 | This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Approximation" – news ·newspapers ·books ·scholar ·JSTOR(April 2013) (Learn how and when to remove this message) |
Anapproximation is anything that is intentionally similar but not exactlyequal to something else.
The wordapproximation is derived fromLatinapproximatus, fromproximus meaningvery near and theprefixad- (ad- beforep becomes ap- byassimilation) meaningto.[1] Words likeapproximate,approximately andapproximation are used especially in technical or scientific contexts. In everyday English, words such asroughly oraround are used with a similar meaning.[2] It is often found abbreviated asapprox.
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 tonumbers, it is also frequently applied to such things asmathematical functions,shapes, andphysical laws.
In science, approximation can refer to using a simpler process or model when the correct model is difficult to use. An approximate model is used to make calculations easier. Approximations might also be used if incompleteinformation prevents use of exact representations.
The type of approximation used depends on the availableinformation,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.
Approximation theory is a branch of mathematics, and a quantitative part offunctional analysis.Diophantine approximation deals with approximations ofreal numbers byrational numbers.
Approximation usually occurs when an exact form or an exact numerical number is unknown or difficult to obtain. However some known form may exist and may be able to represent the real form so that no significant deviation can be found. For example, 1.5 × 106 means that the true value of something being measured is 1,500,000 to the nearest hundred thousand (so the actual value is somewhere between 1,450,000 and 1,550,000); this is in contrast to the notation 1.500 × 106, which means that the true value is 1,500,000 to the nearest thousand (implying that the true value is somewhere between 1,499,500 and 1,500,500).
Numerical approximations sometimes result from using a small number ofsignificant digits. Calculations are likely to involverounding errors and otherapproximation errors.Log tables, slide rules and calculators produce approximate answers to all but the simplest calculations. 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.[3] Approximation can occur when a decimal number cannot be expressed in a finite number of binary digits.
Related to approximation of functions is theasymptotic value of a function, i.e. the value as one or more of a function's parameters becomes arbitrarily large. For example, the sum is asymptotically equal tok. 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.
Theapproximately equals sign,≈, was introduced by British mathematicianAlfred Greenhill in 1892, in his bookApplications of Elliptic Functions.[4][5]
| ≅ ≈ | |
|---|---|
Approximately equal to Almost equal to | |
| In Unicode | U+2245 ≅APPROXIMATELY EQUAL TO (≅, ≅) U+2248 ≈ALMOST EQUAL TO (≈, ≈, ≈, ≈, ≈, ≈) |
| Different from | |
| Different from | U+2242 ≂MINUS TILDE |
| Related | |
| See also | U+2249 ≉NOT ALMOST EQUAL TO U+003D =EQUALS SIGN U+2243 ≃ASYMPTOTICALLY EQUAL TO |
Typical meanings ofLaTeX symbols.
\approx) : approximate equality, like.\not\approx) : inequality, despite any approximation ().\simeq) : function asymptotic equivalence, like.\sim) : function proportionality; the used in\simeq is.\cong) : figure congruence, like.\eqsim) : equal up to a constant.\lessapprox) and (\gtrapprox) : either an inequality holds or approximate equality.Approximate equalities denoted by wavy or dotted symbols.[6]
| U+223C ∼TILDE OPERATOR | Sometimes indicatesproportionality. |
| U+223D ∽REVERSED TILDE | Sometimes indicates proportionality. |
| U+2243 ≃ASYMPTOTICALLY EQUAL TO | Combined "≈" and "=" representingasymptotic equality. |
| U+2245 ≅APPROXIMATELY EQUAL TO | Combined "≈" and "=" representingisomorphism orcongruence. |
| U+2246 ≆APPROXIMATELY BUT NOT ACTUALLY EQUAL TO | |
| U+2247 ≇NEITHER APPROXIMATELY NOR ACTUALLY EQUAL TO | |
| U+2248 ≈ALMOST EQUAL TO | |
| U+2249 ≉NOT ALMOST EQUAL TO | |
| U+224A ≊ALMOST EQUAL OR EQUAL TO | Combined "≈" and "=" representing equivalence or approximate equivalence. |
| U+2250 ≐APPROACHES THE LIMIT | Represents a variable, likey, approaching alimit, for example,.[7] |
| U+2252 ≒APPROXIMATELY EQUAL TO OR THE IMAGE OF | "≈" or "≃" equivalent inJapan,Taiwan, andKorea. |
| U+2253 ≓IMAGE OF OR APPROXIMATELY EQUAL TO | Reversed variant of "≒" (U+2252). |
| U+225F ≟QUESTIONED EQUAL TO | |
| U+2A85 ⪅LESS-THAN OR APPROXIMATE | |
| U+2A86 ⪆GREATER-THAN OR APPROXIMATE |
Approximation arises naturally inscientific experiments. The predictions of a scientific theory can differ from actual measurements. This can be because there are factors in the real situation that are not included in the theory. For example, simple calculations may not include the effect of air resistance. Under these circumstances, the theory is an approximation to reality. Differences may also arise because of limitations in the measuring technique. In this case, the measurement is an approximation to the actual value.
Thehistory of science shows that earlier theories and laws can beapproximations to some deeper set of laws. Under thecorrespondence principle, a new scientific theory should reproduce the results of older, well-established, theories in those domains where the old theories work.[8] The old theory becomes an approximation to the new theory.
Some problems in physics are too complex to solve by direct analysis, or progress could be limited by available analytical tools. Thus, even when the exact representation is known, an approximation may yield a sufficiently accurate solution while reducing the complexity of the problem significantly.Physicists often approximate theshape of the Earth as asphere even though more accurate representations are possible, because many physical characteristics (e.g.,gravity) are much easier to calculate for a sphere than for other shapes.
Approximation is also used to analyze the motion of several planets orbiting a star. This is extremely difficult due to the complex interactions of the planets' gravitational effects on each other.[9] An approximate solution is effected by performingiterations. In the first iteration, the planets' gravitational interactions are ignored, and the star is assumed to be fixed. If a more precise solution is desired, another iteration is then performed, using the positions and motions of the planets as identified in the first iteration, but adding a first-order gravity interaction from each planet on the others. This process may be repeated until a satisfactorily precise solution is obtained.
The use ofperturbations to correct for the errors can yield more accurate solutions. Simulations of the motions of the planets and the star also yields more accurate solutions.
The most common versions ofphilosophy of science accept that empiricalmeasurements are alwaysapproximations — they do not perfectly represent what is being measured.
Within theEuropean Union (EU), "approximation" refers to a process through which EU legislation is implemented and incorporated withinMember States' national laws, despite variations in the existing legal framework in each country. Approximation is required as part of thepre-accession process for new member states,[10] and as a continuing process when required by anEU Directive.Approximation is a key word generally employed within the title of a directive, for example the Trade Marks Directive of 16 December 2015 serves "to approximate the laws of the Member States relating to trade marks".[11] TheEuropean Commission describes approximation of law as "a unique obligation of membership in the European Union".[10]
{{cite book}}:ISBN / Date incompatibility (help)≐ approaches a limit
Media related toApproximation at Wikimedia Commons