Orders of approximation

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Orders of approximation have been used not only in science, engineering, and other quantitative disciplines to make approximations with various degrees of precision but also more generally, and more loosely, to indicate relative precision outside these disciplines in the form of "first level", "second level" and so on, "approximations". In the science and engineering disciplines approximations can be classified based on the order of magnitude of the rounding error involved. It is an application of the concepts in big O notation.

[edit] Usage in science and engineering

Zeroth-order approximation (also 0^th order) is the term scientists use for a first educated guess at an answer. Many simplifying assumptions are made, and when a number is needed, an order of magnitude answer (or zero significant figures) is often given. For example, you might say "the town has a few thousand residents", when it has 3,914 people in actuality. This is also sometimes referred to as an order of magnitude approximation.

A zeroth-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be constant, or a flat line with no slope. For example,

$x=[0,1,4]\,$

$y=[0,1,2]\,$

$y\sim f(x)=1.67\,$

is an approximate fit to the data.

First-order approximation (also 1^st order) is the term scientists use for a further educated guess at an answer. Some simplifying assumptions are made, and when a number is needed, an answer with only one significant figure is often given ("the town has 4,000 residents").

A first-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a straight line with a slope. For example,

$x=[0,1,4]\,$

$y=[0,1,2]\,$

$y\sim f(x)=1.5x\,$

is an approximate fit to the data.

Second-order approximation (also 2^nd order) is the term scientists use for a decent-quality answer. Few simplifying assumptions are made, and when a number is needed, an answer with two or more significant figures ("the town has 3,900 residents") is generally given.

A second-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a parabola. For example,

$x=[0,1,2]\,$

$y=[3,3,5]\,$

$y\sim f(x)=x^2-x+3\,$

is an approximate fit to the data. In this case, with only three data points, a parabola is an exact fit.

While higher-order approximations exist and are crucial to a better understanding and description of reality, they are not typically referred to by number.

A third-order approximation would be required to fit four data points, and so on.

These terms are also used colloquially by scientists and engineers to describe phenomena that can be neglected as not significant (eg., "Of course the rotation of the earth affects our experiment, but it's such a high-order effect that we wouldn't be able to measure it" or "At these velocities, relativity is a fourth-order effect that we only worry about at the annual calibration.") In this usage, the ordinality of the approximation is not exact, but is used to emphasize its insignificance; the higher the number used, the less important the effect.