Today I had need for a non-round number, so that when I resized the spacing of a grid, I wouldn't get any coincident grid points. So I chose 1.23456789, which might seem natural enough. But when I took the old spacing of 1, and divided by my factor, I got 0.810000007371. That seemed oddly round. Was it really supposed to be 0.81 exactly (and some computational round-off error?). I tried again with more accuracy, and sure enough,
there it is with 6 zeroes right near the front. And then 4 digits, and then 2 fewer (4) zeroes, and then 2 more (6) digits, and then 2 fewer (2) zeroes, and then 2 more (8) digits, and then 2 fewer (0) zeroes, and then a movie phone number (555-4571). And beyond that, a suspicious number of triply-repeated digits.
So then, of course, I tried it the other way around. What is 1/0.81? This answer also looks quite non-arbitrary:
So then I was thinking, what's so special about 0.81 that makes it have a reciprocal with such a strange pattern; so close to counting, but without the 8's? And then it hit me…
Bonus points for anybody (without a PhD) who can figure out what's so special about 0.81?
In the meantime, for your enjoyment, here are the first 1000 digits of 1/1.234567890:
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