HP Prime & HP 49G Problem with square roots

06262014, 10:25 AM
(This post was last modified: 06262014 11:33 AM by Gerald H.)
Post: #1




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)+√(12362*√6+2*√(420352402*√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) 

06262014, 10:45 AM
Post: #2




RE: HP 49G Problem with square roots  
06262014, 11:00 AM
Post: #3




RE: HP 49G Problem with square roots
(06262014 10:45 AM)Thomas Klemm Wrote:(06262014 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 cloudcomputing; 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. 

06262014, 11:01 AM
Post: #4




RE: HP 49G Problem with square roots
(06262014 10:25 AM)Gerald H Wrote: <clipped>these two expressions: 50G says no Thanks ~~~~8< Art >8~~~~ PS: Please post more 50G stuff :) 

06262014, 11:03 AM
Post: #5




RE: HP 49G Problem with square roots
(06262014 11:01 AM)CosmicTruth Wrote:(06262014 10:25 AM)Gerald H Wrote: <clipped>these two expressions: Thank you. So now we have two authorities dsagreeing(see post #2)? 

06262014, 11:36 AM
Post: #6




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 2.27373675443E13.
The numerical value for  is certainly wrong. 

06262014, 04:08 PM
Post: #7




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 1e2005, so I'd say they are equal at least up to the first 2000 digits. Claudio 

06262014, 04:25 PM
Post: #8




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 

06262014, 04:29 PM
Post: #9




RE: HP Prime & HP 49G Problem with square roots
(06262014 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. 

06262014, 05:58 PM
(This post was last modified: 06262014 07:06 PM by Manolo Sobrino.)
Post: #10




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+352\sqrt{6}+2\sqrt{\left(352\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{352\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{352\sqrt{6}} \end{equation} If a>b it's trivial to prove that:\begin{equation}\sqrt{a+b}+\sqrt{ab}=\sqrt{2a+2\sqrt{a^2b^2}}\end{equation} In this case: \begin{equation}\sqrt{70+2\sqrt{12254\cdot 6}}=\sqrt{70+2\sqrt{1201}}\end{equation} There you go. (You guys should use paper and pencil more often ) 

06262014, 06:49 PM
Post: #11




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.


06262014, 06:51 PM
Post: #12




RE: HP Prime & HP 49G Problem with square roots
(06262014 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? 

06262014, 07:15 PM
Post: #13




RE: HP Prime & HP 49G Problem with square roots  
06262014, 07:28 PM
Post: #14




RE: HP Prime & HP 49G Problem with square roots
I take my hat off to you,Manolo Sobrino. Bravo!


06262014, 07:37 PM
Post: #15




RE: HP Prime & HP 49G Problem with square roots
Thank you Gerald!


06282014, 11:24 PM
Post: #16




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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