Modular Numbers: Not Accounting for Decimal Portion in

Step 1
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:
${\displaystyle 8\cdot {10}^4+3\cdot {10}^3+5\cdot {10}^2+2\cdot 10+1}$
where we have a sum over powers of 10 times digits from ${\displaystyle \{0,1,2,3,4,5,6,7,8,9\}}$.
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
${\displaystyle 13.1415\dots =1\cdot {10}^1+3+1\cdot {10}^{-1}+4\cdot {10}^{-2}+1\cdot {10}^{-3}+5\cdot {10}^{-4}+\dots }$
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 ${\displaystyle 0.999\dots =1.000\dots }$. 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 ${\displaystyle \{0,1,2\}}$ and generalizes in the same ways)