Hi Brett,
I enjoyed reading your article. However, your first argument relies on the starting assumption that repeating decimals equal fractions. This is the whole thing that we need to prove; we cannot assume it.
Your second argument does the same; it assumes that 0.333… = 1/3.
Your third argument is “if we kept writing out the decimal expansion we could write it out to infinity and get the exact value 0.999….”. But the more terms we write, we have still only written a finite number of terms. We are not getting any closer to ‘infinitely many’, indeed, we are not even approaching ‘infinitely many’. In order to accept your argument, we have to use our imagination and pretend that we can imagine getting closer and actually reaching infinity. It is an argument based on imagination.
Your final argument is that “we have a geometric series that converges to a/(1 – r)”. But this is a formula for finding the constant part of a geometric series (called the ‘limit’ for cases where |r|<1), it does not somehow add up infinitely many non-zero terms. That is to say, the nth partial sum of a geometric series can be described by the expression k – kr^n where k = a/(1 – r). So it is the constant part of the partial sum expression, not the addition of the series terms. The mainstream position is that it is fine to define an ‘infinite sum’ to equal the limit, but this defining 0.999… to equal 1, is not a proof. It is merely claiming they are equal because we have defined them to be equal.
Later you said “The series is getting infinitesimally close to the convergence value. On an infinite scale, convergence becomes equality.”. One of the main reasons that the concept of limits was developed in the 19th century was that many mathematicians were unhappy that the incoherent notion of ‘infinitesimals’ was required for calculus. In order for a sequence to get ‘infinitesimally close’ there must be the inclusion of a single term that changes it from being a finite distance from the limit to being ‘infinitesimally close’. In other words, some finite distance minus some other finite distance must equal an infinitesimally small distance. It does not make logical sense.
After seeing your article I wrote my own article (on Medium) called “The Long and Troubled History of 0.999… = 1”. I’d love to hear your comments on that because 0.999… is a favourite topic of mine.
