Orders of approximation

In science, engineering, and other quantitative disciplines, orders of approximation refer to formal or informal expressions for how accurate an approximation is. In formal expressions, the ordinal number used before the word order refers to the highest term in the series expansion used in the approximation. The choice of series expansion depends on the scientific method used to investigate a phenomenon. The expression order of approximation is expected to indicate progressively more refined approximations of a function in a specified interval. If a quantity is constant within the whole interval, approximating it with a second-order Taylor series will not increase the accuracy. Thus the numbers zeroth, first, second etc. used formally in the above meaning do not directly give information about percent error or significant figures. In science, engineering, and other quantitative disciplines, orders of approximation refer to formal or informal expressions for how accurate an approximation is. In formal expressions, the ordinal number used before the word order refers to the highest term in the series expansion used in the approximation. The choice of series expansion depends on the scientific method used to investigate a phenomenon. The expression order of approximation is expected to indicate progressively more refined approximations of a function in a specified interval. If a quantity is constant within the whole interval, approximating it with a second-order Taylor series will not increase the accuracy. Thus the numbers zeroth, first, second etc. used formally in the above meaning do not directly give information about percent error or significant figures. This formal usage of order of approximation corresponds to the order of the power series representing the error, which is the first first nonzero higher derivative of the error. The expressions: a zeroth-order approximation, a first-order approximation, a second-order approximation, and so forth are used as fixed phrases. The omission of the word order leads to phrases that have less formal meaning. Phrases like first approximation or to a first approximation may refer to a roughly approximate value of a quantity. The phrase to a zeroth approximation indicates a wild guess. The expression order of approximation is sometimes informally used to mean the number of significant figures, in increasing order of accuracy, or to the order of magnitude. However, this may be confusing as these formal expressions do not directly refer to the order of derivatives. Formally, an nth-order approximation is one where the order of magnitude of the error is at most x n + 1 {displaystyle x^{n+1}} , or in terms of big O notation, the error is O ( x n + 1 ) . {displaystyle O(x^{n+1}).} In the case of a smooth function, the nth-order approximation is a polynomial of degree n, which is obtained by truncating the Taylor series to this degree. 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. 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. 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 (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,