The log scale captures an important fact that is true of many quantities in life. Take money for instance. If you have one dollar, then earning another dollar is great because you've doubled your money! If you have a million dollars, earning another dollar does not make much of a difference. Small changes matter less the more you already have.

This is true on a log scale too. On a log scale, 1 is the same distance from 2 as 100 is from 200. The higher you go up, the more the numbers all get smooshed together. What does that mean for the digits from zero to nine?

In the above picture, I show a log scale. And on that scale, I highlighted in blue all the regions where 1 is the first digit of the number. You should see that the blue regions cover more than one tenth of the log scale. In fact, they cover about 30%. And so, if we pick numbers randomly on the log scale, about 30% of those numbers will have 1 as their first digit.

Just for fun, let's apply this concept on the fundamental constants of nature. I will compare two hypotheses:

- The fundamental constants of nature are distributed on a log scale. About 30% of the constants will have 1 as their first digit, 18% will have 2 as their first digit, and so on.
- Each digit from 1 to 9 is equally likely to be the first digit of the fundamental constants of nature.

The caveat is that there is more than one way to choose our 26 fundamental constants! For instance, the mass of the u quark in planck units could be considered a fundamental constant. But instead of this constant, we could use a different constant: the ratio of the mass of the u quark to the mass of the electron. It's possible I could bias my results by "choosing" the fundamental constants that confirm my hypothesis.

Therefore, I will

*not*choose which constants to use. I will simply use the list compiled by John Baez and David Black.* Note that 6 of the constants are unknown, so we'll have to make do with the remaining 20.

*Baez offers two sets of equivalent constants for the CKM matrix. I just used the first set.

**Results**

The frequency of first digits more resembles hypothesis 1 than hypothesis 2. I also tried doing some Bayesian analysis. If our prior belief is that the likelihood of each hypothesis is 1:1, then this evidence increases the ratio to 15.8:1. In other words, if we previously thought they were equally likely, then after seeing this evidence, hypothesis 1 is almost 16 times as likely. As far as evidence goes, this is fairly weak evidence, but you're not going to get much better with only 20 fundamental constants.

Does this really mean fundamental constants are randomly distributed on a log scale? Gee, I don't know. Probably not really. What's your interpretation?

## 11 comments:

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?

Oh Eduard, great question! When I have the chance, I'll try to apply the analysis to Wikipedia's list of mathematical constants.

Okay, here are the results:

[Digit]/[frequency]/[expected frequency]

1/16/12.6

2/9/7.4

3/4/5.2

4/1/4.1

5/3/3.3

6/4/2.8

7/2/2.4

8/2/2.1

9/1/1.9

Bayesian analysis shows that this is 840,000 times more likely than the hypothesis that all first digits are equally likely.

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).

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).

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.

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).

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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.

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.

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. ;)

One last note: I would agree with Eduard in that 1 isn't the universe's only favourite

number(there's pi, e and phi (the golden ratio) as well), but remember that we are talking about digits, not numbers.Alex,

You should check out John Baez's discussion 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.

Ah! I see i've made a bit of a fool of myself.

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.

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.

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.

(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.)

Alex,

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.

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.

On second thought, Miller, you're right; it

isbecause 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.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".

What do you think?

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!

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