If you post on internet forums, this may save years of your life.

According to some friends of mine, approximately 99.9% of "math-related" discussion on popular internet forums consists of arguments over the repeated decimal 0.99999... and whether or not it is equal to the integer 1. Of course, one could make a convincing argument to the contrary, as we must at least define the rationals for any decimal representation to make sense, and then you can always bring up the fact that the integers are not, strictly speaking, contained in the rationals, since the rational numbers actually consist of equivalence classes of pairs of integers. I cannot imagine the private hell that people who work on formal logic projects have to go through to handle all the various identifications that we take for granted.

Unfortunately, that's generally not the grounds on which objections are placed. Most people want numbers to work in some intuitive sense, or to be intimately tied to physical reality in some sense. This is preposterous even for the integers; there is nothing (that I know of) in this world which has the property that no matter how much of it you have, you can always have more. Once we work up to the real numbers, which are defined as equivalence classes of sets of equivalence classes of ordered pairs of integers, any connection to reality is purely coincidental.

Apparently, this camp is generally "refuted" by demonstrations via long division, which are perfectly valid -- if one first shows that long division "works." Which I doubt is happening very often on the internet. In a quick, nonscientific survey, I was completely unable to find a rigorous proof in the form of a simple, linkable image file anywhere on the internet, so I made one myself.

For background: the symbol "N . a₁ a₂ a₃ ...," where N is an integer and each a_i is an integer between 0 and 9, inclusive, is defined to mean the real number given by N + a₁/10 + a₂/100 + a₃/1000... As an infinite sum is defined as the limit of the partial sums, N is the limit as n -> oo of the sequence N + a₁/10 + a₂/100 + ... + a_n/10^n. Now we just use a good, old-fashioned epsilon proof:



There you have it.