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An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.

Approximations may be used because incomplete information prevents use of exact representations. Many problems in physics are either too complex to solve analytically, or impossible to solve using the 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.

For instance, physicists often approximate the shape of the Earth as a sphere even though more accurate representations are possible, because many physical behavious—e.g. gravity—are much easier to calculate for a sphere than for other shapes.

It is difficult to exactly analyze the motion of several planets orbiting a star, for example, due to the complex interactions of the planets' gravitational effects on each other, so 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.

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 solution.

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.

Science[link]

The scientific method is carried out with a constant interaction between scientific laws (theory) and empirical measurements, which are constantly compared to one another.

The approximation also refers to using a simpler process. This model is used to make predictions easier. 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 history of science indicates that the scientific laws commonly felt to be true at any time in history are only approximations to some deeper set of laws. For example, attempting to resolve a model using outdated physical laws alone incorporates an inherent source of error, which should be corrected by approximating the quantum effects not present in these laws.

Each time a newer set of laws is proposed, it is required that in the limiting situations in which the older set of laws were tested against experiments, the newer laws are nearly identical to the older laws, to within the measurement uncertainties of the older measurements. This is the correspondence principle.

Mathematics[link]

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 1.414. Numerical approximations sometimes result from using a small number of significant digits. Approximation theory is a branch of mathematics, a quantitative part of functional analysis. Diophantine approximation deals with approximations of real numbers by rational numbers.

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 around.

See also[link]

Approximately equals sign Approximation error Congruence relation Estimation Fermi estimate Fitness approximation Least squares

Linear approximation Newton's method Numerical analysis Orders of approximation Runge–Kutta methods Successive approximation ADC Taylor series

Look up approximation in Wiktionary, the free dictionary.

The Normal is the recording artist name used by English music producer Daniel Miller, a film editor at the time, who is best known as the founder of the record label Mute Records.

Background[link]

In 1977, Miller had split up with his girlfriend. A friend suggested that he read a book he himself had just finished. The book was Crash (1973) by J.G. Ballard.^[1] He felt that Ballard's writing took him five minutes into the future; the novel was to be a major influence in the music he would produce as The Normal.^[1] Miller was disillusioned by the fact you needed to learn three chords to be in a punk band, so he decided to purchase a synthesiser. His thinking was that you only needed to learn to press one key on a synthesiser.^[1] After buying a Korg 700s synthesiser from Macari's music shop in London,^[1] Miller recorded and released a single under the name The Normal. This was "T.V.O.D."/"Warm Leatherette". Both tracks were minimalist electronic songs influenced by the Crash novel.^[1] He wanted the sound of the recordings to be visual, like driving along a highway between large buildings then going through a tunnel.^[1] The single was recorded in Daniel Miller's house using a TEAC four track tape recorder and the Korg synthesiser.^[2]

"Live at West Runton Pavilion", The Normal's second release, done with another Mute Records act, Robert Rental, wasn't well-received. A strange release, it was a one-sided album (side two was left blank) of improvised electronic noises, in a plain purple dust jacket. Marat Records released the record in Germany as Daniel Miller Robert Rental Live, with a black and white picture sleeve, catalogue No. Marat Rough 017.

Influence on other artists[link]

Songs by The Normal have been covered and performed by many notable artists such as Grace Jones, Chicks on Speed/Hell on a split-7", Giddle and Boyd Rice and in 2006 by Trent Reznor with Jeordie White and Peter Murphy.^[3] A recent cover of "Warm Leatherette" was performed live by Duran Duran in November 2007, and as part of an electro medley during their 2008 tour,^[4] in support of their album Red Carpet Massacre.

Discography[link]

• The Normal - "T.V.O.D." / "Warm Leatherette" (Mute Records 1978)
• Robert Rental & The Normal - Live At West Runton Pavilion, 6-3-79 (Rough Trade 1980)

References[link]

Notes[link]

External links[link]

• The Normal at Mute Records Official site
• The Normal at Allmusic
• The Normal Discography at discogs.com