What you are describing are known as asymptotes. They are usually first encountered when shown the graph of y=1/x.
http://en.wikipedia.org/wiki/Asymptote See the second figure from the top.
This is a similar idea because it is generally stated by mathematicians that at infinity, the line will touch the axis. It is similar because it deals with a concept of infinity that developed due to the internal inconsistencies of the previous concept and this version of infinity is the same as is being used here. However, beyond that, the similarities end.
Hey michel, I am uncertain if you are approximising when you say "0.9999999" but I just want to make sure that you understand that I am not talking about 0.9999999 but "0." followed by an
infinite number of 9s (for which the mathematical notation is 0.9^)
Keeping the above in mind, it is impossible to have 0.9^8 because this would involve placing an 8 at the end of an infinite sequence of 9s. "at the end of infinity" is similar to ideas such as "outside of space" and "before time". It is certainly not a concept that is mathematically supported. Infinity has no end at which the 8 may be placed.
Pi is one of the mathematical nuggets that I have not yet studied in depth. I understand it to be an irrational number (so it cannot be accurately written as a fraction). However, it is also a transcendental number and so it is part of an entire realm of mathematics I have yet to study. I am uncertain how you would go about calculating a transcendental number (the definition of one is sufficiently complex so I will not post it here) but as far as I am aware, it is not possible to write them without approximising.
However, the decimal to a sufficient degree will always be more accurate than the fraction. The fraction is useful for "simple" calculations in which relatively imprecise answers are sufficient (eg understanding the use and impact of Pi does not require an accurate output)
Thank you but I was specifically after a reference that stated that 0.9^ is "a number approaching 1 that can never quite get there". Apologies for my skepticism but I do not believe that you will be able to find a respectable mathematician who will endorse that statement.
You should also know that an "infinite regress", as you have defined it, involves convergent series and that it is well documented that when you sum and infinitely long series made up of convergent terms, you will get a finite number.
http://en.wikipedia.org/wiki/Geometric_series#Infinite_geometric_series
And a piece of cake is a fraction of the entire cake. It is 1/3 of the entire cake and 1/3 is an infinitely recurring decimal. That you have 3 pieces of cake makes sense since 3 x 1/3 = 3/3 = 1 and you have 1 cake.
I agree but "I can't fly" does not equate to "flight is impossible" which appears to be analagous to what is being argued in that particular case.