Approximation: The Real Numbers
The fixing of a reference point or origin, and a unit point on a line constitutes the establishment of a coordinate system on that line. For each choice of a coordinate system on a line there is determined the set of rational points; those measurable usin
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Approximation: The Real Numbers
The fixing of a reference point, or origin, and a unit point on a line constitutes the establishment of a coordinate system on that line. For each choice of a coordinate system on a line there is determined the set of rational points; those measurable using Q, and the set of irrational points such as fl. We have mentioned the idea of using the fact that Q is dense, to "approximate" irrationals by rationals, a notion that we must make more precise. If one is working in a specific practical context (e.g., building a house), that context includes the means to arrive at a judgment of what constitutes "good enough" approximation, or "small enough" error B. Factors affecting this judgment include: the uses to which numbers will be put; the precision of any instruments being used, and the units employed. EXAMPLES. 1000 is a small number of grains of sand on a beach, but a large number in a serving of spinach. Context! 0.01 is a small error in a paycheck expressed in dollars, but a large error in a bank transfer expressed in millions of dollars. Units! It is easy to imagine circumstances in which 1.4 is an adequate fl and 22/7 is a satisfactory n ... the errors would be "small enough." Our point is that the meaning of "small enough" error depends critically on context, including choice of unit. This observation is to be taken together with the observation that
For mathematics to be applicable it is essential that it be free of special context, i.e., that it be abstract. In the absence of a context, the only possible meaning of "small 37
G. Pedrick, A First Course in Analysis © Springer Science+Business Media New York 1994
1. Approximation: The Real Numbers
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enough error" is that arbitrarily small error be achievable: error less than e for any e > O. (Note: For this to say what it intends, the error bound e should be taken from an Archimedean ordered field, where there are arbitrarily small positive numbers: Q, for example.) These comments lead us to a precise, context free, meaning for: a number x is approximated by numbers in a set A. Namely, for any e > 0 there is an a, E A such that Ix - atl < e. Of course the approximating number at might change as the error bound e changes: the notation a, stresses this. The statement expresses the possibility of achieving "good enough" accuracy regardless of the context. One could say that the subject matter of "Analysis" is the study of approximation in this sense. The first step in such a study would be the extension of Q to an Archimedean ordered field that includes all numbers that can be approximated by sets of rationals. Using such "numbers" one could analyze any particular practical context, confident that whatever amounts of various quantities are encountered, they either are rational or are approximable by rationals. The discussion of how to extend Q in this way must begin with a description in terms of Q of those numbers that can be approximated by sets of rationals. We begin by so describing to point the way to more general cases
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