tag:blogger.com,1999:blog-9124539381685751273.post879825857051851155..comments2023-06-19T04:35:06.263-07:00Comments on Skeptic's Play: One: the universe's favorite digitmillerhttp://www.blogger.com/profile/05990852054891771988noreply@blogger.comBlogger11125tag:blogger.com,1999:blog-9124539381685751273.post-19596405004580739282011-05-04T11:18:11.544-07:002011-05-04T11:18:11.544-07:00I was trying to think of a constant earlier that w...I was trying to think of a constant earlier that was just a rational number, but couldn't think of one. "Number of quarks per hadron"--I like that!millerhttps://www.blogger.com/profile/05990852054891771988noreply@blogger.comtag:blogger.com,1999:blog-9124539381685751273.post-55831357350882879962011-05-03T23:08:56.017-07:002011-05-03T23:08:56.017-07:00On second thought, Miller, you're right; it is...On second thought, Miller, you're right; it <i>is</i> because there are more irrationals than rationals. The values of (non-unitless) physical constants depend on our definitions of the base units (the metre, the second, etc), as i myself mentioned earlier, so actual physical constants are never multiples of our units; add to that the fact that pi and e, which appear everywhere, are irrationals, and irrational constants are bound to appear much more often than rational ones. I should have seen this the moment i thought about how many irrational constants there are.<br /><br />There's also the first thing you said; 3 appears all over the place, but we don't call it the "number-of-quarks-per-hadron constant" because it's already called "3".<br /><br />What do you think?Rainhttps://www.blogger.com/profile/02853997873870014947noreply@blogger.comtag:blogger.com,1999:blog-9124539381685751273.post-27469293435080794742011-05-03T17:40:07.571-07:002011-05-03T17:40:07.571-07:00Alex,
I agree that the 30% figure has to do with ...Alex,<br /><br />I agree that the 30% figure has to do with the base 10 system we're using. Under any other base, the frequencies would be different. And there's nothing fundamental about base 10.<br /><br />I think if a physical constant appeared to be rational, we wouldn't bother giving it a name, because it already has one! Or maybe it's because there are just so many more irrational numbers than rational numbers. Or maybe they really are rational, and we just haven't determined them to sufficient precision.millerhttps://www.blogger.com/profile/05990852054891771988noreply@blogger.comtag:blogger.com,1999:blog-9124539381685751273.post-66504099155669432792011-05-03T17:10:03.386-07:002011-05-03T17:10:03.386-07:00Ah! I see i've made a bit of a fool of myself....Ah! I see i've made a bit of a fool of myself.<br /><br />Even though i now see what you mean (and those constants, if i'm reading the table correctly, don't depend on measuring system because they're ratios), i maintain that the fact that 30% of them begin with 1 is only because we use a decimal counting system.<br /><br />I'll illustrate this with the number pi. In a decimal system, pi equals roughly 3.14159265, so the first digit of pi is 3. However, in a binary counting system pi would be a little over 11 and in a tertiary system pi would be a little over 10; in both of these cases the first digit of pi is 1. This is probably not the best example, since writing nonintegers in nondecimal systems is complicated (at least i don't know how to do it), but i hope you get my point.<br /><br />Still, it is interesting to note that, at least in a decimal system, 1 takes up just about as much space as you predicted with your log_10 theory.<br /><br />(Have you noticed that most important numbers in physics (other than 0 and 1) are irrational? I doubt this has anything to do with the cardinality of irrationals as opposed to that of rationals, so it's an interesting phenomenon. It's unrelated to this discussion on first digits, but i noticed it when looking at the fundamental constants of the standard model and thinking about the numbers Eduard mentioned.)Rainhttps://www.blogger.com/profile/02853997873870014947noreply@blogger.comtag:blogger.com,1999:blog-9124539381685751273.post-27191510047566813302011-05-03T16:35:49.006-07:002011-05-03T16:35:49.006-07:00Alex,
You should check out John Baez's discuss...Alex,<br />You should check out <a href="http://math.ucr.edu/home/baez/constants.html" rel="nofollow">John Baez's discussion</a> of what constitutes a fundamental constant of nature. I actually didn't use any constants like c, h-bar, or G. I only used unitless constants.millerhttps://www.blogger.com/profile/05990852054891771988noreply@blogger.comtag:blogger.com,1999:blog-9124539381685751273.post-70786450041164948792011-05-03T15:10:14.109-07:002011-05-03T15:10:14.109-07:00(continued from previous comment)
On top of that,...(continued from previous comment)<br /><br />On top of that, there are many mathematical properties involving the numbers 0, 1, pi, e and phi (or multiples of them). Off the top of my head, (da/db)(db/dc)(dc/da)=-1 for any functions a,b,c (da/db indicates the partial derivative of a with respect to b while c remains constant) and e^(i phi)=-1.<br /><br />I'm sure i could think of some other reasons why 1 is special, but i'm terribly busy this week and i only stopped to read today's article.<br /><br />In short, i'd say the reasons 1 is one of the universe's favourite digits are more mathematical and conceptual than physical. And, so my opinion doesn't seem biased, i'm a physics student, not a maths one. ;)<br /><br />One last note: I would agree with Eduard in that 1 isn't the universe's only favourite <i>number</i> (there's pi, e and phi (the golden ratio) as well), but remember that we are talking about digits, not numbers.Rainhttps://www.blogger.com/profile/02853997873870014947noreply@blogger.comtag:blogger.com,1999:blog-9124539381685751273.post-25377667413996533602011-05-03T15:06:33.889-07:002011-05-03T15:06:33.889-07:00This comment has been removed by the author.Rainhttps://www.blogger.com/profile/02853997873870014947noreply@blogger.comtag:blogger.com,1999:blog-9124539381685751273.post-68986157629712178732011-05-03T15:06:22.835-07:002011-05-03T15:06:22.835-07:00I agree that 1 is the universe's favourite dig...I agree that 1 is the universe's favourite digit, but it hardly has anything to do with the log scale, in my opinion (although 0 is, i believe, the universe's other favourite digit, and log_b(1)=0 for every complex number b... which is rather cool). In fact, there is nothing special about the particular log operator you're using (log_10); we (humans) only use it because having ten fingers and ten toes makes counting in a decimal system easier for us than in any other system (e.g. hexadecimal, tertiary, binary...); if you plotted log_n with n any integer other than 10, the percentage of numbers beginning with 1 would not be 30% (it would still be greater than the percentage of numbers beginning with any other digit, though).<br /><br />I believe 1 and 0 are the universe's favourite digits because of their beautiful mathematical properties, not because of the values of what we call the fundamental constants in a particularly comfortable (for us) measuring system (mks). In fact, i would argue that the so-called "natural system of units", where speed and not length is a fundamental quantity and c (the speed of light), h or hbar (Planck's constant), epsilon0 (vacuum permittivity), mu0 (vacuum permeability) and G (Newton's constant) are all valued 1, is the universe's preferred measuring system; it makes much more sense to call the speed of light 1 and calculate all other speeds from there than to call Earth's circumference 40,000 and calculate all other lengths from there. Same goes for Planck's constant and for every other "fundamental constant" there is (and earlier, when i said that only we call them such, i was referring to the fact that other intelligent species in the universe probably have a different set of "fundamental constants"; it all depends on which fundamental quantities (e.g. length, time, speed, area, power, pressure, force, illuminance, electric charge, etc) we are comfortable with using (mks has, among others, length, time and mass, while the "natural system" has, among others, speed, time and mass, for instance).<br /><br />But on to my point. Leaving 0 aside for another discussion, 1 has many great properties. No matter which measuring system or fundamental quantities you're using, 1 is always the unit, the base number from which all others are constructed, the first integer, the first intuitive or "counting" number. We define everything in terms of 1; we define 2 as 1+1, 3 as 2+1, and so on; we define the multiplicative inverse of a number A as that number whose product with A equals 1; we are able to tell which of two quantities is larger by dividing them and comparing the result with 1 (or subtracting them and comparing with 0, but i said i'd leave 0 aside for now); and so on.<br /><br />Although someone might argue that that's just the way languages evolved on this planet, it seems natural to make a distinction between 1 and all other numbers: 1 is singular, while all integers from 2 onwards are plural (in some (or all?) slavic languages, there is also a distinction between the numbers 2-5 and the numbers 6-9, but 1 is still special on its own).<br /><br />(continued on next comment)Rainhttps://www.blogger.com/profile/02853997873870014947noreply@blogger.comtag:blogger.com,1999:blog-9124539381685751273.post-11653193812546005582011-05-03T12:27:12.094-07:002011-05-03T12:27:12.094-07:00Okay, here are the results:
[Digit]/[frequency]/[e...Okay, here are the results:<br />[Digit]/[frequency]/[expected frequency]<br />1/16/12.6<br />2/9/7.4<br />3/4/5.2<br />4/1/4.1<br />5/3/3.3<br />6/4/2.8<br />7/2/2.4<br />8/2/2.1<br />9/1/1.9<br /><br />Bayesian analysis shows that this is 840,000 times more likely than the hypothesis that all first digits are equally likely.millerhttps://www.blogger.com/profile/05990852054891771988noreply@blogger.comtag:blogger.com,1999:blog-9124539381685751273.post-35964967614489810892011-05-03T11:35:02.766-07:002011-05-03T11:35:02.766-07:00Oh Eduard, great question! When I have the chance...Oh Eduard, great question! When I have the chance, I'll try to apply the analysis to <a href="http://en.wikipedia.org/wiki/Mathematical_constants#Table_of_selected_mathematical_constants" rel="nofollow">Wikipedia's list</a> of mathematical constants.millerhttps://www.blogger.com/profile/05990852054891771988noreply@blogger.comtag:blogger.com,1999:blog-9124539381685751273.post-51876107918598031202011-05-03T01:02:56.374-07:002011-05-03T01:02:56.374-07:00What about mathematical constants like Pi, e, Gamm...What about mathematical constants like Pi, e, Gamma, plastic number, golden section, ... volume to surface ratios of polyhedra, ratios of of cutting length CL to side length in minimal CL problems?Eduardhttp://baumannatmcnet.chnoreply@blogger.com