# Just Some Theorizing for Pi Enthusiasts

1. Jun 13, 2013

### 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->∞
$\sqrt[x]{ψ\pi^{x}}$ = $\pi$

2. Jun 13, 2013

### Curious3141

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
3. Jun 13, 2013

### 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.

4. Jun 13, 2013

### 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.

5. Jun 13, 2013

### Office_Shredder

Staff Emeritus
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

6. Jun 14, 2013

### coolul007

my favorite is still 355/113 = 3.1415929203539823008849557522124

7. Jun 16, 2013

### skiller

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
8. Jun 16, 2013

### coolul007

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

9. Jun 16, 2013

### skiller

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