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 $$ e = 2.718\dots $$
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Writing $e = 2.718 \ldots$ indicates that the first $4$ digits are exactly $2.718$, i.e. that $2.718 \le e < 2.719$. On the other hand, if you use $\approx$ you aren't really making a precise claim that $e$ is in a certain interval; – Robert Israel Nov 21 '23 at 06:12
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1It is correct to write $e=2.718...$. – pancini Nov 21 '23 at 06:21
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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.
H. sapiens rex
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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 $2.718$.
FishDrowned
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