-gon, tt, a thrice-truncated 125,000-gon, ttt, or a four-fold-truncated 62,500-gon, tttt, a five-fold-truncated 31,250-gon, , or a six-fold-truncated 15,625-gon, . A regular megagon has an interior angle of 179°59'58.704" 3.14158637 rad.Darling, David J., The Universal Book of Mathematics: from Abracadabra to Zeno's Paradoxes', John Wiley & Sons, 2004. Page 249. . The area of a regular megagon with sides of length ''a'' is given by :A = 250,000a^2 \cot \frac. The perimeter of a regular megagon inscribed in the unit circle is: :2,000,000 \sin\frac, which is very close to 2π. In fact, for a circle the size of the Earth's equator, with a circumference of 4 ... [...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 computable function that is not 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, the two-argument Ackermann–Péter function is defined as follows for nonnegative integers ''m'' and ''n'': : \begin \operatorname(0, n) & = & n + 1 \\ \operatorname(m+1, 0) & = & \operatorname(m, 1) \\ \operatorname(m+1, n+1) & = & \operatorname(m, \operatorname(m+1, n)) \end Its value grows rapidly, even for small inputs. For example, is an integer o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graham's Number
Graham's number is an 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 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 power towers of the form a ^. How ... [...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] |
Large Numbers
Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical mechanics. They are typically large positive integers, or more generally, large positive real numbers, but may also be other numbers in other contexts. Googology is the study of nomenclature and properties of large numbers. In the everyday world Scientific notation was created to handle the wide range of values that occur in scientific study. 1.0 × 109, for example, means one billion, or a 1 followed by nine zeros: 1 000 000 000. The reciprocal, 1.0 × 10−9, means one billionth, or 0.000 000 001. Writing 109 instead of nine zeros saves readers the effort and hazard of counting a long series of zeros to see how large the number is. Examples of large numbers describing everyday real-world objects include: * The ... [...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 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] |
Iterated Function
In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of 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 in the function as input, and this process is repeated. For example on the image on the right: :with the circle‑shaped symbol of function composition. Iterated functions are objects of study 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 (a notation introduced by Hans Heinrich Bürmann and John Frederick William Herschel), where ''n'' is a non-negative integer, by: f^0 ~ \stackrel ~ \operatorname_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Notation
Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathematics, science, and engineering for representing complex concepts and properties in a concise, unambiguous and accurate way. For example, Albert Einstein's equation E=mc^2 is the quantitative representation in mathematical notation of the mass–energy equivalence. Mathematical notation was first introduced by François Viète at the end of the 16th century, and largely expanded during the 17th and 18th century by René Descartes, Isaac Newton, Gottfried Wilhelm Leibniz, and overall Leonhard Euler. Symbols The use of many symbols is the basis of mathematical notation. They play a similar role as words in natural languages. They may play different roles in mathematical notation similarly as verbs, adjective and nouns play different roles in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Association (mathematics)
In mathematics, the associative property is a property of some binary operations, which means 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
