Approximation
Approximation is a method used in both everyday contexts and mathematical calculations, where a value or object is close to but not exactly equal to another. It serves to simplify computations, providing quicker results at the cost of accuracy. For instance, rounding numbers such as using 10 as an approximation for 9.878677 can facilitate easier decision-making, as seen when friends agree to meet "around 9:00." While approximation can be practical, especially when exact values are difficult to obtain, it inherently introduces a margin of error.
In mathematics, there is a dedicated field known as approximation theory, which focuses on how functions can be represented through simpler forms and evaluates the implications of these simplifications on accuracy. This branch also explores the concept of significant digits—numbers that indicate the precision of an approximation. Various degrees of approximation exist, classified as orders, with zero order being the least accurate and higher orders providing more precision but becoming increasingly complex to define. Understanding these concepts can enhance how people approach both daily estimations and rigorous mathematical challenges.
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Approximation
Approximation can be both a process and an element used in calculation. In general, it refers to any value or object that is close to, but not precisely equal to, another object or value. For example, the value 10 is an approximation of the value 9.878677, because it is close to having an equal value but differs by a small amount.
Approximation is used in mathematics to make calculations faster or to reduce the amount of effort required to arrive at an exact value for a calculation. The drawback to using approximation is that in exchange for the benefits of speed and simplicity, accuracy is sacrificed. In some situations the reduction in accuracy is acceptable given the purpose of the calculations, as when friends agree to meet at a restaurant for dinner at "about 9:00."
In other cases, the need to approximate is caused by the unavailability of exact values or the extreme amount of time or effort that would be required to obtain exact values. An example of the latter occurs in calculations requiring the use of π, where the convention has become to use 3.14 as an approximation of the value, since irrational numbers such as π cannot be precisely expressed with a finite value.
Overview
Many of the approximations people use on a daily basis, and even some of those used in mathematics, involve visual inspection of various phenomena, such as the measurement one takes when using a ruler or a non-digital thermometer. These assessments are not precise, but for ordinary purposes are frequently satisfactory. While approximation is used frequently in mathematics and in everyday life according to its colloquial meaning, there is also a formal branch of mathematics devoted to approximation theory. Approximation theory studies functions, how they can be approximated by the use of simpler functions, and what the consequences of this use of approximation are for accuracy and similar factors. Approximation theory also tries to define as accurately as possible the errors that will be introduced by approximation. One indication of the accuracy of an approximation is the number of significant digits it contains, with significant digits being all digits of a number that are not equal to zero, as well as zeros occurring between these digits. The more significant digits present in a number, the more precise that number is.
Different degrees of approximation vary according to how close they are to the true value. Each degree is referred to as an order of approximation. The lowest—that is, least accurate—degree of approximation is zero order approximation. Above this, at a higher degree of accuracy, is first order approximation, followed by second order approximation, and so forth. Usually, scientists and mathematicians consider zero order and first order approximations to be educated guesses at the solution to a calculation. In most cases, there is no need to refer to specific orders of approximation beyond the second order, so these are usually grouped together under the umbrella of "higher order" approximations, which rapidly become difficult if not impossible to define in precise values.
Bibliography
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Bustamante, González J. Algebraic Approximation: A Guide to Past and Current Solutions. Basel: Birkhäuser, 2012.
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De Villiers, Johan. Mathematics of Approximation. Amsterdam: Atlantis, 2012.
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Trefethen, Lloyd N. Approximation Theory and Approximation Practice. Philadelphia: Soc. for Industrial and Applied Mathematics, 2013.