6 ms·

The Efimov constant is only approximately ≅ 22.7. It would be a little surprising to have a physical constant end in exactly 0.7, since it would imply that the

by ipsin 12y ago

The Efimov constant is only approximately ≅ 22.7. It would be a little surprising to have a physical constant end in exactly 0.7, since it would imply that there's something special about base 10 and bases that are a multiple of 10. Section 3 of this paper talks about the actual transcendental equation that provides the constant. http://rsta.royalsocietypublishing.org/content/369/1946/2679.full http://rsta.royalsocietypublishing.org/content/369/1946/2679...

infogulch 12y ago
Thank you! I was thinking the same thing about the constant being exactly representable in base 10.
3rd3 12y ago
Rational numbers tend to have easily recognizable patterns in their fractional digits in all bases (at least as long the numbers aren't too big), so it wouldn't imply there is something special about base 10. In fact, you can always choose the base as q for a rational number p/q and get rather simple fractional digits.
- ipsin 12y ago
  It's not repeating digits I'm talking about, though, it's termination. If it were exactly 22.7, that's 227/10. The question would be "Why is the denominator 10?"
  - akiselev 12y ago
    Because you used the denominator "10." If the constant is precisely 22.7 then it can be represented in any other base with just as much precision, you'd just use a different format for the number before and after the decimal point.
    - thaumasiotes 12y ago
      You're not paying attention to the comment you're pretending to reply to. From above: > It's not repeating digits I'm talking about, though, it's termination. 227/10 will not terminate in any numerical base which isn't a multiple of ten. Combining this with the idea that no numerical base should be privileged over another, we can rephrase this to say that it's surprising for a physical constant to be a rational (but not integral) number. It's not surprising for a physical constant to be an integer because we really believe that integers have special qualities. It's not surprising for a physical constant to be irrational because that's just the way numbers are. If it's rational but not integral, you want to justify the denominator somehow.
      - wlievens 12y ago
        For an example of an integral constant, consider the exponent in Newton's law of universal gravitation: the distance is squared (exponent two).
        thaumasiotes 12y ago
        ...so? That one is a nice case where not only is the number clean and simple, but we have a solid theoretical basis for why it's 2 and not some other number.
      - keithnoizu 12y ago
        There are many constants out there. Sooner or later it seems natural that you'll find one that happens to dive nicely into base 10. Why does that imply anything special about base 10. It would simply be a coincidence.
        thaumasiotes 12y ago
        It's a staggeringly unlikely coincidence; the odds of getting a rational number when selecting uniformly from any bounded interval are 0% -- not approximately zero, but exactly zero. In contrast, the odds of getting an irrational number are 100%. It's not possible to select uniformly over an unbounded interval, but rational numbers don't get any easier to find. (Technical note: though the figures of 0% and 100% are correct and exact, they actually do not mean that selecting a rational number is impossible. For a similar idea, note that if I select a number uniformly from the interval [0,1], the prior odds of choosing whatever number I did choose were also zero, but by the construction of the problem I had to end up with something.) This means that if you find a rational number, you should immediately suspect that it's like that for a good reason. You're much more likely to win the jackpot of every lottery you ever enter, even if you buy a ticket every day, than to encounter a rational number by coincidence. Or using a different analogy, the world population is roughly 7 billion. It would be extremely unlikely, so unlikely that we can essentially rule it out beforehand on theoretical grounds, for any of them to be a rational number of centimeters tall, or to mass a rational number of grams. That sort of thing can't happen.
  - stephengillie 12y ago
    227/10 = x/π = y/e = z/n
  - 3rd3 12y ago
    It's just that it's a non-sequitur to say "oh, that looks rather simple and it has a 10 in it, there must be something special about our number system" (which is what you say in your first comment). I implicitly subsumed termination under 'simple pattern' since repeating sequences can be described in a finite way too. The only surprising property of 22.7 in this context is that it's rational.
snorkel 12y ago
I vaguely recall that if a circular disc with a circumference of pi (3.14 ...) is held at an angle of 22.7 degrees relative to a flat surface, it projects an ellipse that has a circumference of 3.000 Coincidence? Eh, probably. Might've also been 26.8 degrees.
- ColinDabritz 12y ago
  One common association of those digits with PI is the handy approximation of 22/7 accurate to three digits, off by just over .0004. I stress that this is an approximation only! I had to explain about PI being irrational rather than exactly equal to the shortcut approximation to a couple co-workers, explaining some poor behaviors in their rotation function.
- igonvalue 12y ago
  It looks like it's only accurate to two decimal places; the circumference is approximately 3.02. http://www.wolframalpha.com/input/?i=ellipse+1%2F2%2C+cos%2822.7+degrees%29%2F2 http://www.wolframalpha.com/input/?i=ellipse+1%2F2%2C+cos%28...
deleted 12y ago
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by ipsin 12y ago

infogulch 12y ago

Thank you! I was thinking the same thing about the constant being exactly representable in base 10.

3rd3 12y ago

Rational numbers tend to have easily recognizable patterns in their fractional digits in all bases (at least as long the numbers aren't too big), so it wouldn't imply there is something special about base 10. In fact, you can always choose the base as q for a rational number p/q and get rather simple fractional digits.

ipsin 12y ago
It's not repeating digits I'm talking about, though, it's termination. If it were exactly 22.7, that's 227/10. The question would be "Why is the denominator 10?"
- akiselev 12y ago
  Because you used the denominator "10." If the constant is precisely 22.7 then it can be represented in any other base with just as much precision, you'd just use a different format for the number before and after the decimal point.
  - thaumasiotes 12y ago
    You're not paying attention to the comment you're pretending to reply to. From above: > It's not repeating digits I'm talking about, though, it's termination. 227/10 will not terminate in any numerical base which isn't a multiple of ten. Combining this with the idea that no numerical base should be privileged over another, we can rephrase this to say that it's surprising for a physical constant to be a rational (but not integral) number. It's not surprising for a physical constant to be an integer because we really believe that integers have special qualities. It's not surprising for a physical constant to be irrational because that's just the way numbers are. If it's rational but not integral, you want to justify the denominator somehow.
    - wlievens 12y ago
      For an example of an integral constant, consider the exponent in Newton's law of universal gravitation: the distance is squared (exponent two).
      - thaumasiotes 12y ago
        ...so? That one is a nice case where not only is the number clean and simple, but we have a solid theoretical basis for why it's 2 and not some other number.
    - keithnoizu 12y ago
      There are many constants out there. Sooner or later it seems natural that you'll find one that happens to dive nicely into base 10. Why does that imply anything special about base 10. It would simply be a coincidence.
      - thaumasiotes 12y ago
        It's a staggeringly unlikely coincidence; the odds of getting a rational number when selecting uniformly from any bounded interval are 0% -- not approximately zero, but exactly zero. In contrast, the odds of getting an irrational number are 100%. It's not possible to select uniformly over an unbounded interval, but rational numbers don't get any easier to find. (Technical note: though the figures of 0% and 100% are correct and exact, they actually do not mean that selecting a rational number is impossible. For a similar idea, note that if I select a number uniformly from the interval [0,1], the prior odds of choosing whatever number I did choose were also zero, but by the construction of the problem I had to end up with something.) This means that if you find a rational number, you should immediately suspect that it's like that for a good reason. You're much more likely to win the jackpot of every lottery you ever enter, even if you buy a ticket every day, than to encounter a rational number by coincidence. Or using a different analogy, the world population is roughly 7 billion. It would be extremely unlikely, so unlikely that we can essentially rule it out beforehand on theoretical grounds, for any of them to be a rational number of centimeters tall, or to mass a rational number of grams. That sort of thing can't happen.
- stephengillie 12y ago
  227/10 = x/π = y/e = z/n
- 3rd3 12y ago
  It's just that it's a non-sequitur to say "oh, that looks rather simple and it has a 10 in it, there must be something special about our number system" (which is what you say in your first comment). I implicitly subsumed termination under 'simple pattern' since repeating sequences can be described in a finite way too. The only surprising property of 22.7 in this context is that it's rational.

snorkel 12y ago

I vaguely recall that if a circular disc with a circumference of pi (3.14 ...) is held at an angle of 22.7 degrees relative to a flat surface, it projects an ellipse that has a circumference of 3.000 Coincidence? Eh, probably. Might've also been 26.8 degrees.

ColinDabritz 12y ago
One common association of those digits with PI is the handy approximation of 22/7 accurate to three digits, off by just over .0004. I stress that this is an approximation only! I had to explain about PI being irrational rather than exactly equal to the shortcut approximation to a couple co-workers, explaining some poor behaviors in their rotation function.
igonvalue 12y ago
It looks like it's only accurate to two decimal places; the circumference is approximately 3.02. http://www.wolframalpha.com/input/?i=ellipse+1%2F2%2C+cos%2822.7+degrees%29%2F2 http://www.wolframalpha.com/input/?i=ellipse+1%2F2%2C+cos%28...

deleted 12y ago

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