TITLE: What exactly is "approximation"?
QUESTION [8 upvotes]: There are a  lot of great "approximations" that exist in the mathematical field:$$\dfrac{22}{7} \approx \pi$$
$$e \approx \left(1 + \dfrac{1}{n}\right)^n$$But the fact that I have yet to know what these statements strictly mean makes me rather uncomfortable.  My question would be:

What is the definition of "approximation" in terms of calculus or algebra? Are there any constraints to whether a number can be "approximate" to another or not? If not, can we say $1 \approx 2$ just because they seem to be close? And if that is not the case, then what allows us to assert $1 \approx 1.001$?

REPLY [2 votes]: Professor Arnold Ross, of the Ohio State University, used to ask, "What is an approximation to $5$?" and then answer, "Any number other than $5$."