Discussion of the Question 05/00
THE FIRST DIGITThe question was:
Back side of the front cover of the Handbook of Chemistry and Physics lists 18 important fundamental (dimensional) constants and non-standard units such as speed of light, charge of an electron, Stephan-Boltzmann constant, electron volt, etc. Values of SEVEN out of 18 constants begin with digit 1.
Why is digit 1 so prevalent in that list?
Will the situation change if instead of MKSA (SI) units we will use, say, inch, second, pound and electrostatic charge unit?
(5/00) Y. Kantor: we got several enlightening e-mails on the subject. Below we present edited extracts from these e-mails.
(3/5/2000)Dr. Fred Goesmann from Max-Planck-Institut fuer Aeronomie (Katlenburg-Lindau) (e-mail goesmann@linmpi.mpg.de) wrote:
If numbers are equidistributed on a logarithmic scale, which is not unreasonable to assume, since they vary over so many orders of magnitude, numbers starting with a 1 should account for about one third of all numbers. (see, for example log graph paper). 7 out of 18 is not far out. Hence this phenomenon should really not depend on the system of units employed.
(3/5/2000)Mike Salem from Case Western Reserve University (e-mail mps5@po.cwru.edu) wrote:
Last year, I wrote a program which multiplied each of the numbers from 1 to 1000 by each other, and each of the resulting products by each number from 1 to 1000, for a total of a billion products. Fractional multiples of arbitrary units don't make any difference since we are only concerned with the leading (or first non-zero) digit. With this simple test, I found that 31-33% (I can't remember the actual fraction) of the products had one as a leading digit, confirming what my roommate suggested. Idealy, one would find the product of all numbers, an infinite number of times (I suppose), however my experiment, which was also conducted with the numbers from 1 to 100 and with 1 to 1000 only multiplied by eachother once, suggested that the fraction converges to a value somewhere in the low 30 percentile.
(5/5/2000)Ansgar Esztermann from University of Duesseldorf (e-mail ansgar@thphy.uni-duesseldorf.de) wrote:
It all comes down to chosing the correct distribution function. If we pick some numbers at random (without any constraints, such as order of magnitude), their relative distances should be about constant, so (x1-x2)/x1=const=:1/C
Since there is one number within a "length" of x1-x2, the density of numbers is
rho(x)=1/(x1-x2)=C/x
The probability of a number starting with 1 is then obtained by integrating:
p=int12 rho(x) dx / int110 rho(x) dx
=(ln(2)-ln(1))/(ln(10)-ln(1)) =log10(2) =~ 0.3
Thus, a logarithmic distribution (as proposed by F. Goesmann) is obtained.
(4/5/2000)Andy Frohmader from Case Western Reserve University (e-mail adf3@po.cwru.edu) sugested the following argument (which was send to us by his friend Mike Salem):
The definitions of the units are entirely arbitrary... Thus, the probability that any constant is any particular value should be equal to the probability that the constant is twice that value...
Let the probability that the mantissa of a constant is on the interval [m, n) be p([m, n)). Then p([1, 2)) = p([2, 4)) = p([4, 8)) = p([8, 10) U [1, 1.6)), and so on. Adding n of these terms gives np([1, 2)) = p([1, 2)) + p([2, 4)) + ... + p([mantissa(2(n-1)), mantissa(2n)) (or the last term may have to be broken up into a union of 2 intervals as before). Note that the mantissa must be somewhere on the interval [1, 10), so p([1, 10)) = 1. The above sum covers the entire interval [1, 10) [log(2n)] times (where [x] is the greatest integer which is not greater than x; also, the logarithms are in base 10) and part of an additional time. Hence, [log(2n)] < np([1, 2)) < [log(2n)] + 1. Rearranging this gives [nlog(2)] < np([1, 2)) < [nlog(2)] + 1, or [nlog(2)]/n < p([1, 2)) < ([nlog(2)] + 1)/n. Taking the limit as n goes to infinity here gives that p([1, 2)) = log(2). But p([1, 2)) is just the probability that the mantissa starts with a 1, which is, of course, the probability that any physical constant begins with a 1. Therefore, the probability of any constant beginning with a 1 is log(2) = .301...
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