In article <firstname.lastname@example.org>, wishing to remain anonymous <email@example.com> wrote:
> How can .9999999999999999999... Be equal to one >... > And, just because there is nothing between .999999... and 1, it >doesn't men that they are one and the same. I believe that there is >nothing between .9999... and one because it can never be subtracted > from it. If you think about it, you may come to think that >.99999999999... cannot exist.
Your thoughts are on the right track, they just require a little illumination.
Your statement ".999... cannot exit" indicates that you are working under the assumption that mathematical notations have a life outside our heads. This is not true.
There was a time, as recently as a few centuries ago, when the string of symbols ".999..." indeed had no meaning whatsoever -- it had never been written down. Then somebody had the need to express a particular concept and invented the notation ".999..." for that purpose. Note that I said _invented_, not _discovered_. The expression ".999..." had not existenced before that invention.
In the same way, the string of symbols ".18.104.22.168..." does not have a meaning today. I just invented it. Of course this does not mean a thing unless I TELL you what I mean by this string of symbols.
When we invent a newfangled notaion such as ".999..." or ".22.214.171.124...", we'd better TELL very carefully what we mean by it, otherwise the audience will have no clue what we're taking about.
Have you been told the meaning of ".999..."?
There is a very preceise meaning attached to that notation. Unfortunately the description of the meaning requires a somewhat sophisticated mathematics therefore it is not covered in elementray math courses in schools. Consequently people often try to read a meaning into ".999..." which is not there.
It makes no sense to say ".999... cannot exist" if you don't know what that notation is supposed to mean. It is the same as saying ".126.96.36.199... cannot exist". Why can't it? I can make it to mean whatever I want it to mean! All I need to do is to TELL you what I have in mind for that sequence of symbols.
At any rate, if you want to take my word for it, the precise definition of ".999..." (which I have not told you what it is) makes it clear that .999... = 1. You will see this with various levels of rigor in college courses such as calculus or analysis.