## Problem: Multiples of pi

Hi everyone,

About 15 minutes ago I came up with a problem... What whole number multiple of pi would result in a number closest to a whole number?

Does a single whole number multiple exist, and can we... prove it?

-Daniel
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 Quote by danyo Hi everyone, About 15 minutes ago I came up with a problem... What whole number multiple of pi would result in a number closest to a whole number? Does a single whole number multiple exist, and can we... prove it? Thanks for help in advance! -Daniel
There is no solution to that problem. The thing is that we can get $n\pi$ as close to an integer as we like. This is basically Kroneckers density theorem. Of course, a nonzero multiple of $\pi$ can never actually equal an integer (since that would imply that $\pi$ is rational), but it can be arbitrary close.

The number $\pi$ is not special here, it works for any irrational number.

 Quote by micromass There is no solution to that problem. The thing is that we can get $n\pi$ as close to an integer as we like. This is basically Kroneckers density theorem.
Thank you for the reply, micromass! I was not familiar with Kronecker's density theorem, but its logic clarifies this problem. I wonder if there's any pattern in what integers $n$ would bring us closer to a whole number...

Ah, well there goes my bedtime tonight! Thanks for the direction :D

## Problem: Multiples of pi

 Quote by danyo I wonder if there's any pattern in what integers $n$ would bring us closer to a whole number...
That is a much more interesting problem (to me). For certain types of irrational numbers, there is indeed a pattern (you can check out Pell's equation and Continued Fractions to find ways to very closely approximate square roots).

However, here is how you would find such integers for pi. We know the close approximation of 22/7 for pi. Then we have:

22/7≈pi
22≈7pi

And verifying, we have 7*pi≈21.99114858

Another close approximation is 333/106:

333/106≈pi
333≈106pi

and 106pi≈333.0088213...

I hope this proves useful!
 You might also be interested in the following thread from the wu riddle site; "Say I am given a number X = A*[sqrt]2 + B*[pi], where A and B are integers. Given X, how can you find A and B, without using brute force?" It comes with a long discussion. see http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi
 I ran a quick computer program just for interest sake. 78256779 103767361 129277943 131002976 156513558 180299107 182024140 183749173 205809689 207534722 209259755 233045304 234770337 236495370 258555886 260280919 262005952 Those numbers if multiplied with pi will give you a number so close to a integer that the decimal part can't fit in a double precision floating point. As stated above you can't actually get a integer from multiplying a integer with pi (except 0)

 Tags brown university, multiple, number theory, whole number