Approximation

For the sound change, seeLenition.

"Approximate" redirects here. Not to be confused withApproximant.

Anapproximationis anything that is intentionally similar but not exactlyequalto something else.

Etymology and usage

The wordapproximationis derived fromLatinapproximatus,fromproximusmeaningvery nearand theprefixad-(ad-beforepbecomes ap- byassimilation) meaningto.Words likeapproximate,approximatelyandapproximationare used especially in technical or scientific contexts. In everyday English, words such asroughlyoraroundare used with a similar meaning.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 incompleteinformationprevents 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.

Mathematics

Approximation theoryis a branch of mathematics, and a quantitative part offunctional analysis.Diophantine approximationdeals with approximations ofreal numbersbyrational 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 × 10⁶means 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 approximationssometimes result from using a small number ofsignificant digits.Calculations are likely to involverounding errorsand 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.Approximation can occur when a decimal number cannot be expressed in a finite number of binary digits.

Related to approximation of functions is theasymptoticvalue of a function, i.e. the value as one or more of a function's parameters becomes arbitrarily large. For example, the sum⁠$k/2+k/4+k/8+\cdots +k/2^{n}$⁠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.

Typography

See also:Glossary of mathematical symbols § Equality, equivalence and similarity

One of the first uses of the symbol (≈) "Approximately equal to." -Alfred Greenhill(1892)

Theapproximately equals sign,≈,was introduced by British mathematicianAlfred Greenhillin 1892, in his bookApplications of Elliptic Functions.

≅ ≈
Approximately equal to Almost equal to
InUnicode	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

LaTeX symbols

Symbols used inLaTeXmarkup.

• $\approx$(\approx), usually to indicate approximation between numbers, like$\pi \approx 3.14$.
• $\not \approx$(\not\approx), usually to indicate that numbers are not approximately equal ($1\not \approx 2$).
• $\simeq$(\simeq), usually to indicate asymptotic equivalence between functions, like$f(n)\simeq 3n^{2}$.
  • So writing$\pi \simeq 3.14$would be wrong under this definition, despite wide use.
• $\sim$(\sim), usually to indicate proportionality between functions, the same$f(n)$of the line above will be$f(n)\sim n^{2}$.
• $\cong$(\cong), usually to indicate congruence between figures, like$\Delta ABC\cong \Delta A'B'C'$.
• $\eqsim$(\eqsim), usually to indicate that two quantities are equal up to constants.
• $\lessapprox$(\lessapprox) and$\gtrapprox$(\gtrapprox), usually to indicate that either the inequality holds or the two values are approximately equal.

Unicode

See also:Unicode mathematical operators

Symbols used to denote items that are approximately equal are wavy or dotted equals signs.

U+223C∼TILDE OPERATOR	Which is also sometimes used to indicateproportionality.
U+223D∽REVERSED TILDE	Which is also sometimes used to indicate proportionality.
U+2243≃ASYMPTOTICALLY EQUAL TO	A combination of "≈" and "=", which is used to indicateasymptotic equality.
U+2245≅APPROXIMATELY EQUAL TO	Another combination of "≈" and "=", which is used to indicateisomorphismorcongruence.
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	Another combination of "≈" and "=", used to indicate equivalence or approximate equivalence.
U+2250≐APPROACHES THE LIMIT	Which can be used to represent the approach of a variable,y,to alimit;like the common syntax,$\lim _{x\to \infty }y(x)\doteq 0$.
U+2252≒APPROXIMATELY EQUAL TO OR THE IMAGE OF	Which is used like "≈"or"≃"inJapan,Taiwan,andKorea.
U+2253≓IMAGE OF OR APPROXIMATELY EQUAL TO	A reversed variation ofU+2252≒APPROXIMATELY EQUAL TO OR THE IMAGE OF.
U+225F≟QUESTIONED EQUAL TO
U+2A85⪅LESS-THAN OR APPROXIMATE
U+2A86⪆GREATER-THAN OR APPROXIMATE

Science

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 scienceshows that earlier theories and laws can beapproximationsto 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.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.Physicistsoften approximate theshape of the Earthas asphereeven 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.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 ofperturbationsto 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 scienceaccept that empiricalmeasurementsare alwaysapproximations— they do not perfectly represent what is being measured.

Law

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 processfor new member states,and as a continuing process when required by anEU Directive.Approximationis 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".TheEuropean Commissiondescribes approximation of law as "a unique obligation of membership in the European Union".

See also

• Approximation algorithm– Class of algorithms that find approximate solutions to optimization problems
• Approximate computing– Computation of nearly accurate results
• Approximations of π– Varying methods used to calculate pi
• Binomial approximation– Approximation of powers of some binomials
• Congruence relation– Equivalence relation in algebra
• Double tilde (disambiguation)– Various meanings of ~~ or ≈
• Estimation– Process of finding an approximation
• Fermi problem– Estimation problem in physics or engineering education
• Idealization (philosophy of science)– Process by which a scientific model is simplified by assuming strictly false facts to be true
• Least squares– Approximation method in statistics
• Linear approximation– Approximation of a function by its tangent line at a point
• Newton's method– Algorithm for finding zeros of functions
• Order of approximation– Expressions for approximation accuracy
• Rough set– Approximation of a mathematical set
• Runge–Kutta methods– Family of implicit and explicit iterative methods
• Significant figures– Any digit of a number within its measurement resolution, as opposed to spurious digits
• Small-angle approximation– Simplification of the basic trigonometric functions
• Successive-approximation ADC– Type of analog-to-digital converter
• Taylor series– Mathematical approximation of a function
• Tolerance relation– Math relation that is reflexive and symmetric
• Intuition– Ability to acquire knowledge without conscious reasoning

External links

• Media related toApproximationat Wikimedia Commons