Parents quickly discover when they first have children that the year after the second birthday is often unpleasant. This is so marked that child rearing books, and presumably, child psychologists, refer to this as the terrible twos.
For physicists, its a bit different. The terrible twos refer to the number two. And they’re terrible because we keep getting the algebra wrong by a factor of two.
OK, occasionally its another integer, but usually its two. This appears to be related to Benford’s Law, [Link] which is the frequency distribution of occurrence of the digits 1-9. Most treatments of the law ignore zero, for some reason.
Since its a frequency distribution (integer or discrete probability density functionish – I’ll explain what a “functionish” is some other time,) its sum (integral) has to be one if it completely spans the state space or is properly normalized. We would suppose the digits 0-9 to be distributed more or less uniformly in their occurrence in numbers.
If we go to a maths handbook and examine (e.g.,) a logarithm table (we don’t care if it is natural or common,) we find that the distribution is not uniform. In fact, if we ignore the zero, which is a very classical thing to do in a Roman or Greek sense, and only look at the digits 1-9, then we find that the digit 1 is the most common, followed in succession of frequency sequentially.
Now something akin to this occurs in physics. Obviously, we don’t care if we make a mistake in a “1”, since its a one, after all. But we do care if we’re off by a factor of “2” or “3”.
But this distribution isn’t Benford’s because the “2” mistake is really the most common, much more than predicted by Benford’s Law.
If you know what the proper distribution is, LMK. If I get time to worry about it enough to figure it out, I’ll LYK.