Q:

Decimals and equivalence relations

I am told that decimals set up an equivalence relation on the Reals and that decimal numbers and the Reals are not the same thing. I believe this also clarifys the famous $.\bar{9}=1$. That $1$ and $.\bar{9}$ are not equal but in the same equivalence class. If the statements I just said are true, what is this relation that decimals set up and why are decimal numbers not the same as the Real numbers?
Thank you.

A:

To answer my own question. There is an equivalence relation among the decimal numbers that results in 1 and .9 repeating being infinitesimally close and thus equivalent. This relation can then be imposed on the Reals once decimal numbers are shown to be part of the reals . The relation is, after shown to be transitive symmetric and reflexive, $a $ is equivalent to $b $ given $a$ is less than or equal to $b $ if there exists some $c $ such that $a+c=b $. If we take a set of decimal numbers bounded by some maximum decimal number in order to discover the least maximum $c $ that can be added to $a$ to form $b $, this set is empty since there is nothing between $1$ and $.9$ repeating thus under this equivalence $c $ is 0 for $1$ and $.9$ repeating and they are equal under normally equality not just this relation. 

