ForumTechnical Corner ► Approximating Irrational Numbers w/ Other Irrational Numbers
2√3 ~= pi
  
even 2√2 is closer to π than 2√3
  
shhh
  
It's not, though.
  
sqrt2+sqrt3 is pretty close
  
sorry for the double post, but 5√306 is also quite close


edit: 9√29809
  
5√306 is very good. Not quite as elegant as 3√31 but it is a lot closer
  
its not elegant/simple but it is accurate to 12 digits: 29√261424513280000
  
ln(-1)/i = π(2k+1) ∀ k ∈ Z (restrict the natural log's complex extrapolation to [0, 2π)*i for best results)
Alternatively use infinite sums to find both!
6*√(Σ1/k²) = π (k all positive integers)
Use this formula for pi later
e^(ln(π²/6))=π²/6
ln(π²/6) = Σ(1/k*1/(pk)) (p is all primes, k is all positive integers)
So e = (π²/6)1/ln(π²/6)
If you want to approximate, stop at any prime/integer, but be warned that the way to approximate the natural log converges super slowly and is incredibly intensive on your processor

I get that at no point except in the limit do you ever get an irrational number, but I still like this one
  
Forum > Technical Corner > Approximating Irrational Numbers w/ Other Irrational Numbers