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Just Some Theorizing for Pi Enthusiasts

  1. Jun 13, 2013 #1
    So I heard someone say that the cube root of 31 is a good approximation for ∏. This led me to thinking about higher roots and approximations.

    ψ denotes the nearest whole number

    [itex]\pi[/itex][itex]^{2}[/itex] [itex]\approx[/itex] 9.86960440109
    ψ=10
    [itex]\sqrt[2]{10}[/itex]=3.16227766017

    [itex]\pi[/itex][itex]^{3}[/itex] [itex]\approx[/itex] 31.00627668029
    ψ=31
    [itex]\sqrt[3]{31}[/itex]=3.14138065239

    [itex]\pi[/itex][itex]^{4}[/itex] [itex]\approx[/itex] 97.40909103400
    ψ=97
    [itex]\sqrt[4]{97}[/itex]=3.13828899272

    The percent error gets smaller as the root increases.

    Is it then valid to say:

    limit as x->∞
    [itex]\sqrt[x]{ψ\pi^{x}}[/itex] = [itex]\pi[/itex]
     
  2. jcsd
  3. Jun 13, 2013 #2

    Curious3141

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    Homework Helper

    There's an error in this. Did you mean this: ##\lim_{x \to \infty}[\pi^x]^{\frac{1}{x}} = \pi##, where ##[]## represents the "nearest integer function"?

    If so, it's true, but also trivial. You can represent ##[\pi^x]## as ##\pi^x + \delta##, where ##\delta## is a small positive or negative real number such that ##0<\delta<1## and of course ##\delta <<\pi^x##. After that, use a binomial expansion to see this. But there is no mathematical utility here - there is no guarantee that increasing x will necessarily improve the accuracy of the approximation, because you haven't proven it.
     
    Last edited: Jun 13, 2013
  4. Jun 13, 2013 #3

    Mark44

    Staff: Mentor

    It's better to say this:
    $$\sqrt[n]{\pi ^n} = \pi$$

    This is true for all integers n ≥ 2, which makes it also true in the limit as n → ∞. Rounding before raising to the power inside the radical makes the result a little off from what you would get in the formula above.
     
  5. Jun 13, 2013 #4
    Here is another approximation I found that i have not seen anywhere.

    462*(e^pi) almost equals 10691

    so ln(10691/462) almost equals pi to around 8 places if i remembered correctly.
     
  6. Jun 13, 2013 #5

    Office_Shredder

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    Staff Emeritus
    Science Advisor
    Gold Member

    You had to memorize eight digits (10691 and 462) and need to calculate a logarithm. You were better off just memorizing the eight first digits of pi.

    Your OP claim is correct but it's not clear how you would use it in practice - if I want to take pi to the 100th power for example and round it to the nearest integer, I need to know what pi is to extraordinary precision already
     
  7. Jun 14, 2013 #6
    my favorite is still 355/113 = 3.1415929203539823008849557522124
     
  8. Jun 16, 2013 #7
    It may be your favourite but it's still wrong by the time you get to the 7th decimal place, beyond which we were taught at school!
     
    Last edited: Jun 16, 2013
  9. Jun 16, 2013 #8
    Of course it's wrong, It's still a good practical approximation. a one kilometer diameter circular arc would be off by

    1,000,000 mm x pi = 3141592.6535897932384626433832795
    1,000,000 mm x 355/113 = 3141592.9203539823008849557522124

    with a difference of 0.2667641890624223123689328864963 mm

    a practical error I can live with
     
  10. Jun 16, 2013 #9
    Of course it's a good practical measure - but is it a good enough one...!?
     
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