[Wilson is using 'v' to mean square-root. I typically use q or r. --rcs] This sort of thing happens in some (all?) extension fields based on pure radicals. In your example, imagine running through the sign changes on +-v8 and +-v5. If you flip v5, your equation changes into 1 / (v8 - v5 - 5) = 1/4 * (-5v10 - 7v5 + 11v2 + 15) = small The small term is small because the denominator v8-v5-5 is no longer close to 0. Similar things happen when you flip v8 to -v8, and when you flip both of them. Now add up signed combinations of these results: A straight Add cancels all terms except the +15, which isn't affected by the sign flips: The sum is 1/4 * (15+15+15+15) = 15 = 15.505053+ + small + small + small. True! Next, instead of Adding, use Subtract with the equations where v5 is flipped: Various things cancel out, except the v5 flipped terms: 1/4 * (7v5 + 7v5 + 7v5 + 7v5) = 7v5 = 15.505053 - small + small - small so 7v5 = roughly 15.5. The other terms 11v2 and 5v10 behave similarly, with other combinations of sign flips. I think this generalizes to fields with more square roots. For cube roots, such as cbrt2, instead of sign flips +1 -> -1, substitute +1 -> w and w2, with w & w2 being the two complex cube roots of 1. Rich --------- Quoting David Wilson <davidwwilson@comcast.net>:
By happenstance, I noticed that v8 + v5 = 5.064495+ is just over 5, so I computed
1 / (v8 + v5 - 5) = 1/4 * (5v10 + 7v5 + 11v2 + 15) = 15.505053+
I noticed that the terms
5v10 = v250 = 15.811388+ 7v5 = v245 = 15.652475+ 11v2 = v242 = 15.556349+ 15 = v225 = 15.000000
are all in the same ball park.
Is there some reason this should be the case? Can other similar examples be formulated?
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