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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.
Usage in Science & Engineering Zeroth-order approximation (also 0th 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, is an approximate fit to the data. First-order approximation (also 1st 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, is an approximate fit to the data. Second-order approximation (also 2nd 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, 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. General Usage "To start with, there are two different groups, we can get into more detail, but at the first level of approximation, there's two targets for propaganda. One is what's sometimes called the political class. There's maybe twenty percent of the population which is relatively educated, more or less articulate, plays some kind of role in decision-making. They're supposed to sort of participate in social life -- either as managers, or cultural managers like teachers and writers and so on. They're supposed to vote, they're supposed to play some role in the way economic and political and cultural life goes on. Now their consent is crucial. So that's one group that has to be deeply indoctrinated. Then there's maybe eighty percent of the population whose main function is to follow orders and not think, and not to pay attention to anything -- and they're the ones who usually pay the costs." Manufacturing Consent: Noam Chomsky and the Media* | ||||||||
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