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Approximations
An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ''ad-'' (''ad-'' before ''p'' becomes ap- by assimilation) meaning ''to''. Words like ''approximate'', ''approximately'' and ''approximation'' are used especially in technical or scientific contexts. In everyday English, words such as ''roughly'' or ''around'' are used with a similar meaning. It is often found abbreviated as ''approx.'' The term can be applied to various properties (e.g., value, quantity, image, description) that are nearly, but not exactly correct; similar, but not exactly the same (e.g., the approximate time was 10 o'clock). Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws. In science, approximation can refer to usi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Approximation Theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' will depend on the application. A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials. One problem of particular interest is that of approximating a function in a computer mathematical library, using operations that can be performed on the computer or calculator (e.g. addition and multiplication), such that the result is as close to the actual function as possible. This is typically done with polynomial or rational (ratio of polynomials) approximations. The objective is to make the approximation as close as possible to the actual function, typically with an accuracy close to that of the underlying computer's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Diophantine Approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number ''a''/''b'' is a "good" approximation of a real number ''α'' if the absolute value of the difference between ''a''/''b'' and ''α'' may not decrease if ''a''/''b'' is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of continued fractions. Knowing the "best" approximations of a given number, the main problem of the field is to find sharp upper and lower bounds of the above difference, expressed as a function of the denominator. It appears that these bounds depend on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number is larger than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of is , or . The logarithm of to ''base'' is denoted as , or without parentheses, , or even without the explicit base, , when no confusion is possible, or when the base does not matter such as in big O notation. The logarithm base is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e (mathematical constant), as its base; its use is widespread in mathematics and physics, because of its very simple derivative. The binary logarithm uses base and is frequently used in computer science. Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Equality (mathematics)
In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. The equality between and is written , and pronounced equals . The symbol "" is called an "equals sign". Two objects that are not equal are said to be distinct. For example: * x=y means that and denote the same object. * The identity (x+1)^2=x^2+2x+1 means that if is any number, then the two expressions have the same value. This may also be interpreted as saying that the two sides of the equals sign represent the same function. * \ = \ if and only if P(x) \Leftrightarrow Q(x). This assertion, which uses set-builder notation, means that if the elements satisfying the property P(x) are the same as the elements satisfying Q(x), then the two uses of the set-builder notation define the same set. This property is often expressed as "two sets that have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Round-off Error
A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are due to inexactness in the representation of real numbers and the arithmetic operations done with them. This is a form of quantization error. When using approximation equations or algorithms, especially when using finitely many digits to represent real numbers (which in theory have infinitely many digits), one of the goals of numerical analysis is to estimate computation errors. Computation errors, also called numerical errors, include both truncation errors and roundoff errors. When a sequence of calculations with an input involving any roundoff error are made, errors may accumulate, sometimes dominating the calculation. In ill-conditioned problems, significant error may accumulate. In short, there are two major facets of roundoff e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Korean Language
Korean (South Korean: , ''hangugeo''; North Korean: , ''chosŏnmal'') is the native language for about 80 million people, mostly of Koreans, Korean descent. It is the official language, official and national language of both North Korea and South Korea (geographically Korea), but over the past years of political division, the North–South differences in the Korean language, two Koreas have developed some noticeable vocabulary differences. Beyond Korea, the language is recognised as a minority language in parts of China, namely Jilin, Jilin Province, and specifically Yanbian Korean Autonomous Prefecture, Yanbian Prefecture and Changbai Korean Autonomous County, Changbai County. It is also spoken by Sakhalin Koreans in parts of Sakhalin, the Russian island just north of Japan, and by the in parts of Central Asia. The language has a few Extinct language, extinct relatives which—along with the Jeju language (Jejuan) of Jeju Island and Korean itself—form the compact Koreanic l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Taiwanese Mandarin
Taiwanese Mandarin, ''Guoyu'' ( zh, s=, t=國語, p=Guóyǔ, l=National Language, first=t) or ''Huayu'' ( zh, s=, t=華語, p=Huáyǔ, first=t, l=Mandarin Language, labels=no) refers to Mandarin Chinese spoken in Taiwan. A large majority of the Taiwanese population is fluent in Mandarin, though many also speak Taiwanese Hokkien, commonly called ''Minnanyu'' ( ''Mǐnnányǔ'') or Southern Min, a variety of Min Chinese. This language has had significant influence on Mandarin as spoken on the island. ''Guoyu'' is not the indigenous language of Taiwan. Chinese settlers came to Taiwan in the 16th century, but spoke other Chinese languages, primarily Southern Min. Japan annexed Taiwan in 1895 and governed the island as a colony for the next 50 years, during which time Japanese was introduced and taught in schools, while non-Mandarin languages were spoken at home. With the defeat of Imperial Japan in World War II, Taiwan was returned to the Republic of China under the Kuomintang (KMT ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Japanese Language
is spoken natively by about 128 million people, primarily by Japanese people and primarily in Japan, the only country where it is the national language. Japanese belongs to the Japonic or Japanese- Ryukyuan language family. There have been many attempts to group the Japonic languages with other families such as the Ainu, Austroasiatic, Koreanic, and the now-discredited Altaic, but none of these proposals has gained widespread acceptance. Little is known of the language's prehistory, or when it first appeared in Japan. Chinese documents from the 3rd century AD recorded a few Japanese words, but substantial Old Japanese texts did not appear until the 8th century. From the Heian period (794–1185), there was a massive influx of Sino-Japanese vocabulary into the language, affecting the phonology of Early Middle Japanese. Late Middle Japanese (1185–1600) saw extensive grammatical changes and the first appearance of European loanwords. The basis of the standard dial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Limit (mathematics)
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. In formulas, a limit of a function is usually written as : \lim_ f(x) = L, (although a few authors may use "Lt" instead of "lim") and is read as "the limit of of as approaches equals ". The fact that a function approaches the limit as approaches is sometimes denoted by a right arrow (→ or \rightarrow), as in :f(x) \to L \text x \to c, which reads "f of x tends to L as x tends to c". History Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work ''Opus Geometricum'' (1647): "The ''terminus'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Congruence Relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation. Basic example The prototypical example of a congruence relation is congruence modulo n on the set of integers. For a given positive integer n, two integers a and b are called congruent modulo n, written : a \equiv b \pmod if a - b is divisible by n (or equivalently if a and b have the same remainder when divided by n). For example, 37 and 57 are congruent modulo 10, : 37 \equiv 57 \pmod since 37 - 57 = -20 is a multiple of 10, or equivalently since both 37 and 57 have a remainder of 7 when divided by 10. Congruence modulo n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping 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 isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". 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 be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Proportionality (mathematics)
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constant. Two sequences are inversely proportional if corresponding elements have a constant product, also called the coefficient of proportionality. This definition is commonly extended to related varying quantities, which are often called ''variables''. This meaning of ''variable'' is not the common meaning of the term in mathematics (see variable (mathematics)); these two different concepts share the same name for historical reasons. Two functions f(x) and g(x) are ''proportional'' if their ratio \frac is a constant function. If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a proportion, e.g., (for details see Ratio). Proportionality is closely r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |