- #1

- 14

- 0

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

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- Thread starter danyo
- Start date

- #1

- 14

- 0

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

- #2

- 22,089

- 3,296

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 [itex]n\pi[/itex] as close to an integer as we like. This is basically Kroneckers density theorem. Of course, a nonzero multiple of [itex]\pi[/itex] can never actually equal an integer (since that would imply that [itex]\pi[/itex] is rational), but it can be arbitrary close.

The number [itex]\pi[/itex] is not special here, it works for any irrational number.

- #3

- 14

- 0

There is no solution to that problem. The thing is that we can get [itex]n\pi[/itex] 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 [itex]n[/itex] would bring us closer to a whole number...

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

- #4

- 33

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I wonder if there's any pattern in what integers [itex]n[/itex] would bring us closer to a whole number...

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!

- #5

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"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

- #6

- 4

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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)

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