Orders of approximation

Often in science, engineering, or other quantitative disciplines, it is necessary to make approximations with various degrees of precision. These approximations can be classified based on the order of magnitude of the rounding error involved.

Table of contents

1 Zeroth order approximation 2 First order approximation 3 Second order approximation

Zeroth order approximation

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 no 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. For example,

is an approximate fit to the data.

First order approximation

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. 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,

is an approximate fit to the data.

Second order approximation

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 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,

is an approximate fit to the data. In this case, with only three data points, a parabola is an exact fit. A third order approximation would be required to fit four data points, and so on.