HP50g simplifing a root
10-12-2020, 03:17 AM (This post was last modified: 10-13-2020 12:08 PM by Albert Chan.)
Post: #25
 Albert Chan Senior Member Posts: 2,699 Joined: Jul 2018
RE: HP50g simplifing a root
(10-11-2020 06:28 PM)Albert Chan Wrote:  Example, simplify ³√(36 + 20i√7)

>>> from mpmath import *
>>> A, B, k = 36, 20, -7
...
>>> c = cbrt(A*A-B*B*k); print c
16.0

Here is a novel way to solve for r instead, then get a.
(Note: b can be halves, just like a)

R/r = (B/b)² = (3c + 4r)² = integer, where r = b²k

For this case, we hit the jackpot. If b=±1, 3c+4r = 3c+4k = B

r = k = -7
a = A / (c+4r) = 36/ (16-28) = -3

3*arg(a±√r) = 3*atan2(±√7, -3) ≈ ±2.3098 pi

We need to match above angle to arg(A+√R), which is on the first quadrant.
+2.3098 pi - 2 pi = 0.3098 pi matches.

³√(36 + 20i√7) = (-3 + i√7) / (-1)^(2/3)

This is just for fun. No need to solve the cubics.
Directly calculate ³√(A+√R) (previous post) is preferred.

Comment: Solving for b instead, we can take square root of both side

B/b = 3c + 4r = 3c + 4kb² = integer

This removed the ambiguity of b = ±1 to b = B/(3c + 4kb²) = 1
Above example need to match angles anyway, so I skipped this.
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 Messages In This Thread HP50g simplifing a root - peacecalc - 09-29-2020, 09:22 PM RE: HP50g simplifing a root - Albert Chan - 09-29-2020, 11:47 PM RE: HP50g simplifing a root - Albert Chan - 09-30-2020, 02:22 AM RE: HP50g simplifing a root - Albert Chan - 09-30-2020, 10:50 PM RE: HP50g simplifing a root - Albert Chan - 10-01-2020, 07:31 AM RE: HP50g simplifing a root - peacecalc - 09-30-2020, 05:33 AM RE: HP50g simplifing a root - peacecalc - 10-01-2020, 02:20 PM RE: HP50g simplifing a root - Albert Chan - 10-01-2020, 05:22 PM RE: HP50g simplifing a root - peacecalc - 10-04-2020, 06:05 PM RE: HP50g simplifing a root - Albert Chan - 10-04-2020, 11:48 PM RE: HP50g simplifing a root - peacecalc - 10-04-2020, 07:36 PM RE: HP50g simplifing a root - peacecalc - 10-05-2020, 11:36 AM RE: HP50g simplifing a root - Albert Chan - 10-05-2020, 05:01 PM RE: HP50g simplifing a root - peacecalc - 10-06-2020, 05:25 AM RE: HP50g simplifing a root - Albert Chan - 10-06-2020, 09:40 AM RE: HP50g simplifing a root - Albert Chan - 10-06-2020, 12:06 PM RE: HP50g simplifing a root - Albert Chan - 10-06-2020, 04:13 PM RE: HP50g simplifing a root - Albert Chan - 10-07-2020, 06:12 PM RE: HP50g simplifing a root - Albert Chan - 10-09-2020, 12:20 AM RE: HP50g simplifing a root - Albert Chan - 10-09-2020, 02:31 PM RE: HP50g simplifing a root - Albert Chan - 10-11-2020, 06:28 PM RE: HP50g simplifing a root - Albert Chan - 10-12-2020 03:17 AM RE: HP50g simplifing a root - Albert Chan - 10-24-2020, 02:19 PM RE: HP50g simplifing a root - Albert Chan - 10-12-2020, 10:54 PM RE: HP50g simplifing a root - CMarangon - 10-12-2020, 11:45 PM RE: HP50g simplifing a root - grsbanks - 10-13-2020, 06:46 AM RE: HP50g simplifing a root - Albert Chan - 10-09-2020, 05:21 PM RE: HP50g simplifing a root - Albert Chan - 10-10-2020, 03:58 PM RE: HP50g simplifing a root - Albert Chan - 10-10-2020, 04:49 PM RE: HP50g simplifing a root - peacecalc - 10-12-2020, 08:49 PM RE: HP50g simplifing a root - peacecalc - 10-13-2020, 06:30 AM RE: HP50g simplifing a root - peacecalc - 10-13-2020, 06:36 AM

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