Let n = 10, s = 4.
[s] = {s' $\in \mathbb{z}$ | s' $\sim$ s} = {s' $\in \mathbb{z}$ | 10 | s' - 4} = {10k+4 | k $\in \mathbb{z}$ } = {s' $\in \mathbb{z}$ | decimal expansion of s' ends in a 4 if s' is pos, ends in a 6 if s' is neg}
How does this last line make sense? We know that 4 $\sim$ 14. But, from my understanding of decimal expansions, isn't the decimal expansion of 4 4.0000 (repeating) and 3.9999 (repeating). So how can we say that decimal expansion of 4 ends in a 4? Similarly, we know that -6 $\sim$ -16. But shouldn't the decimal expansion of -6 be -6.0000 (repeating) or -5.99999 (repeating)? So how can we say that decimal expansion of -6 ends in a 6?
It seems like my professor is stopping at the ones digit and not accounting for the "decimal" portion. I think the problem is that I lack a clear understanding of what a "decimal expansion" is.
Edit: Is my teacher just considering one decimal representation out of two?
The term "decimal expansion" here is just the way you would write an integer as $267456$ or $127$ or any other string of digits - where there are no decimal points at all to worry about. It's the ordinary way of writing an integer.
Formally, the idea is that the notation $83521$ is really just shorthand for: $$8\cdot 10^4 +3\cdot 10^3 + 5\cdot 10^2 + 2\cdot 10 + 1$$ where we have a sum over powers of $10$ times digits from $\{0,1,2,3,4,5,6,7,8,9\}$. Every non-negative integer can be uniquely expressed in this form (and every negative integer is just the negation of some positive integer) - and, when you're talking only about integers, this is what is meant by "decimal expansion."
There's a generalization of this that applies to all real numbers, where something like $$13.1415\ldots = 1\cdot 10^1 + 3 + 1\cdot 10^{-1} + 4\cdot 10^{-2}+1\cdot 10^{-3}+5\cdot 10^{-4}+\ldots$$ where we allow negative exponents of $10$ as well (indicated by the decimal point) and also allow infinitely many terms with these negative exponents if we desire. Some numbers can be represented in multiple ways in this notation since $0.999\ldots = 1.000\ldots$. If people are talking about real numbers, this is what "decimal expansion" would mean - but it's not what your professor is referring to.
(Note: The word "decimal" here refers to the fact that we write a number as a sum of powers of ten - which is probably what you were going to do anyways. Other bases can be indicated in the same terminology - for instance, the term "ternary expansion" refers to writing a number as a sum of powers of $3$ times values in $\{0,1,2\}$ and generalizes in the same ways)