Real number

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As Infinite Decimals
Any Rational number of the form $$\frac{m}{n}$$, where n is of the form $$2^a5^b$$ can we written as a decimal in terminating decimal. All other fractions repeat after finite many decimals. But are there other numbers? Simon Stevin (1548-1620), who invented modern decimal notation, began the theory of Real numbers as decimals. However, he took an engineering approach, and recognized no one would ever need Pi or surds beyond five or six decimal places, though he acknowledged they went on "forever". What does the "forever" or the ellipses at the end of such numbers mean??

Choice
One approach to infinite decimals is to say that each next decimal place is arrived at by choice. This is also called the quasi-combinatorial approach. Each digit is chosen arbitrarily and independently.

Advantages of this approach are several. A) It resembles our approach to finite decimals, where we imagine picking each digit are random, with only ten choices to choose between. B) Apart from infinite nines being equivalent to the next number (e.g. 0.09999... = 0.1000...), every number is unique. C) It turns a process into a number. D) In Analytic Geometry, if a graph looks like it has a solution, then it does. E) Similarly, any polynomial can be factored into linear and quadratic terms. F) Every truncation is a rational approximation. G) A single number can record uncountably infinite amounts of information.

Unfortunately, there are several fatal flaws in this approach. A) No infinite number can be exhibited by human beings. B) Basic operations – such as addition and multiplication – are no longer well-defined, or even performable. C) Hence, arithmetic itself is no longer possible. D) Even rational numbers require infinite representations, since they cannot be defined algorithmically. E) There exist no method for tell if two Real numbers are equal. F) Number theoretical questions are now routinely ignored, as mathematicians cease to ask about extending fields to find new solutions to problems, in favor of decimal approximations. G) Similarly, all trigonometry problems are thought to have decimal approximation answers, instead of doing the work to find theoretical ones. G) Humility is over. Now humans believe they can manipulate infinite objects, when they cannot.

Despite these difficulties, this approach is the accepted one today in set theoretical frameworks, justified by a lot of smoke and mirrors, called axioms. First and foremost, the Axiom of Choice was invented to justify the idea of Real numbers as infinite decimals.

Algorithmic
Second, some mathematicians advocate a finite algorithmic approach. That is, rather than exhibiting every one of the infinitely many digits, they say there is a finite algorithm which can generate the successive digits, if carried on forever. This ties in well with Computer Science, a young branch, which does not necessarily have a universally agreed upon theory of algorithms.

Advantages of this approach are several. A) This is how almost all irrational numbers arose historically. B) It is related to the mathematical notion of an infinite (convergent) series. C) It is related to the definition of many functions. D) It is related to the definition of many integrals.

Unfortunately for this approach, there is not always a way of comparing algorithms. Some algorithms rely on other algorithms (such as roots, logs, irrational exponents, etc.), meaning the problem is nested. Many Ph.D's have been granted to someone who proves such-and-such algorithm is equivalent to a more famous one, but there is no end to these comparisons, and there are no canonical forms to compare. Additionally, there is not a clear setup or beginning place. Can we define basic operation for algorithms (addition, subtraction, etc.)? What is the run-time? How do we know? How do we know when two algorithms are the same? What would the canonical form of an algorithm be? We cannot tell most of the time when two are the same. As before, arithmetic has been tautalogical.

Incoherent
Finally, one possibility is that we should throw up out hands and recognize that infinite decimals are impossible to make well-defined and usable.

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