Choosing a decimal representation in a given domain of discourse.

Question

Say I limit myself to a domain of discourse in the the natural numbers, would it be incorrect to use a decimal representation like $2.000$ as opposed to just $2$ or $02$? Does the use of the decimal place imply the existence of $0.1$, $0.01$ etc? Do we slightly adjust our decimal system to account for this? If we don't directly discuss the numbers outside of $N$ it doesn't seem problematic, but I do not want to be caught out on this, If we consider the real number and natural number $1$ to be the same, is it also incorrect to use $0.999$ as it's proof implies addition of numbers not available to me?

Answer 1

I think this is more an issue of communication than one of "correct math". Do whatever you think transfers your idea from your head to your reader's head in the best possible way. Note that what counts as "best" is mostly up to you (within reason). Do you want to be compact and efficient? Use as few letters and digits as possible. Do you want to convey accuracy? Use decimals to your heart's content. You have other concerns? prioritize them exactly as much as you feel like.

Natural numbers are completely accurate without using decimals. So writing $2.000$ is not needed for accuracy, is bad for efficiency, and as it will likely be somewhat unexpected, could confuse your reader. These concerns all pull in the direction of using $2$ instead. But $2.000$ is still correct.