HP Prime & HP 49G Problem with square roots
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06-26-2014, 10:25 AM
(This post was last modified: 06-26-2014 11:33 AM by Gerald H.)
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HP Prime & HP 49G Problem with square roots
Some (irrelevant to this thread) calculations produced these two expressions:
√1201+√(70+2*√1201) & √(35+2*√6)+√(1236-2*√6+2*√(42035-2402*√6)) which should be equal. I have checked with the Longfloat Lib on the HP 49G, & the two are equal to many decimal positions, but are they in fact exactly equal? Assistance appreciated. (Title edited to include HP Prime, 13:33, 26/6) |
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06-26-2014, 10:45 AM
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RE: HP 49G Problem with square roots | |||
06-26-2014, 11:00 AM
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RE: HP 49G Problem with square roots
(06-26-2014 10:45 AM)Thomas Klemm Wrote:(06-26-2014 10:25 AM)Gerald H Wrote: are they in fact exactly equal? Thank you. I believe the result is correct & dislike relying on the authority of some cloud-computing; Why should I trust Wolframalpha if I have no means of checking the result? My version of Maple is antiquated & can't deal with the question, nor can the HP 49G - & even if they did return an intelligible answer, I'd still want to know how. |
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06-26-2014, 11:01 AM
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RE: HP 49G Problem with square roots
(06-26-2014 10:25 AM)Gerald H Wrote: <clipped>these two expressions: 50G says no Thanks ~~~~8< Art >8~~~~ PS: Please post more 50G stuff :) |
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06-26-2014, 11:03 AM
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RE: HP 49G Problem with square roots
(06-26-2014 11:01 AM)CosmicTruth Wrote:(06-26-2014 10:25 AM)Gerald H Wrote: <clipped>these two expressions: Thank you. So now we have two authorities dsagreeing(see post #2)? |
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06-26-2014, 11:36 AM
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RE: HP Prime & HP 49G Problem with square roots
I have now tried the expressions on HP Prime CAS: for == a zero is returned & for - a value of -2.27373675443E-13.
The numerical value for - is certainly wrong. |
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06-26-2014, 04:08 PM
Post: #7
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RE: HP Prime & HP 49G Problem with square roots
My 2 cents:
I ran it in the newRPL demo at 2007 digits precision, and the difference between both expressions came out 1e-2005, so I'd say they are equal at least up to the first 2000 digits. Claudio |
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06-26-2014, 04:25 PM
Post: #8
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RE: HP Prime & HP 49G Problem with square roots
To get an algebraic proof, I don't have the time but I think the key is:
Code:
Claudio |
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06-26-2014, 04:29 PM
Post: #9
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RE: HP Prime & HP 49G Problem with square roots
(06-26-2014 04:08 PM)Claudio L. Wrote: My 2 cents: Thank you for the confirmation - I hadn't tested to such precision. The means by which the two expressions arose implies, I believe, equality & I'm not bright enough to demonstrate this equality. More precision will (hopefully) corroborate equality, but a convincing reasoning would settle the matter. |
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06-26-2014, 05:58 PM
(This post was last modified: 06-26-2014 07:06 PM by Manolo Sobrino.)
Post: #10
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RE: HP Prime & HP 49G Problem with square roots
OK, first let's notice that 1201 is prime, 42035=35*1201 and 1236=35+1201. Now rewrite the longer expression:
\begin{equation} \sqrt{35+2\sqrt{6}}+\sqrt{1201+35-2\sqrt{6}+2\sqrt{\left(35-2\sqrt{6}\right)1201}} \end{equation} That is the square of a sum:\begin{equation} \sqrt{35+2\sqrt{6}}+\sqrt{\left(\sqrt{1201}+\sqrt{35-2\sqrt{6}}\right)^2} \end{equation} You don't need to worry about the absolute value, it's simply:\begin{equation}\sqrt{1201}+\sqrt{35+2\sqrt{6}}+\sqrt{35-2\sqrt{6}} \end{equation} If a>b it's trivial to prove that:\begin{equation}\sqrt{a+b}+\sqrt{a-b}=\sqrt{2a+2\sqrt{a^2-b^2}}\end{equation} In this case: \begin{equation}\sqrt{70+2\sqrt{1225-4\cdot 6}}=\sqrt{70+2\sqrt{1201}}\end{equation} There you go. (You guys should use paper and pencil more often ) |
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06-26-2014, 06:49 PM
Post: #11
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RE: HP Prime & HP 49G Problem with square roots
This reminds me of Dedekind's Theorem that \(\sqrt{2} \sqrt{3} = \sqrt{6}\). A very readable account is in the article "Dedekind's Theorem: ..." by Fowler in The American Mathematical Monthly Vol. 99, No. 8, Oct., 1992, p.725.
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06-26-2014, 06:51 PM
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RE: HP Prime & HP 49G Problem with square roots
(06-26-2014 05:58 PM)Manolo Sobrino Wrote: OK, first let's notice that 1201 is prime, 42035=35*1201 and 1236=35+1201. Now rewrite the longer expression: The last calculation line is a typo? |
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06-26-2014, 07:15 PM
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RE: HP Prime & HP 49G Problem with square roots | |||
06-26-2014, 07:28 PM
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RE: HP Prime & HP 49G Problem with square roots
I take my hat off to you,Manolo Sobrino. Bravo!
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06-26-2014, 07:37 PM
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RE: HP Prime & HP 49G Problem with square roots
Thank you Gerald!
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06-28-2014, 11:24 PM
Post: #16
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RE: HP Prime & HP 49G Problem with square roots
HP50G calculator for sale or trade for good pencil and paper pad.
| v hehe jk Thanks ~~~~8< Art >8~~~~ PS: Please post more 50G stuff :) |
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