I guess I see. My intuition was that if you changed signs in terms on the left, you would end up with entirely different terms on the right, but as you tacitly imply below, term sign changes on the left result only in term sign changes on the right. In this case, there are 4 linearly independent sign changes on the left, and 4 (I assume linearly independent) term changes on the right. One of the sign changes on the left results in a small right-hand value, the others in large right hand values. Solving the right side for term magnitudes leads to terms of about the same size. I assume this would work for any similar example (where the left side has one large-valued sign change). I also presume the fact that there are four sign changes on the left and 4 terms on the right is not coincidence.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of rcs@xmission.com Sent: Thursday, July 06, 2017 7:10 PM To: math-fun@mailman.xmission.com Cc: rcs@xmission.com Subject: Re: [math-fun] Algebraic number question
[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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