Embedded powers of two (was Re: have a pint)

[Mouse <mouse%Rodents-Montreal.ORG@localhost>]

> TLDR: the "(US) survey foot" is 1200/3937 SI meters, and the
> "(international) foot" is 0.3048 SI meters, a ratio of about
> 1.000002000004000:1 or 1:0.999998.

Fascinating.

(1200/3937)/.3048 -> 1.000002000004000008000016000032000064000128000256000512001024002048004096008192016384032768065536131072262144524289048578097156...

in which I see a curious string of overlapping powers of two.

(1200/3937)/(3048/10000)
 = (1200 * 10000) / (3937 * 3048)
 = (2^8 * 3 * 5^6) / (2^3 * 3 * 31 * 127^2)
 = (2^5 * 5^6) / (31 * 127^2)
 = 500000 / 499999

which, upon doing long division, makes it obvious what's going on.
(And makes it clear that the reciprocal, .999998, is exact.)

And there are modifications such as 50000000/49999999; that one
produces similar strings of powers of two with more zeros between them.
And there are related fractions, such as 100000000/99999997 (which
generates powers of three).  Obvious once pointed out, but I'd never
seen them before.

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