# 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

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

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.

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

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)