In science, engineering, and other quantitative disciplines, orders of approximation refer to formal or informal terms for how precise an approximation is, and to indicate progressively more refined approximations: in increasing order of precision, a zeroth order approximation, a first order approximation, a second order approximation, and so forth.
Formally, an nth order approximation is one where the order of magnitude of the error is at most n; in terms of big O notation, with error bounded by O(x^{n}). In suitable circumstances, approximating a function by a Taylor polynomial of degree n yields an nth order approximation, by Taylor's theorem: a first order approximation is a linear approximation, and so forth.
The term is also used more loosely, as detailed below.
Zerothorder 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 zerothorder 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: a polynomial of degree 0. For example,
is an approximate fit to the data.
Firstorder 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 firstorder approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a linear approximation, straight line with a slope: a polynomial of degree 1. For example,
is an approximate fit to the data.
Secondorder approximation (also 2^{nd} order) is the term scientists use for a decentquality 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 secondorder approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a quadratic polynomial, geometrically, a parabola: a polynomial of degree 2. For example,
is an approximate fit to the data. In this case, with only three data points, a parabola is an exact fit.
While higherorder approximations exist and are crucial to a better understanding and description of reality, they are not typically referred to by number.
A thirdorder 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 highorder effect that we wouldn't be able to measure it" or "At these velocities, relativity is a fourthorder 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.
