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Distribution of named and important mathematical constants

  1. Sep 6, 2014 #1
    I've noticed that the vast majority of named or important mathematical constants, are what you might call small numbers. Their modulus lies very often in the range [0,5]. Here's two examples of tables:



    If you were to plot a histogram of all those numbers, the distribution would cluster very tightly in the above range. The major exception seems to be numbers which are astronomically large - but it's hard to say how "important" some of those are, since they may only appear in a handful of esoteric proofs. The numbers which are ubiquitous throughout mathematics seems to be of quite ordinary magnitude.

    Is there something to this? Anyone know any discussion about this? Anyone know of important mathematical (i.e. independent of units) constants which are in the range of say 100-1000?
  2. jcsd
  3. Sep 6, 2014 #2
    Although there may be other, more "fundamental" reasons, I think that one explanation is that a lot of these constants come from a general set of numbers, of which we have selected the first few, or the values of certain functions at small integers. For example, ln(2) and sqrt(2) are interesting simply because they are among the first that come up in the sequences of ln(n) or sqrt(n).

    Also, many constants are defined by simple combinations of other constants (your second link has a lot of those, like i^i and e^π). So there isn't much room for the values to get very large, using only addition, multiplication and exponentiation of small constants.

    Finally, a lot of these constants are somehow related to the distribution of primes or other "statistical" properties of numbers. These might by definition be small, since primes are rare, and thus expressions such as the sum of the reciprocals of primes can't really be large.

    Of course what I said is merely a cynical guess, but maybe there is indeed a deeper reason for some of these constants.
  4. Sep 6, 2014 #3


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    I'm not sure how you'd find an explanation for this, but it is an interesting observation.
  5. Sep 6, 2014 #4
    I'm not really talking about things like ln(2) or e^pi/2 - moreso numbers like [itex] e, \pi, \gamma, \varphi, \delta [/itex] etc... if you look at the section of the second link called "Classical, named math constants".
  6. Sep 6, 2014 #5
    Here's a picture:

  7. Sep 7, 2014 #6
    One explanation might be that many of these were named in ancient times. In those times the computation of numeric values was time consuming and so the ones that were evaluated tended to be of low value.
  8. Sep 7, 2014 #7
    Here's a philosophical take on it: Our current scheme of mathematics is a little arbitrary, and there's definitely many nearly-equivalent formulations of mathematics (Which I think Godel's theorem implies). Like how the base we normally used is defined by the number of our fingers, the fundamental constants we use as the foundation of our mathematical system are also low and easily understood.

    For example, the arguably most famous constant, pi, is arbitrary; We could write out new sets of formulas for a new value of pi' = 1000*pi and everything would still work fine. (As is in fact the case with tau-lovers, who use tau=2pi instead if I remember correctly.

    bhillyard also provides a salient point: There very well was a bias in research towards low numbers until we had the computational tools to handle larger numbers.

    I think the fundamental physical constants; plank length, the elementary charge, the speed of light, etc are also relevant to the discussion. The constants in actual physical reality are way off the scale of normal human experience, in direct contrast to mathland. I don't think that's a coincidence.
    Last edited: Sep 7, 2014
  9. Sep 7, 2014 #8
    What I said here is actually false, since the sum of the reciprocals of primes diverges. But the sum of the reciprocals of twin primes does not (converges to Brun's constant).

    I admit these seem harder to explain. I agree with the idea that humans are biased towards small constants, though.
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