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(formal)

~ (informal)

symbols representing
approximation.

An approximation is an inexact representation of something that is 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. 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 behaviours — e.g. gravity — are much easier to calculate for a sphere than for less regular shapes.

The problem consisting of two or more planets orbiting around a sun has no exact solution. Often, ignoring the gravitational effects of the planets gravitational pull on each other and assuming that the sun does not move achieve a good approximation. 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 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

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

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.

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

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 approximation to real numbers by rational numbers. The symbol "≈" means "approximately equal to".