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.
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.