Just Some Theorizing for Pi Enthusiasts

[ohshiznit422]

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

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

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

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

The percent error gets smaller as the root increases.

Is it then valid to say:

limit as x->∞
\sqrt[x]{ψ\pi^{x}} = \pi

 

[Curious3141]

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

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

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

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

The percent error gets smaller as the root increases.

Is it then valid to say:

limit as x->∞
\sqrt[x]{ψ\pi^{x}} = \pi

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.

 

[Mark44]

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.

 

[guysensei1]

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.

 

[Office_Shredder]

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

 

[coolul007]

my favorite is still 355/113 = 3.1415929203539823008849557522124

 

[skiller]

my favorite is still 355/113 = 3.1415929203539823008849557522124

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!

 

[coolul007]

It may be your favourite but it's still wrong by the time you get to the 7th decimal place, beyond which we we taught at school!

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

 

[skiller]

Of course it's wrong, It's still a good practical approximation.

Of course it's a good practical measure - but is it a good enough one...!?