Abstract
In this chapter we introduce the system of real numbers and study the basic structural properties of this space. We will meet among others the following three principles: The concept of compactness, the principle of the supremum, and the Baire Category method.
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Notes
- 1.
Observe that 0 is not considered a natural number.
- 2.
There is an agreement not to consider 1 a prime number —nor composite. One of the reasons is to keep the formulation of the fundamental theorem of the arithmetic as in Theorem 8. Indeed, should 1 be considered prime, uniqueness of the expansion of any number as a product of primes—disregarding order—will fail, as 1 can be added—or suppressed—from any such expansion without changing the product.
- 3.
Intervals will be introduced in Definition 33. Here we only need the definition of a closed and bounded interval, i.e., a subset of \({\mathbb R}\) of the form \([a,b]:=\{x\in{\mathbb R}:\ a\le x\le b\}\), where a and b are real numbers such that \(a\le b\). In particular, [0, 1], called the unit interval, is the set of all real numbers greater than or equal to 0 and, simultaneously, smaller than or equal to 1.
- 4.
The proof, given below, of this last statement is, somehow, incomplete. Its pretension is just to provide a practical procedure to find the fraction associated to a given expansion in some base. The reader may note that we are assuming (without justification) that the algebraic operations of sum and product on \({\mathbb Q}\), when applied to their expansions, give the usual rules for manipulating them. This has a simple proof when the expansions are finite, since finally they represent just a finite sum. However, to justify them in the case of nonterminating (hence periodic) expansions we need to ultimately rely on the concept of the sum of a series. This will be done, properly, in Example 171.1.
- 5.
We distinguish by the context between the role of the symbol \(|\cdot|\) when applied to an interval —denoting then its length, see Definition 35— and when applied to a real number —denoting then its absolute value (Definition 37).
- 6.
The question about the existence of a least upper bound for a bounded above subset of \({\mathbb Q}\) in the number system \({\mathbb Q}\) has a negative answer in general. For example, remaining ourselves in \({\mathbb Q}\)}, the set \(\{q\in{\mathbb Q}:\ q^2<2\) has no least upper bound. This is exactly what was proved in Theorem 18.
- 7.
If \({\alpha}=0\) and \({\beta}>0\), then \({\alpha}^{\beta}\) is defined to be \(0\), while \(0^0\) is undefined. If \({\alpha}<0\) we stumble into another serious problem, whose solution cannot be found, in general, in the framework of the theory of real analysis: another extension, the complex analysis theory, is needed for that.
- 8.
The family of the open sets in \({\mathbb R}\) is much larger than the family of the open intervals, and it is the model, in an abstract setting, for families of sets that will be called “topologies” (their elements generically named “open sets”). The basic properties (O1) to (O4) of the family of all open sets in \({\mathbb R}\), isolated in Proposition 71, are the requirements for the abstract definition of a “topology” on a set (see also Remark 105).
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© 2015 Springer International Publishing Switzerland
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Montesinos, V., Zizler, P., Zizler, V. (2015). Real Numbers: The Basics. In: An Introduction to Modern Analysis. Springer, Cham. https://doi.org/10.1007/978-3-319-12481-0_1
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