Without spoiling it, here is another (mythical) set of "metric conversion units" with the same decimal magic properties. Readers can look for the common thread: Instead of 2.54 and 39.37, our "units" are 13.82 and 723.589. Their reciprocals are: 1/13.82 = 0.0723589 001447178 002894356 005788712 ... 1/723.589 = 0.001382 000002764 000005528 000011056 ... Digit-groups are 9 digits, instead of David's 6-digit groups. For the "geometric mean compromise", instead of "39.37/254=0.155" we calculate 13.82/723.589 = 0.01909924 003819848 007639696 015279392 ... Then we take the square root of this, and of its reciprocal. These are the "compromise metric conversion factors": sqrt(13.82/723.589) = 0.1382 000001382 000002073 000003455 000006046 2500... sqrt(723.589/13.82) = 7.235890 007235890 010853835 018089725 ... The digit groups show the same relative ratios that we saw in David's example: 1, 1, 3/2, 5/2, 35/8, ... For example, 7.235890 * 5/2 is 18.089725. On Sun, Feb 5, 2012 at 01:54, Michael Reid <reid@gauss.math.ucf.edu> wrote:
I think there's more than this, if I get what David was hinting at.
Rot13, so as not to give it away:
Qnivq'f ahzore snpgbef nf n cebqhpg bs gjb fznyyre ahzoref, juvpu fgvyy funer n pbzzba snpgbe. Zberbire, nyy bs vgf cevzr snpgbef ner Zrefraar cevzrf!
Truly remarkable, David!
David Wilson wrote:
But I promised fun with sequence embedded decimals, so... Prior to 1975, a centimeter would have been an exact 0.3937 inches. This means that an inch would have been 1/0.3937 = 2.54 000508 001016 002032 004064 ... centimeters. Observe the nice doubling pattern of the 6-digit blocks. After 1975, the inch was precisely 2.54 centimeters. Now the centimeter is 1/2.54 = 0.3937 007874 015748 031496 062992 ... inches. Almost amazingly, we find the same 6-digit block doubling pattern here. Prior to the conversion, a centimeter was 0.3937 inches, post-conversion it was 1/2.54 centimeters. Perhaps, instead of letting either the avoirdupois or metric system dictate the conversion factor, we should adopt a compromise system whose conversion factors which are the geometric mean of the conversion factors of the two competing systems. In this new system, a centimeter would be sqrt(0.3937 * (1/2.54)) = sqrt(0.155) = 0.39370 039370 059055 098425 ... inches, and again we see curious relationships between blocks of 6 digits. Similarly, in our new system, the inch would be 1/sqrt(0.155) = 2.5400 025400 038100 063500 111125 ... centimeters. All this mathematical magic actually stems from an interesting factorization of a number near 500000. Perhaps you can find it.
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