Related concepts and fundamentals:
An approximation is anything that is similar but not exactly equal to
1 Etymology and usage
5 LaTeX Symbols
6 See also
8 External links
Etymology and usage
The word approximation is derived from
Latin approximatus, from
proximus meaning very near and the prefix ap- (ad- before p) 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 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 incomplete information prevents use of exact representations.
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.
Approximation theory is a branch of mathematics, a quantitative part
of functional analysis.
Diophantine approximation deals with
approximations of real numbers by rational 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. It also is used when a number is not rational,
such as the number π, which often is shortened to 3.14159, or √2 to
Numerical approximations sometimes result from using a small number of
significant digits. Calculations are likely to involve rounding errors
leading to approximation. 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.
occur when a decimal number cannot be expressed in a finite number of
Related to approximation of functions is the asymptotic value 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)+...(k/2^n) is asymptotically equal to k.
Unfortunately no consistent notation is used throughout mathematics
and some texts will use ≈ to mean approximately equal and ~ to mean
asymptotically equal whereas other texts use the symbols the other way
As another example, in order to accelerate the convergence rate of
evolutionary algorithms, fitness approximation—that leads to build
model of the fitness function to choose smart search steps—is a good
Approximation arises naturally in scientific 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.
The history of science shows that earlier theories and laws can be
approximations to some deeper set of laws. Under the correspondence
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
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
Physicists often approximate the shape of the
Earth as a sphere 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.
An approximate solution is effected by performing iterations. 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 of perturbations 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 of philosophy of science accept that
empirical measurements are always approximations—they do not
perfectly represent what is being measured.
The error-tolerance property of several applications (e.g., graphics
applications) allows use of approximation (e.g., lowering the
precision of numerical computations) to improve performance and energy
efficiency. This approach of using deliberate, controlled
approximation for achieving various optimizations is referred to as
Unicode mathematical operators
Symbols used to denote items that are approximately equal are wavy or
dotted equals signs.
≈ (U+2248, almost equal to)
≃ (U+2243), a combination of "≈" and "=", also used to indicate
asymptotically equal to[clarification needed]
≒ (U+2252), which is used like "≃" in both Japanese and Korean
≓ (U+2253), a reversed variation of "≒"
≅ (U+2245), another combination of "≈" and "=", which is used to
indicate isomorphism or sometimes congruence
≊ (U+224A), yet another combination of "≈" and "=", used to
indicate equivalence or approximate equivalence
∼ (U+223C), which is also sometimes used to indicate proportionality
∽ (U+223D), which is also sometimes used to indicate proportionality
≐ (U+2250, approaches the limit), which can be used to represent the
approach of a variable, y, to a limit; like the common syntax,
displaystyle scriptstyle lim _ xto inf y(x)
≐ 0
(approx), usually to indicate approximation between numbers, like
displaystyle pi approx 3.14
(simeq), usually to indicate asymptotic equivalence between
displaystyle f(n)simeq 3n^ 2
. So writing
displaystyle pi simeq 3.14
would be wrong, despite wide use.
(sim), usually to indicate proportionality between functions, the
of the line above will be
displaystyle f(n)sim n^ 2
(cong), usually to indicate congruence between figures, like
displaystyle Delta ABCcong Delta A'B'C'
Approximately equals sign
Orders of approximation
Successive approximation ADC
^ The Concise Oxford Dictionary, Eighth edition 1990,
^ Longman Dictionary of Contemporary English, Pearson Education Ltd
2009, ISBN 978 1 4082 1532 6
^ Numerical Computation Guide
^ Encyclopædia Britannica
^ The three body problem
^ Mittal, Sparsh (May 2016). "A Survey of Techniques for Approximate
Computing". ACM Comput. Surv. ACM. 48 (4): 62:1–62:33.
^ "Mathematical Operators – Unicode" (PDF). Retrieved
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