Does approximation usually exclude equality?

Question

The Wikipedia article on Approximation says that "An approximation is anything that is similar but not exactly equal to something else." Is excluding exact equality (i.e. $x = y \implies x \not\approx y$) standard when using the terminology "approximate"? For what it's worth, the ISO 80000-2 standard specifies that "Equality is not excluded" for the symbol U+2248 “≈”. I recently came across a use of the word "approximation" in a context that greatly confused me until I realized that they were implicitly excluding the case of equality, which was crucial for making their claim correct.

Answer 1

I don't think it matters all that much. I would use "approximation" in a situation where I could prove that something is close to another thing but I'm not bothering to check either way whether they are exactly equal. It may be that I end up approximating a function $f(x)$ by another function $g(x)$ which happens to be equal to $f$ for some values of $x$, but that wouldn't disqualify $g(x)$ from being an approximation. I would be very surprised if they were always equal, since generally I pick $g(x)$ deliberately to be much simpler than $f(x)$, so I would expect that to basically never happen anyway, e.g. for Stirling's approximation.

As another example, I would use the phrase "Taylor approximation" to describe approximating a function near a point by truncating its Taylor series near that point even if the function is a polynomial, so that the approximation is exact at some point. I don't think this sort of usage is particularly uncommon either; google the phrase "approximation is exact" for more examples in this vein.

If you have a question about a particular case you should just ask about that.