Is there a reason why most of the important mathematical constants are close to zero on the number line? : askscience

iCookBaconShirtless

The problem with addressing this question is that there is no natural and intrinsic way to decide whether a non-zero number is "close to zero" or "far from zero." The constants you listed are closer to zero than 10¹⁰ but further from zero than 10^-10 . Who is to say where we should place to cutoff between numbers that are "close to" or "far from" from zero?

They may strike you as being close to zero, but this is a question of psychology not mathematics. Perhaps we deal with numbers that are further from zero than pi more often than numbers that are closer to zero than pi, so pi seems close to zero?

I'll add that the precise values of those constants might not be as fundamental as you think. The ratio of the surface area to the diameter of a sphere in n dimensions, for example, is only pi when n=2. However, it can be written in terms of pi for other dimensions. In fact, many people argue that tau=2*pi should be thought of as more fundamental than pi. I personally believe that any rational multiple of pi is equally "fundamental", whatever that means. We just settled on one of them out of convenience and for historical reasons.

As for your second question, the reason that small primes are denser than large primes is fairly well understood. Seethe prime number theorem[1] .

Edit: Think about it this way. OP is intrigued that so many constants are smaller than 5 in magnitude. But any finite list of numbers has an upper bound. The list given by OP happens to have 5 as an upper bound. So what is so surprising about 5 versus some other upper bound? It's tempting to answer that a number smaller than 5 is unlikely to occur if you uniformly draw a random positive number. But there is actually no mathematical way to make sense of drawing a real number uniformly. The lack of scale is the real culprit. You cannot claim that a number is intrinsically "large" or "small" in any meaningful way without choosing something to compare that number to.

EDIT 2: To anyone who still thinks that the list of constants given by OP (or any other finite list of numbers) is intrinsically small in magnitude, please provide a list of number that you think is not small in magnitude.

I am spending a lot of time responding to commenters that might be glossing over some subtleties of my argument. If you wish to debate my comment, please first answer the question I posed in EDIT 2. I think the doomed effort to answer this question will reveal some of the subtleties of the point I am making.

EDIT 3: I am in no way trying to discredit OP's question. I agree with OP that the fact that so many named constants are less than 5 is surprising in a psychological sense. But I contend that there is no way to answer the question of whether it is surprising mathematically. The reasons for this are actually a bit more subtle than they first appear. The problem is that every finite list of numbers has an upper bound. So how surprising is it that there are upper bounds on OP's list that are less than 5? Answering this would require defining some sense of a probability distribution on the positive real numbers. But every probability distribution on the positive real numbers artificially imposes a scale because there is no uniform distribution on the positive reals. So you can only really ask whether a list of numbers is surprisingly close to zero with respect to some arbitrarily chosen scale.