nglasser (01/27/83)
It is true, and well known, that 0.9999... is equal to 1. But what does it mean to say 0.000...1? At what decimal place does the 1 occur? 0.9999... is well defined, but 0.000..1 does not make any sense. It is like saying, "Take an infinite list of numbers, and at the 'end' add another number." This is obviously obsurd. We could add another number at any given point in the list, but there is no end to the list, so we cannot add a number to the end of it. Another point about 0.9999 = 1. The proofs of the form x = 0.99999... 10x = 9.99999... 9x = 9 x = 1 are not valid. Otherwise you could use a similar argument to show that the series ln2 = 1 - 1/2 + 1/3 - 1/4 + ... converges to any number you like, or even diverges. You must add some more details to the proof, such as the fact that the series in question converges absolutely, that addition of series term by term is valid, etc. to make it valid. The easiest proof is the one already given in this newsgroup evaluating this number as the sum of an infinite geometric series. As to infinity, most books on set theory will be able to give you a good picture of cardinalities of infinite sets, and whether sets are countable or uncountable. - Nathan Glasser ..decvax!yale-comix!nglasser