ForumTechnical Corner ► Approximating Irrational Numbers w/ Other Irrational Numbers
I decided it would be a good use of my time to approximate e in terms of pi and approximate pi in terms of e

I challenge you to make a closer approximation than mine:

RULES:
Can use any operations
Choose 1 irrational number to use
Choose 1 irrational number to represent
You can only use that one irrational number to approximate the second one you chose
Get as close as you can without going over
Along with your approximation, provide a percentage in the form of 100(approximation)/(actual number)
  
here are my 2 entries so far

approximation of pi in terms of e
approximation of e in terms of pi

  
I improved your approximation of pi in terms of e

edit: improved e in terms of pi too

edit 2: why approximate pi with e when you can find its exact value with e? arccos(e - e - e/e) is exactly equal to pi. Should we limit the set of legal operations?
  
I feel I might have over-approximated
Accurate to 16 digits
  
improved on the approximation of e in terms of pi

working on the pi in terms of e

ninjad damn
  

holy shit, mines only accurate to 4
  
(Maclaurin series might be cheating)
  

How did you manage this? (ninja'd, nvm)
  
Maclaurin series. If you keep adding pi's over the first sigma, it'll keep converging (really fucking slowly, though).
  
why did I think this was a good idea
it didn't even work
  
Pi, accurate to seven digits.
There was a trick I'm blanking on that you could use to make the pi series converge faster. Something to do with tangent?
aprzn123 said:
why did I think this was a good idea

Me when I look at how long that equation scrolls for:
  
I see your Taylor Series and raise you a continued fraction. It's already accurate enough, but you can make the pi^pi a proper power tower (pi^pi^pi^…) to make it converge real fast (and also very difficult impossible to actually compute, because it's kinda overkill).
  
This has now passed above my understanding
  
I can do my best to explain further it if you'd like, but the gist is that continued fractions are great and e has a decently simple continued fraction.
  
Can you make a continued fraction for pi?
  
I see your continued fraction and raise you by basically cheating (screenshot b/c I'm out for lunch and on my phone).

That's it, we've done it. It literally cannot be approximated any further.
  
That's not an approximation to pi, that's just pi. Although I suppose you could just make it a Riemann Sum without too much effort.

And I did do a continued fraction version for pi, but it does not converge quickly at all.
  
Technically, an integral is an already-converged rsum if you thing about it.
  
I just mean that if you view getting it exact as cheating then it would be trivial to get arbitrarily close.
  
KylIjoy said:
aprzn123 said:
why did I think this was a good idea

Me when I look at how long that equation scrolls for:

I was trying to implement Ramanujan's approximation but it had a four digit prime number
  
This thread when it started: Fun ways of trying different combinations to come within a few hundredths of irrational numbers

This thread now: Nerdy arms race of applying progressively more convoluted methods in order to create impressively accurate approximations
  
Maybe we should pick some numbers that are a little bit less well-researched than the two most famous transcendental numbers. Like, maybe we try approximating the cosine fixed point in terms of the Euler-Mascheroni constant.
  
Maybe we should pick some numbers that are a little bit less well-researched than the two most famous transcendental numbers. Like, maybe we try approximating the cosine fixed point in terms of the Euler-Mascheroni constant.

I like that idea

here is my final submissions for approximating e
https://www.desmos.com/calculator/pmmgrmukgp
  
it feels like cheating to use a number less than 1: approximating dottie number using sqrt(i^i)
  
Here is an extremely simple one

π = 3√31

99.993% accuracy

I challenge you all to find an approximation as simple with a better accuracy

Edit: I have just realized that this forum is like 3 months old and I don't even care. I demand that the abyss answer my call.
  
Forum > Technical Corner > Approximating Irrational Numbers w/ Other Irrational Numbers