Mathematical Magic — A Beautiful Approximation
$$e\approx\Big(1+3^{-2^{85}}\Big)^{9^{4^{6\times 7}}}$$
This approximate expression for e is quite beautiful, and it happens to use exactly the nine digits 1 through 9. But that's not the only beautiful thing about it — go ahead and guess how many significant digits of accuracy it gives. 10 digits? 100 digits? 1000 digits? 10000 digits? more
The result is frankly terrifying — its degree of approximation goes far beyond what we might imagine. It's accurate to 1,315,266,887,768,832,673,579,363 decimal places!
Obviously this can't just be a coincidence — or rather, it is a coincidence, but one with a hidden reason. The secret lies here: $e=\lim\limits_{n\to\infty}(1+1/n)^n$, and $9^{4^{6\times 7}}$ happens to equal exactly $3^{2^{85}}$. The consequences of this are quite severe — the resulting exponent is so enormous that Mathematica simply reports an Overflow. That's precisely why it manages to match e to so many decimal places.
Reportedly, this god-tier approximate expression originally comes from here.
Because this piece of mathematical magic is so beautiful, in order to prevent it from being lost, I've deliberately saved a verbatim copy of the webpage's contents. It's available on this site at: http://kexue.fm/sci/Math-Magic/Math-Magic.htm
This article is reposted from: Mathematics Research Forum (bbs.emath.ac.cn)
Translated automatically with claude-sonnet-5; all equations are reproduced verbatim from the source. Copyright remains with the original author.