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No consistent notation is used throughout mathematics and some texts use ≈ to mean approximately equal and ~ to mean asymptotically equal whereas other texts use the symbols the other way around.Īpproximation arises naturally in scientific experiments. For example, the sum ( k/2)+( k/4)+( k/8)+.( k/2^ n) is asymptotically equal to k. the value as one or more of a function's parameters becomes arbitrarily large.
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Related to approximation of functions is the asymptotic value of a function, i.e. Approximation can occur when a decimal number cannot be expressed in a finite number of binary digits. The results of computer calculations are normally an approximation expressed in a limited number of significant digits, although they can be programmed to produce more precise results.
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Log tables, slide rules and calculators produce approximate answers to all but the simplest calculations. Calculations are likely to involve rounding errors and other approximation errors. Numerical approximations sometimes result from using a small number of significant digits. It also is used when a number is not rational, such as the number π, which often is shortened to 3.14159, or 1.414 as the shortened form of √ 2. For example, 1.5 × 10 6 means that the approximation 1,500,000 has been measured to the nearest hundred thousand (the actual value is somewhere between 1,450,000 and 1,550,000), this is in contrast to the notation 1.500 × 10 6 which measures 1,500,000 to the nearest thousand (therefore giving a true value somewhere between 1,499,500 and 1,500,500). However some known form may exist and may be able to represent the real form so that no significant deviation can be found. Diophantine approximation deals with approximations of real numbers by rational numbers.Īpproximation usually occurs when an exact form or an exact numerical number is unknown or difficult to obtain. The type of approximation used depends on the available information, the degree of accuracy required, the sensitivity of the problem to this data, and the savings (usually in time and effort) that can be achieved by approximation.Īpproximation theory is a branch of mathematics, a quantitative part of functional analysis. Approximations might also be used if incomplete information prevents use of exact representations. An approximate model is used to make calculations easier. In science, approximation can refer to using a simpler process or model when the correct model is difficult to use. 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).Īlthough approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws. In everyday English, words such as roughly or around are used with a similar meaning. Words like approximate, approximately and approximation are used especially in technical or scientific contexts. 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.