Abstract
The questions raised in the introductory chapter stem from a single problem: bridging the gap between the discrete and the continuous. Discreteness is exemplified by the positive integers 1,2,3,4,..., which arise from counting but also admit addition and multiplication. Continuity is exemplified by the concept of distance on a line, which arises from measurement but also admits addition and multiplication. The problem is to find a concept of real number that embraces both counting and measurement, and satisfies the expected laws of addition and multiplication.
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Notes
- 1.
A specific way to do this is to choose rational s and t that are close together on either side of lub(K). For example, find s ∈ K, t∉K with t < 2 and \(t - s < \frac{2-r} {4}\). Then we have
$$\displaystyle{{t}^{2} - {s}^{2} = (t + s)(t - s) < 4 \cdot \frac{2 - r} {4} = 2 - r,}$$and hence \(r < {s}^{2} < 2 < {t}^{2}\).
- 2.
In particular, when the bounded set consists of the cuts L r where r 2 < 2, its least upper bound L is the cut representing \(\sqrt{2}\).
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Stillwell, J. (2013). From Discrete to Continuous. In: The Real Numbers. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-01577-4_2
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