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As Brad Morris has pointed out, this sort of knife-edge rounding issue is inconsequential in celestial navigation and indeed in most situations. There are many different rules for rounding because none of them are right. There's no "right" rounding. Rather there are optional rounding systems. And of course there is a huge Wikipedia article on those options. They exist, and some people have to worry about them, becaus they are standards that are applied for auditing purposes --so that we can go back later and determine the exact algorithmic origin of every digit. But that's only necessary when auditing is essential.

Among the rules for rounding of the knife edge case, one that I was advocating is simply "take your pick" or "toss a coin" (see random tie-breaking in the Wikipedia article). This has the disadvantage of lacking a repeatable audit trail, but otherwise it's every bit as good as rules like "banker's rounding". But, if we want to get picky (just for laughs now), random rounding has another subtler disadvantage which is that we do not make choices at random if there is any psychological input because we don't believe runs (like ten coins tossed "heads" in a row) are properly random. So we avoid certain patterns which paradoxically imprints patterns on the result! The next problem then is how to generate proper random numbers. There are lots of excellent pseudo-random number generators, but once again, when auditing enters the game, you have to generate perfectly random numbers. And the best options here are quantum decays and things like noise from Brownian motion that derive from quantum phenomena. Or, in my joke example, decaying neutrons. Decaying particles are believed to be a pure source of randomness. Neutrons are impractical, but the concept is quite valid.

I said I was sending a pound of neutrons, and that they would arrive in 15 hours. Here's why: a gram of anything contains about 6e+23 nucleons (total number of neutrons and protons, ignoring small binding energy differences and counting electrons as insignificant. Half a pound, or roughly 500g, then amounts to about 3e+26 neutrons. That's a lot of random decays! But you have to watch closely! Free neutrons decay with a half-life of about 10.2 minutes. So after 102 minutes or ten half-lives I will have reduced my original sample by a factor of 1024 (I've also done some major damage with radiation, but hey, that's not my department). Roughly every 100 minutes, the amount of remaining neutrons falls by a factor of 1000. Those 15 hours are equal to 900 minutes, so that means we divide the remaining sample by 1000 nine times. And at the end of 15 hours? You open the box, and there's only one neutron left. Or maybe zero neutrons, depending on your luck. All the coins have already been flipped. :)

Again, just to emphasize, there is no right way to do rounding. ...unless you need an audit trail. In celestial navigation, if you get to a point where something ends in a 5 and you need to round it up/down to the nearest more significant digit, indulge yourself! Pick a direction at random. Live a little. And relax in the assurance that it doesn't matter. :)

Frank Reed