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Approximation
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 u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Of Approximation
In science, engineering, and other quantitative disciplines, order of approximation refers to formal or informal expressions for how accurate an approximation is. Usage in science and engineering In formal expressions, the ordinal number used before the word order refers to the highest power in the series expansion used in the approximation. The expressions: a ''zeroth-order approximation'', a ''first-order approximation'', a ''second-order approximation'', and so forth are used as fixed phrases. The expression a ''zero-order approximation'' is also common. Cardinal numerals are occasionally used in expressions like an ''order-zero approximation'', an ''order-one approximation'', etc. The omission of the word ''order'' leads to phrases that have less formal meaning. Phrases like first approximation or to a first approximation may refer to ''a roughly approximate value of a quantity''. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 comput ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Approximation Error
The approximation error in a data value is the discrepancy between an exact value and some ''approximation'' to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute error divided by the data value). An approximation error can occur because of computing machine precision or measurement error (e.g. the length of a piece of paper is 4.53 cm but the ruler only allows you to estimate it to the nearest 0.1 cm, so you measure it as 4.5 cm). In the mathematical field of numerical analysis, the numerical stability of an algorithm indicates how the error is propagated by the algorithm. Formal definition One commonly distinguishes between the relative error and the absolute error. Given some value ''v'' and its approximation ''v''approx, the absolute error is :\epsilon = , v-v_\text, \ , where the vertical bars denote the absolute value. If v \ne 0, the relative error is : \eta = \frac = \left ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Approximation
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical Law
Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term ''law'' has diverse usage in many cases (approximate, accurate, broad, or narrow) across all fields of natural science (physics, chemistry, astronomy, geoscience, biology). Laws are developed from data and can be further developed through mathematics; in all cases they are directly or indirectly based on empirical evidence. It is generally understood that they implicitly reflect, though they do not explicitly assert, causal relationships fundamental to reality, and are discovered rather than invented. Scientific laws summarize the results of experiments or observations, usually within a certain range of application. In general, the accuracy of a law does not change when a new theory of the relevant phenomenon is worked out, but rather the scope of the law's application, since the mathematics or statement representin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Analysis
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as becomes very large, the term becomes insignificant compared to . The function is said to be "''asymptotically equivalent'' to , as ". This is often written symbolically as , which is read as " is asymptotic to ". An example of an important asymptotic result is the prime number theorem. Let denote the prime-counting function (which is not directly related to the constant pi), i.e. is the number of prime numbers that are less than or equal to . Then the theorem states that \pi(x)\sim\frac. Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is often expressed there in terms of big O notation. Definition Formally, given functions and , we define a binary relation f(x) \sim g(x) \q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called ''numerals''; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number. The most common numeral system is the Hindu–Arabic numeral system, which allows for the representation of any number using a combination of ten fundamental numeric symbols, called digits. In addition to their use in counting and measuring, numerals are often used for labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, a ''numeral'' is not clearly distinguished from the ''number'' th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 errors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Significant Figures
Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something. If a number expressing the result of a measurement (e.g., length, pressure, volume, or mass) has more digits than the number of digits allowed by the measurement resolution, then only as many digits as allowed by the measurement resolution are reliable, and so only these can be significant figures. For example, if a length measurement gives 114.8 mm while the smallest interval between marks on the ruler used in the measurement is 1 mm, then the first three digits (1, 1, and 4, showing 114 mm) are certain and so they are significant figures. Digits which are uncertain but ''reliable'' are also considered significant figures. In this example, the last digit (8, which adds 0.8 mm) is also considered a significant figure even though ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being '' incommensurable'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
