Editing Order of approximation (section)

=== Zeroth-order ===
''Zeroth-order approximation'' is the term [[scientist]]s use for a first rough answer. Many [[Approximation#Science|simplifying assumptions]] are made, and when a number is needed, an order-of-magnitude answer (or zero [[significant figure]]s) is often given.  For example, "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|order-of-magnitude]] approximation. The zero of "zeroth-order" represents the fact that even the only number given, "a few", is itself loosely defined. 

A zeroth-order approximation of a [[function (mathematics)|function]] (that is, [[mathematics|mathematically]] determining a [[formula]] to fit multiple [[data point]]s) will be [[Constant (mathematics)|constant]], or a flat [[line (mathematics)|line]] with no [[slope]]: a polynomial of degree 0. For example,

: <math>x = [0, 1, 2],</math>
: <math>y = [3, 3, 5],</math>
: <math>y \sim f(x) = 3.67</math>

could be – if data point accuracy were reported – an approximate fit to the data, obtained by simply averaging the ''x'' values and the ''y'' values. However, data points represent [[Unit_of_observation#Data_point|results of measurements]]  and they do differ from [[Point_(geometry)#Points_in_Euclidean_geometry|points in Euclidean geometry]]. Thus quoting an average value containing three significant digits in the output with just one significant digit in the input data could be recognized as an example of [[false precision]]. 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&nbsp;±&nbsp;2.0 in the interval of ''x'' from −0.5 to 2.5, considering the [[standard deviation]]. 

If the data points are reported as 
: <math>x = [0.00, 1.00, 2.00],</math>
: <math>y = [3.00, 3.00, 5.00],</math>
the zeroth-order approximation results in
: <math>y \sim f(x) = 3.67.</math>
The accuracy of the result justifies an attempt to derive a multiplicative function for that average, for example,

: <math>y \sim x + 2.67.</math>

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 (mathematics)|interval]], which may be a large part of it. This means that ''y'' could have another component which equals 0 at the ends and in the middle of the interval. A number of functions having this property are known, for example ''y'' = sin π''x''. [[Taylor series]] are useful and help predict [[Closed-form expression|analytic solutions]], but the approximations alone do not provide conclusive evidence.