In formal expressions, the ordinal number used before the word order refers to the highest power in the series expansion used in the approximation.
The omission of the word order leads to phrases that have less formal meaning.
However, this may be confusing, as these formal expressions do not directly refer to the order of derivatives.
The choice of series expansion depends on the scientific method used to investigate a phenomenon.
The choice of order of approximation depends on the research purpose.
One may wish to simplify a known analytic expression to devise a new application or, on the contrary, try to fit a curve to data points.
For example, if a quantity is constant within the whole interval, approximating it with a second-order Taylor series will not increase the accuracy.
The formal usage of order of approximation corresponds to the omission of some terms of the series used in the expansion.
used above meaning do not directly give information about percent error or significant figures.
For example, in the Taylor series expansion of the exponential function,
Zeroth-order approximation is the term scientists use for a first rough answer.
Many simplifying assumptions are made, and when a number is needed, an order-of-magnitude answer (or zero significant figures) is often given.
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: a polynomial of degree 0.
With the implied accuracy of the data points of ±0.5, the zeroth order approximation could at best yield the result for y of ~3.7 ± 2.0 in the interval of x from −0.5 to 2.5, considering the standard deviation.
If the data points are reported as the zeroth-order approximation results in The accuracy of the result justifies an attempt to derive a multiplicative function for that average, for example, One should be careful though, because the multiplicative function will be defined for the whole interval.
If only three data points are available, one has no knowledge about the rest of the interval, which may be a large part of it.
Taylor series is useful and helps predict an analytic solution, but the approximation alone does not provide conclusive evidence.
[3] Some simplifying assumptions are made, and when a number is needed, an answer with only one significant figure is often given ("the town has 4×103, or four thousand, residents").
A first-order 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.
In this example there is a zeroth-order approximation that is the same as the first-order, but the method of getting there is different; i.e. a wild stab in the dark at a relationship happened to be as good as an "educated guess".
Few simplifying assumptions are made, and when a number is needed, an answer with two or more significant figures ("the town has 3.9×103, or thirty-nine hundred, residents") is generally given.
As in the examples above, the term "2nd order" refers to the number of exact numerals given for the imprecise quantity.
A second-order 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.
However, the data points for most of the interval are not available, which advises caution (see "zeroth order").
While higher-order approximations exist and are crucial to a better understanding and description of reality, they are not typically referred to by number.
Continuing the above, a third-order approximation would be required to perfectly 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 (e.g. "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.
The terminology, in this context, represents a high level of precision required to account for an effect which is inferred to be very small when compared to the overall subject matter.