User:Sławomir Biały/0.999... RfC diff

Although these arguments demonstrate that 0.999… = 1, they fall short of being valid mathematical proofs, and it takes considerable effort to make these arguments rigorous: that requires, in particular, a proper definition of the real number system and a derivation of its basic properties. According to Peressini and Peressini (p.186), simple arguments like these fail to "explain why this equality holds." They note that such an explanation involves the distinction between numbers and their decimal representations, the concept of infinity, and the Cauchy completeness property.

For someone with no knowledge of the detailed properties of the real number system, a plausible reading of the first equation in the first proof $1/9=0.111\ldots$ is "the division of one into nine leaves one-tenth, with a remainder leaving one-hundredth, and a remainder leaving one-thousandth, and so forth". Based on this reading, the equation is not an equality of numbers, but reporting the result of a computation that can be carried out indefinitely: what Byers (p. 40) identifies as a process rather than an object. In order to make sense of $1/9=0.111\ldots$ as an equation of numbers, it is necessary to have a conception of the decimal $0.111\ldots$ itself as an object rather than a process.

According to Fred Richman (p. 396), the first argument "gets its force from the fact that most people have been indoctrinated to accept the first equation [$1/9=0.111\ldots$] without thinking." However, as Byers notes, for someone without knowledge of the real number system, the number $0.999\ldots$ may make sense only as process rather than an object, and so the equation $0.999\ldots =1$ is difficult to resolve, because it appears to be a category error: one cannot have a process (a verb) equal to an object (a noun). He suggests that a student who agrees that 0.999… = 1 because of the above proofs, but hasn't resolved this ambiguity, doesn't really understand the equation (Byers pp. 39–41).

The completeness axiom of the real number system is what allows infinite decimals like $0.111\ldots$ and $0.999\ldots$ to be regarded as objects (real numbers) in their own right, independently of their realization as common fractions. Once the real number system has been formally defined, its properties can be used to establish the decimal representation of real numbers. The properties of the decimal representation can then be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999… and 1.000… both represent the same real number. This is done below.