Irrationals yet as ordered pairs of rationals?

Question

Let's consider the naturals $\mathbb{N}$ as being given. Then we have the following sequence of extensions:

1. The whole numbers $\mathbb{Z}$ are defined as ordered pairs of naturals (: Wikipedia)
2. The rational numbers $\mathbb{Q}$ are defined as ordered pairs of whole numbers (: Wikipedia)
3. The real numbers $\mathbb{R}$ are not defined as ordered pairs of rational numbers (: Wikipedia)
4. The complex numbers $\mathbb{C}$ are defined as ordered pairs of real numbers (: Wikipedia)

All extensions of the number system are such that many important properties of the previous number type are inherited. Three out of four extensions are done by the formation of ordered pairs, with proper definitions of the four basic operations: addition, substraction, multiplication and division. The step from the rational numbers to the real numbers, however, differs essentially from the other extensions. According to standard mathematics, it cannot be done by considering pairs of rational numbers. But let's be stubborn and nevertheless try the following
Definition. A real number $r$ is an ordered pair of rationals $(a,b)$ such that $a < b$. The absolute difference $|a-b|$ is called the error of the real number $r$. The main problem is to minimize the error (i.e. preferrably there is a limit that makes it zero). The idea is that irrational numbers can be understood as the number squeezed between ever more converging pairs of rational numbers. It is noticed, however, that this idea is rather an idealization of a not-so-nice reality, where errors do really exist.
Example 1. Let the real number $\,e\,$ for any natural $n$ be defined by $$e = \left(\left[1+\frac{1}{n}\right]^n , \left[1+\frac{1}{n}\right]^{n+1}\right)$$ Where it is noticed that there is a lot of theory in $\mathbb{Q}$ preceding this. The error can be made as minimal as desired with help of a theory about limits in $\mathbb{Q}$, which is assumed to be fully developed: $$\lim_{n\to\infty} \left[ \left(1+\frac{1}{n}\right)^{n+1}-\left(1+\frac{1}{n}\right)^n \right] =\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n\frac{1}{n} = \lim_{n\to\infty}\frac{e}{n}=0$$ Example 2. Let the real number $\,\pi\,$ for any natural $n$ be defined by Leibniz formula: $$ \pi = \left( 4\sum_{k=0}^{2n-1} \frac{(-1)^k}{2k+1} , 4\sum_{k=0}^{2n} \frac{(-1)^k}{2k+1}\right) $$ Again, the error can be made as minimal as desired: $$ \lim_{n\to\infty} \left[ 4\sum_{k=0}^{2n} \frac{(-1)^k}{2k+1} - 4\sum_{k=0}^{2n-1} \frac{(-1)^k}{2k+1} \right] = \lim_{n\to\infty} 4 \frac{(-1)^{2n}}{4n+1} = 0 $$ Does the above make sense?

Answer 1

The things your considering aren't real numbers. They approximate real numbers, both in the sense that each real number can be represented by one with as small error as is liked and in the sense that each real number can be expressed as the limit of a sequence of such objects; but this doesn't mean they're actually the reals.

I think your question ultimately breaks down into two pieces: "Do approximate reals (either as you describe them, or some other way) form a reasonable/interesting/useful number system?" and "Is this number system the reals?" As you yourself say, the answer to the second question is of course "no" (and I think in light of this your title question is slightly misleading); however, the answer to the first question is certainly "yes". For instance, you might also look at fuzzy numbers if you're interested in versions of the real numbers which accomodate some amount of uncertainty.

Note that even in error-permitting contexts, we usually consider number systems which are uncountable. Inside these, a countable set of "basic" elements usually lurks and is "dense" in an appropriate sense. So it's worth contrasting cardinality issues with error issues.

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It's worth noting also that, while some irrationals like $\pi$ and $e$ admit "aribtrarily good" descriptions as approximated reals - e.g. (as you write) $e$ is always in $([1+{1\over n}]^n, [1+{1\over n}]^{n+1})$ - most irrationals don't, since most irrationals are not computable. So even if you try to look beyond specific approximated reals to "arbitrarily well"-approximated reals, you still won't capture the whole real line.

That is: although every real can be approximated arbitrarily well by rationals, there are reals (in fact, all but countably many reals) which have no arbitrarily-improving approximation which can be described by a computer program. Put another way: most real numbers do not have computable Cauchy sequences.

Answer 2

The reals can not be defined as ordered pair of rationals because the reals are uncountable and the ordered pairs of rationals are countable.

Answer 3

I think you have not understood the basic difference between extension from $\mathbb{Q}$ to $\mathbb{R}$ and other extensions (like from $\mathbb{N}$ to $\mathbb{Z}$).

The real numbers essentially require some concept of infinity in order to describe them in terms of simpler structures. All the other extensions can be dealt with in a finite manner. A real number can never be described by using a finite number of rationals.

To express more concretely real numbers transcend the powers of algebraic operations $+, -, \times, /, =$ and instead rely on the idea of order relations $<, >$. It is useless to think of them in terms of algebraic notions and instead focus on order relations between rationals and use it to develop the structure of real numbers.

Your definition of real numbers as a pair of rationals is clearly inadequate. What you are in fact trying to do is define real numbers as a sequence of ordered pairs of rationals. This makes sense and is essentially the Dedekind's construction where he used not a sequence of ordered pairs of rationals, but rather an ordered pairs of sets $A, B$ such that $A$ has rationals less than the real number being defined and $B$ has all the other rationals.