e \approx \left(1 + 3^{-2^{85}}\right)^{9^{4^{6 \times 7}}}
This approximate expression for e is very beautiful; it happens to use the nine digits from 1 to 9. And its beauty is not limited to just that. Guess how many significant digits it has? 10? 100? 1000? 10,000?
The result is startling; its degree of approximation far exceeds our imagination—it is accurate to 1,315,266,887,768,832,673,579,363 decimal places!
Clearly, this is absolutely not a coincidence, or rather, it is also
a coincidence. Its secret lies in the fact that e = \lim_{n \to \infty} (1 + 1/n)^n, and
9^{4^{6 \times 7}} happens to be
exactly equal to 3^{2^{85}}.
Consequently, the result is quite substantial; this exponent is so large
that Mathematica directly reports an Overflow, which is why
it can be accurate to so many decimal places of e.
It is said that this god-like approximate expression originally came from here.
Because this magic is so beautiful, to prevent the content from being lost, I have specifically saved the webpage content exactly as it was. This website provides the address: http://kexue.fm/sci/Math-Magic/Math-Magic.htm
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