cbostrum (01/29/83)
all of this discussion about whether .999... = 1 is getting very irritating. or perhaps it should be encouraging: a good demonstration that even people ignorant of basic mathematics can effectively use computers. grade nine proofs of the form "x=.9999... so 10*x=9.999... so 9*x=9 so x=1 so .999...=1" are simply not valid, for one thing. they are exactly the same as "x=1+2+4+8... so 2*x=2+4+8+16... so 2*x-1=1 so x=1 so 1=infinity". the point is that .999... is really an infinite sum, and the regular grade nine addition does not work on all sums. it is well known how certain types of sum are given values; this happens when the sequence of partial sums has a limit (which is a simple but deep concept that allows us to extend the definition of sum to include .999...). once this is done it becomes a trivial fact that .999...=1. now you may wish to define infinite sums in some other way, thinking you know better than all those who have come before you with strong intuitions, but at least be aware of what they have done and why...