Is it incorrect to use ellipses when writing an irrational number's approximate value with a $\approx$?

Question

For example, is it incorrect to write that Euler's number is $$ e \approx 2.718\dots $$ or is it more proper to write $$ e \approx 2.718 $$ or, maybe I should just write $$ e = 2.718\dots $$

Answer 1

It is incorrect to write $e\approx 2.718...$ as the entirety of the expansion is implied by the "$...$" after the digits. Since this is $e$, what you'd be saying is that $e\approx e$, which is obviously nonsense. However, $e\approx 2.718$ is perfectly fine. If you want to use an ellipsis, you need to use the equals sign.

Answer 2

The only time approximately should be used is when an approximation is done. Additionally, ellipses imply continuation (literal meaning in math: "and so forth"), therefore the ellipses in $2.718...$ imply the rest of the digits in Eulers number should follow. So it would be correct to write $e \approx 2.718$ since $e$ is being approximated from $2.718281828...$ to $2.718$.