Thank you to everyone that responded to correct my thinking on this topic. I apologize for the delayed reply. I now understand why you say 1=0.999...! This short discourse has given me a opportunity for an overdue, albeit brief, review of limit and number theory. THANK YOU!
I would like to review my thinking in order to make sure my thought is correct.
Using vanesch's suggestion, it appears that I was constructing both numbers as follows:
For 1.0..., I was thinking
S(m)_{(1.0...)} = \sum^m_{k=0}10^{z_{0}-k}n(z_{0}-k)
where
n_{1} has z_{0} = 1 and n_{1}(1) = "0", n_{1}(0) = "1", n_{1}(-1) = "0", n_{1}(-2) = "0"...
and for 0.9..., I was thinking
S(m)_{(0.9...)} = \sum^m_{k=0}10^{z_{0}-k}n(z_{0}-k)
where
n_{2} has z_{0} = 0 andn_{2}(0) = "0", n_{2}(-1) = "9", n_{2}(-2) = "9",...
Of course the flaw in my thinking had more to do with the fact that it was finite and I was only doing a patial sum. Hence why I said 'it is obvious', at least so I thought, that:
S(m)_{(1.0...)} \neq S(m)_{(0.9...)}
because
S(m)_{(1.0...)} - S(m)_{(0.9...)}\neq 0
which is true if you are only considering a finite or partial sum. This led me to conclude:
S(m)_{(1.0...)} - S(m)_{(0.9...)} = p^m
Where we would could say for this case that p=0.1, which would be true as long as the sum was partial. But this is not the case, for as pointed out by vanesch. For in order to qualify these as real numbers, the limit of m had to go to infinity, or
R(n)=lim_{m \rightarrow \infty} S(m)
in which the difference that I was seeing would go to zero, or
p^m \rightarrow 0
I suspect that this number p^m is correlated to the nested intravel http://home.comcast.net/~integral50/Math/proof2a.pdf" that Integral so thoughtfully provided(BTW, thanks Integral, this really helped me wrap my mind around this nugget plus it was quite neat!
).
So there you have it, hopefully I have corrected my thinking. If you think that further correction is needed please post your thoughts!
There is one thing I did come across that would finalize it for me, which is the following:
Does
lim_{m \rightarrow \infty} [1^m = (0.9...)^m]?
Other thoughts:
Regarding the trisection of angle, after some thought I came to realize what had been nagging me. Though one can bisect an angle it seems that the only way to trisect an angle is to use a limiting proceedure of bisections. If anyone has any thoghts please let me know. I am also curious as to wether any of the greeks had worked out limiting proceedures to any extent.
As for the physical case of the speed of light and Einstein's wish to go that fast, I am curious how fast he would have to go in order to be in that intravel such that nature agrees to say c=0.99...*c. Are these the same energies needed to create a black hole? Is this a feasible thought? I realize that this is another topic that belongs elsewhere but I had to finish the thought.
Once again THANK YOU everyone that helped in this exercise!
http://simpler-solutions.net/pmachinefree/thinkagain/comments.php?id=1589_0_3_0_C"
