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Old 2006-01-02, 06:05   #1
ixfd64
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Default smallest number used in a mathematical proof?

We know that the largest number used in any serious proof is Graham's number.

But what about the smallest number used in a mathematical proof (excluding zero and infinitesimals)?

I know that there's physics-related Planck units, but I'm sure there's numbers much smaller.
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Old 2006-01-02, 06:31   #2
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Possibly some paper involving the ABC conjecture would be the answer, in my opinion.

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Old 2006-01-02, 08:55   #3
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Quote:
Originally Posted by ixfd64
We know that the largest number used in any serious proof is Graham's number.

But what about the smallest number used in a mathematical proof (excluding zero and infinitesimals)?

I know that there's physics-related Planck units, but I'm sure there's numbers much smaller.
The reciprocal of Graham's number ought to be a good candidate for the honor.

Last fiddled with by cheesehead on 2006-01-02 at 08:57
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Old 2006-01-02, 10:32   #4
Orgasmic Troll
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I'm guessing that it's some nonstandard analysis proof that uses an infinitesimal

"infinitesimal" isn't very easy to type after a few glasses of wine.
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Old 2006-01-02, 10:56   #5
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I don't think there's a good answer to this question. You can always recast equations so that constants that appear in it are smaller or larger. Besides, what exactly qualifies as a constant in this context? Are the elements of a series that tends to zero constants?

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Old 2006-01-02, 14:17   #6
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This GreenHodge is not authoratitive, just a bloke with a blog, but he lists some numbers he thinks are interesting. After the Planck length, 1.6160*10^{-35) the next smallest number he lists is 0.412454... which he calls the Thue-Morse constant. So maybe there aren't that many interesting small numbers.
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Old 2006-01-02, 15:34   #7
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Quote:
Originally Posted by Numbers
This GreenHodge is not authoratitive, just a bloke with a blog, but he lists some numbers he thinks are interesting. After the Planck length, 1.6160*10^{-35) the next smallest number he lists is 0.412454... which he calls the Thue-Morse constant. So maybe there aren't that many interesting small numbers.
The Planck length has units, which makes the magnitude of the number itself somewhat arbitrary. I could decide to express it in parsecs or angstroms and get vastly different values for the exact same property. Therefore, I wouldn't consider it a true 'number' in the mathematical sense in the same way that Graham's number is a very large number.

It's an interesting question, but I'm afraid I can't offer any other small constants.

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Old 2006-01-02, 17:40   #8
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Quote:
Originally Posted by drew
The Planck length has units, which makes the magnitude of the number itself somewhat arbitrary. I could decide to express it in parsecs or angstroms and get vastly different values for the exact same property. Therefore, I wouldn't consider it a true 'number' in the mathematical sense in the same way that Graham's number is a very large number.

It's an interesting question, but I'm afraid I can't offer any other small constants.

Drew
There are sundry small and dimensionless quantities in physics. A famous one is the fine structure constant, which is approximately 1/137

Another and much smaller quantity is the ratio of the strengths of the gravitational and lectromagnetic interactions.

Whether physical constants have much to do with the question as originally asked is another question entirely.

Paul
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Old 2006-01-04, 02:33   #9
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What about the difference between neutrons and protons in weight, even taking account of neutrinos and electrons. Or on a similar note, what about the anticipated atomic mass and it's actual value for specific elements/isotopes? For example, helium atoms are actually a little lighter then 3(He I) or 4(He II) times the mass of a hydrogen atom.
What about the probability of a broken cup suddenly reassembling itself in a tornado? :)
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Old 2006-01-04, 17:18   #10
mfgoode
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Question smallest number in a mathematical proof?


Would -459.67 qualify ?
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Old 2006-01-04, 22:20   #11
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Quote:
Originally Posted by mfgoode
Would -459.67 qualify ?
"Smallest" refers to absolute magnitude, not algebraic value, or else ixfd64's "(excluding zero and infinitesimals)" wouldn't make sense.
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