It's a staggeringly unlikely coincidence; the odds of getting a rational number when selecting uniformly from any bounded interval are 0% -- not approximately zero, but exactly zero. In contrast, the odds of getting an irrational number are 100%. It's not possible to select uniformly over an unbounded interval, but rational numbers don't get any easier to find.
(Technical note: though the figures of 0% and 100% are correct and exact, they actually do not mean that selecting a rational number is impossible. For a similar idea, note that if I select a number uniformly from the interval [0,1], the prior odds of choosing whatever number I did choose were also zero, but by the construction of the problem I had to end up with something.)
This means that if you find a rational number, you should immediately suspect that it's like that for a good reason. You're much more likely to win the jackpot of every lottery you ever enter, even if you buy a ticket every day, than to encounter a rational number by coincidence.
Or using a different analogy, the world population is roughly 7 billion. It would be extremely unlikely, so unlikely that we can essentially rule it out beforehand on theoretical grounds, for any of them to be a rational number of centimeters tall, or to mass a rational number of grams. That sort of thing can't happen.
True, but due to precision limits in measurements you're never actually dealing with this scenario. You always have some number of significant digits which implies you're always going to see a rational number for your measurements.
Further there are strict levels of granularity on a number of measurements, for example you run against the plancks constant when measuring a number of things. So this argument for looking at all of [0,N] falls apart if we aren't dealing with measurements that are continuous.
Using probability theory for the definition of interesting numbers is a daring thing to do. There is a rather surprising mathematical result (I unfortunately don't remember the context) that contained the number 27/32 (or something like that). Does that mean there is something extraordinary spacial about base 32? Difficult to say.
Well, sort of by definition a mathematical result will tell you why you got the result you got.
I haven't argued that a physical constant which terminates in base 10 implies that 10 is more special than other bases. I've argued to support the idea that a physical constant terminating in base 10 is surprising. It is, superlatively so. In the words of the OP, the question you need to answer, if you think you have a constant exactly equal to 227/10, is "why is the denominator 10?". It could be that 10 is a holy number, or it could be that the 10 represents some other physical quantity that you weren't paying attention to before. But assuming it's a coincidence is a strange thing to do, and in fact it is not natural to think that if you just looked at enough numbers of course you'd eventually see a rational one. Unless you're choosing numbers by a process that biases you heavily towards getting rationals, you are not likely to see a rational until long after you've chosen an infinite number of irrationals.
The whole genesis of string theory was, first, observing that a sequence generated by a badly-understood phenomenon was the same sequence given by a mathematical result about strings; and then, second, concluding "aha! There must be strings involved in this!" Plenty of people argue that string theory isn't going anywhere, but nobody says the conclusion "there must be some strings around here somewhere" was unjustified. The "it's a coincidence" theory persuades no one.
(Technical note: though the figures of 0% and 100% are correct and exact, they actually do not mean that selecting a rational number is impossible. For a similar idea, note that if I select a number uniformly from the interval [0,1], the prior odds of choosing whatever number I did choose were also zero, but by the construction of the problem I had to end up with something.)
This means that if you find a rational number, you should immediately suspect that it's like that for a good reason. You're much more likely to win the jackpot of every lottery you ever enter, even if you buy a ticket every day, than to encounter a rational number by coincidence.
Or using a different analogy, the world population is roughly 7 billion. It would be extremely unlikely, so unlikely that we can essentially rule it out beforehand on theoretical grounds, for any of them to be a rational number of centimeters tall, or to mass a rational number of grams. That sort of thing can't happen.