Explanation for $\pi^3\approx \frac{3650e+999}{1000}+e^3$

Question

After some fooling around, I found that $\pi^3\approx \frac{3650e+999}{1000}+e^3$ which seemed like a not too bad approximation of $\pi^3$. Is there some nice reason why this transcendental power of $\pi$ can be approximated by some linear combination of rational powers of the transcendental number $e$? Or can these approximations be found for e.g. any power of $\pi$?

Answer 1

It's not a particularly good approximation: the error is about $1.1\times 10^{-5}$. A better one is $$ \pi^3 \approx \frac{103 e^3 +5 e -98}{64} $$ where the error is about $7.3 \times 10^{-8}$.

Any real number can be approximated arbitrarily well by rational linear combinations of any nonzero real number. There is no particular significance to this one.

Answer 2

I know where you're coming from - I too did this a lot when I was younger - and any number can be approximated by other combinations of numbers, it still is kinda weird/cool finding an approximation like this.

My favourite is $pi = (2791563950)^{1/19}$, even if it kinda sucks in terms of accuracy