by Corry Shores
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[The following is summary. I am not a mathematician, so please consult the original text instead of trusting my summarizations, which are possibly mistaken and probably inelegantly articulated. Bracketed comments and subsection divisions are my own. Proofreading is incomplete, so please forgive my mistakes.]
Summary of
Edna Ernestine Kramer
The Nature and Growth of Modern Mathematics
2
Mathematical Method and Main Streams Are Launched
2.x
[selections on decimal expansions as approximating intervals of rational numbers]
(pp.33-34)
Brief summary:
(2.x.1) A real number can be considered as a series of approximating intervals, getting smaller and smaller, and converging upon a particular point on the number line (and thus to an exact value), even if the decimals are nonterminating. Each new decimal, when taken along with the decimal value of one higher, creates an interval, with each one being nested within the prior one and all shrinking down to a particular point.
(p.34)
[Real Number Decimal Expansion as Nested, Convergent Approximating Intervals.]
Summary
[Real Number Decimal Expansion as Nested, Convergent Approximating Intervals.]
[A real number can be considered as a series of approximating intervals, getting smaller and smaller, and converging upon a particular point on the number line (and thus to an exact value), even if the decimals are nonterminating. Each new decimal, when taken along with the decimal value of one higher, creates an interval, with each one being nested within the prior one and all shrinking down to a particular point.]
[Our purpose here is to understand the mathematical notion of “approximating interval” in terms of decimal expansions of real numbers. This is in the context of Cauchy sequences. As we saw in Wildberger’s Math Foundations 111, The basic nature of the Cauchy sequence is that it is an increasing series of rational numbers, but as you go down along the sequence, they tend toward a certain value progressively. So the first one will be very high for instance from the ultimate value, then some other one later will be very low (but still closer), soon one will be a little high, and another a little low. And presumably they converge upon a value. Wildberger, in Math Foundations 111.6, shows this gradual, interchanging convergence of the values with this diagram:
(Image from Wildberger, in Math Foundations 111.6. [Video page])
The green line is the value that the series of rationals are tending toward. The idea was that no matter how small an interval you choose, you will be able to find a place in the sequence after which the gaps between successive values (the space above and below the green line) will be less than that arbitrarily small interval. This implies that it is always moving toward some specific value (the green line). Wildberger also explains algorithmic techniques for decimal expansion in his in Math Foundations 92 [Video page]. At 12.25, he shows a method for obtaining a series of approximations to e. What he explains there might not, however, be a Cauchy sequence. I do not know. But it gives us the idea of a series of successive approximations with a decimal expansion that gets closer and closer to the appropriate value. Kramer speaks similarly of narrowing approximating intervals that converge upon a real number value. But her example simply gives each successive decimal point, with the interval being between that smallest decimal number and the one just larger than it, because the value being converged upon will always be found within such an interval.]
It remains to indicate that this definition makes it possible to establish a one-to-one correspondence between the real numbers and the points of a number line, that is, there must be a point for every non terminating decimal and such a decimal | for every point. Suppose, then, that a real number is defined by some nonterminating decimal which we can carry out to as many places as we please, and that the first few places are given by 2.6314 ... . The decimal gives us a sequence of rational approximations to the real number, namely, 2, 2.6, 2.63, 2.631, 2.6314, ... . In other words, the first approximation in the sequence places the real number in the interval (2, 3), and then 2.6 gives the approximating interval (2.6, 2.7), etc. Thus we have the sequence of nested intervals, (2, 3), {2.6, 2.7), (2.63, 2.64), ... , illustrated in Figure 2.9.
The adjective nested describes the fact that each interval lies within the preceding one. We observe also that the lengths of successive intervals are. 1, 0.1, 0.01, 0.001, 0.0001, ... . Since we are considering a nonterminating decimal, the nest of intervals will ultimately contain an interval of length 0.000 000 001 and then there will be still smaller intervals, so that interval length shrinks toward zero. As the innermost intervals get smaller and smaller, one can imagine their bounding walls approaching collision or, at any rate, getting close enough to “trap” a point of the number line. It is postulated, that is, assumed, that there is a unique point contained in all intervals of the nest. If there is such a point, we see that it must be unique, for if there were another distinct point, it would be separated from the first by some distance, 0.000 01, say. But ultimately some interval of the nest will be smaller than that number, and the first point must be contained in that very small interval. Then the second point would be too far away to be inside the interval and hence would not be contained in every interval of the nest. Since every nonterminating decimal will give rise to a sequence of nested intervals like the one described, there will always be a unique point of the number line corresponding to every real number.
(33-34, boldface and underlining are mine)
Kramer, Edna. The Nature and Growth of Modern Mathematics. Princeton University, 1981.
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