Numerology.Sometimes mathematics yields surprises. There are folks, addicted to "mystical mathematics" who see these as highly significant, indicating some underlying universal theme. The internet has many websites devoted to such things, such as the notion that Fibonacci numbers are the underlying basis of nature's laws. If that were so, then why can you search physics and biology textbooks and not find even one index entry for "Fibonacci", and not one basic law of nature that uses Fibonacci numbers or the Fibonacci ratio, phi?
Several years ago such a mathematical curiosity popped up on the internet. If you take the digits 0 through 9 and arrange them in alphabetical order, which is a completely artificial order, you get eight, five, four, nine, one, seven, six, three, two, zero, or 8549176320. Divide this by five and you get 1709835264, another ten-digit number in which all ten digits appear.
Your calculator might not handle ten digits. So enter the calculation 8549176320/5 into the Google search field. Some people haven't yet discovered that Google is a handy calculator, as well as a unit converter. Try entering "furlongs per fortnight in centimeters per second" into the search field and you will be surprised that Google actually understands and gives the correct answer: 1 furlong per fortnight = 0.0166309524 centimeters per second. Very useful. But I digress.
Does this tell us that our names for the digits (in English, of course) are pre-determined by a mystical ordering in the mathematics of the universe? And why divide by 5? Well 5 is the average of the nine nonzero digits. (Ask a silly question...) Warp your mind calculating the probability of this being the result of random chance.
The digit 5 is indeed special. Try dividing our number by any other single digit. None of the results have non-repeating digits. Only division by 5 results in nine digits with none repeating. What is the chance of that? Please don't say "one in ten".
But wait, there's more. Take the last result (of division by five) and divide it by five again. You get 341967052.8. All ten digits are present and none are repeated. Now what is the chance of that occurring twice? Try it again. Go ahead; you can't resist. But you are disappointed. The pattern doesn't persist.
These oddities illustrate the fact that there are totally useless facts in mathematics and in nature—surprising relations with no significance whatsoever. Unfortunately our brains are wired so that we all too often suppose that anything that seems unexpected must also be significant. And there's the related fallacy that correlations necessarily imply causality.
As for the probability of these results, one must ask "compared to what?" All sorts of nonsense arises when people try to calculate chance or probability inappropriately.
What is the probability that of all the people in the world, you are reading this right now? Well it's exactly 1, for you are reading this. What's the probability that the universe exists, exactly as it is and no other way? Exactly 1, for it's the only universe we know of. There are addicts of speculative mystical metaphysical philosophy who imagine multiple universes, but those exist only in their minds.
But what is a "significant" or "surprising" or "meaninful" fact? If our original "special" nine digit number had yielded a result with non-repeating digits in more cases than the "division by 5" case, we'd have considered that even more significant. Is it significant that "5" happens to be a Fibonacci number? Is "significance" more in the questions we ask, rather than inherent in mathematics? Ask enough questions and you surely will find some that seem significant.
Is it significant that all the classical force laws (gravity, electrical and magnetic) can be formulated as inverse square laws? And does this relate to the fact that energy radiating from an isotropic point source falls off as 1/R2 and that the area of a sphere is proportional to R2? Maybe. Why does "2" appear in all of these, and not "3"?
One can always ask the question "Why?". But most such questions have no answer, and are in fact meaningless. It is said that science doesn't answer "Why?" questions, only "How?" questions. Sometimes we should have the courage to answer "That's just the way it is." When we attempt more we run the great risk of going from physics into speculative armchair philosophy or pseudoscience.
There's an old joke in mathematics. Every number and every number relation is interesting in some way. If there were an exception to this rule, that fact would make it very interesting.
More numeric coincidences. Back in the dark ages (1955) when I was an undergraduate in physics at the University of Iowa, our only calculating tool was still the slide rule. Engineering students proudly wore theirs in leather holsters hanging from their belts, usually fancy and expensive slide rules with many mysterious graduated scales they never used. Most had no idea what those scales were for or how they worked. Physics students didn't flaunt their slide rules. Physics profs advised buying the least expensive slide rule (at about $5), as being good enough for homework and exams. Some of us found inexpensive plastic circular rules that would fit in a shirt pocket. But most professors constructed their exams so that all calculations were simple enough to be done "in your head". Two significant figures were "good enough" they said. In fact, most didn't pose numeric problems at all. Algebra and calculus were enough; the numeric calculation of a result was "too trivial to bother with". Today, students use calculators, computers and software that they don't necessarily understand to obtain answers to far greater precision than the data warrants. At least slide rules didn't require batteries and didn't generate wasted precision.
One astronomy professor startled us one day when doing a calculation requiring the number of seconds in a year. He wrote π×107. How in the world does π figure into the length of a year? It took us a while to figure out that 60 sec/min × 60 min/hr × 24 hr/day × 365 days/yr = 3.1536×107. Close enough to π×107 for "back of the envelope" calculations.
Another professor, John Eldridge, used to say that π or 2π figures into nearly everything that goes around in circles or exhibits strict time periodicity. He also introduced us to dimensional analysis of physical equations, a subject much neglected today. Most equations in freshman physics courses are rather simple, and we were expected to use those often enough that we'd remember them as if they were our own names. My memory has always been like a sieve. Prof. Eldridge illustrated dimensional analysis by showing us how to remember the formula for the period of a simple pendulum in case we forgot it. It is one of the few equations in freshman physics that uses a square root.
We look at the situation, and conclude that there aren't many things the pendulum period might depend upon: pendulum suspension length, the acceleration due to gravity, and the mass of the pendulum bob. These have dimensions of length, length×time-2 and mass, abbreviated, L, LT-2 and M. There's only one way these can be combined into an expression f(ℓ,g,m) that will have dimensions of time. That is √(ℓ/g). There's no way mass can fit in this to balance the dimensions, so we conclude that the period must not depend on the mass! Of course, there may be constants thrown in there somewhere, but we probably remember that they aren't under the radical. Also, this is periodic motion, so there must be a 2π in the equation. So we stick it in front of the radical, giving us T = 2π√(ℓ/g). Why 2π and not 1/(2π)? Well, if we did that, the period of a 1 meter long pendulum would be about 0.05 seconds, which would be absurd. Oh, the lengths a desperate student will go to on an exam when memory fails!
The only significant point of this exercise is the fact that dimensional analysis tells us that the period of the simple pendulum does not depend on its mass. Now, that's profound. It is not intuitively obvious. There are other interesting intuitive ways to reach this same conclusion, but I'll save that for another time.