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Virgil science forum Guru
Joined: 24 Mar 2005
Posts: 5536

Posted: Fri Jun 24, 2005 3:04 am Post subject:
Re: Order relation between two radicals....(Ramanujan )



In article <1119513201.511307.300500@z14g2000cwz.googlegroups.com>,
"Alex. Lupas" <alexandru.lupas@ulbsibiu.ro> wrote:
Quote:  Other interesting/magic/ pairs
(A(i),B(i)), to be compared, are [S.Ramanujan] :
A(1):= sqrt[3]{sqrt[3]{2}1}
B(1):= sqrt[3]{1/9}  sqrt[3]{2/9}+sqrt[3]{4/9}
A(2):= 3*sqrt{sqrt[3]{5}sqrt[3]{4}}
B(2):= sqrt[3]{2} + sqrt[3]{20} sqrt[3]{25}
A(i)=B(i) ? , i in {1,2} .

If you are going to keep this up. using curt(x), or (x)^(1/3) for the
cube root of x is a lot more intelligible than sqrt[3]{x} 

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Alex. Lupas science forum Guru Wannabe
Joined: 06 May 2005
Posts: 245

Posted: Thu Jun 23, 2005 5:53 am Post subject:
Re: Order relation between two radicals....(Ramanujan )



Other interesting/magic/ pairs
(A(i),B(i)), to be compared, are [S.Ramanujan] :
A(1):= sqrt[3]{sqrt[3]{2}1}
B(1):= sqrt[3]{1/9}  sqrt[3]{2/9}+sqrt[3]{4/9}
A(2):= 3*sqrt{sqrt[3]{5}sqrt[3]{4}}
B(2):= sqrt[3]{2} + sqrt[3]{20} sqrt[3]{25}
A(i)=B(i) ? , i in {1,2} . 

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Virgil science forum Guru
Joined: 24 Mar 2005
Posts: 5536

Posted: Wed Jun 22, 2005 2:27 am Post subject:
Re: Order relation between two radicals....(Ramanujan )



In article <1119343520.392485.194310@f14g2000cwb.googlegroups.com>,
"Alex. Lupas" <alexandru.lupas@ulbsibiu.ro> wrote:
Quote:  Which is the order relations between $ A, B $ where
A=sqrt[6]{7\sqrt[3]{20}19} and B= sqrt[3]{5/3} sqrt[3]{2/3} .
I have denoted sqrt[p]{X}:= X^{1/p} , p>= 2 .

They are equal.
Let 'curt(x) ' denote the cube root of x, then
A^6 = 7*curt(20)  19, trivially, and
B^6 = (curt(5/3)  curt(2/3))^6 = 7*curt(20)  7, though it is a good
deal less trivial to do the simplification. 

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Christopher Night science forum beginner
Joined: 30 May 2005
Posts: 29

Posted: Wed Jun 22, 2005 1:32 am Post subject:
Re: Order relation between two radicals....(Ramanujan )



Andreas wrote:
Quote:  "William Elliot" <marsh@hevanet.remove.com> skribis:
On Tue, 21 Jun 2005, Alex. Lupas wrote:
Which is the order relations between $ A, B $ where
A=sqrt[6]{7\sqrt[3]{20}19} and B= sqrt[3]{5/3} sqrt[3]{2/3} .
I have denoted sqrt[p]{X}:= X^{1/p} , p>= 2 .
First I suspected what the problem is:
Grab a calculator, I thought. But then I used different calculators myself:
TI66:
A = 0.312050634263
B = 0.312050636755 => A < B
TI89:
A = 0.31205063679722
B = 0.3120506367604 => A > B
Being a physicists I would claim them equal, though
Andreas

Which, of course, they are. Interesting that the TI89's CAS gets it
wrong unless you give it a hint:
(7*20^(1/3)19)^(1/6)=(5/3)^(1/3)(2/3)^(1/3)
returns false.
7*20^(1/3)19=((5/3)^(1/3)(2/3)^(1/3))^6
returns true. 

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Henry1 science forum addict
Joined: 15 May 2005
Posts: 58

Posted: Tue Jun 21, 2005 8:17 pm Post subject:
Re: Order relation between two radicals....(Ramanujan )



On Tue, 21 Jun 2005 15:37:57 +0200, "Andreas"
<andreas_fritsche@gmx.net> wrote:
Quote:  Being a physicist I would claim them equal, though

You could take the sixth power of each and find the answer quite
easily. 

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Arturo Magidin science forum Guru
Joined: 25 Mar 2005
Posts: 1838

Posted: Tue Jun 21, 2005 4:26 pm Post subject:
Re: Order relation between two radicals....(Ramanujan )



In article <Pine.BSI.4.58.0506210249130.2642@vista.hevanet.com>,
William Elliot <marsh@hevanet.remove.com> wrote:
Quote:  On Tue, 21 Jun 2005, Alex. Lupas wrote:
Which is the order relations between $ A, B $ where
A=sqrt[6]{7\sqrt[3]{20}19} and B= sqrt[3]{5/3} sqrt[3]{2/3} .
I have denoted sqrt[p]{X}:= X^{1/p} , p>= 2 .
Yicks, how cryptic. B = sqr 3?

No; A is the sixth root of 7(20)^{1/3}19, and B is the difference
between the cubic roots of 5/3 and of 2/3.

======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
 Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu 

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Andreas science forum beginner
Joined: 09 Jun 2005
Posts: 3

Posted: Tue Jun 21, 2005 11:37 am Post subject:
Re: Order relation between two radicals....(Ramanujan )



"William Elliot" <marsh@hevanet.remove.com> skribis:
Quote:  On Tue, 21 Jun 2005, Alex. Lupas wrote:
Which is the order relations between $ A, B $ where
A=sqrt[6]{7\sqrt[3]{20}19} and B= sqrt[3]{5/3} sqrt[3]{2/3} .
A = Howis\beingused?
B = (5/3)^(1/3)  (2/3)^(1/3)
I have denoted sqrt[p]{X}:= X^{1/p} , p>= 2 .
Bah. Even unrelated to square root.
Can you come up with a better presentation?
The presentation gives absolutely correct LaTeX if you set the "\" in front 
of each sqrt.
First I suspected what the problem is:
Grab a calculator, I thought. But then I used different calculators myself:
TI66:
A = 0.312050634263
B = 0.312050636755 => A < B
TI89:
A = 0.31205063679722
B = 0.3120506367604 => A > B
Being a physicists I would claim them equal, though
Andreas

( 2+2=5 for small values of 5 and big values of 2) 

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William Elliot science forum Guru
Joined: 24 Mar 2005
Posts: 1906

Posted: Tue Jun 21, 2005 9:39 am Post subject:
Re: Order relation between two radicals....(Ramanujan )



On Tue, 21 Jun 2005, Alex. Lupas wrote:
Quote:  Which is the order relations between $ A, B $ where
A=sqrt[6]{7\sqrt[3]{20}19} and B= sqrt[3]{5/3} sqrt[3]{2/3} .

A = Howis\beingused?
B = (5/3)^(1/3)  (2/3)^(1/3)
Quote:  I have denoted sqrt[p]{X}:= X^{1/p} , p>= 2 .
Bah. Even unrelated to square root. 
Can you come up with a better presentation? 

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William Elliot science forum Guru
Joined: 24 Mar 2005
Posts: 1906

Posted: Tue Jun 21, 2005 7:51 am Post subject:
Re: Order relation between two radicals....(Ramanujan )



On Tue, 21 Jun 2005, Alex. Lupas wrote:
Quote:  Which is the order relations between $ A, B $ where
A=sqrt[6]{7\sqrt[3]{20}19} and B= sqrt[3]{5/3} sqrt[3]{2/3} .
I have denoted sqrt[p]{X}:= X^{1/p} , p>= 2 .
Yicks, how cryptic. B = sqr 3? 


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Alex. Lupas science forum Guru Wannabe
Joined: 06 May 2005
Posts: 245

Posted: Tue Jun 21, 2005 6:45 am Post subject:
Order relation between two radicals....(Ramanujan )



Which is the order relations between $ A, B $ where
A=sqrt[6]{7\sqrt[3]{20}19} and B= sqrt[3]{5/3} sqrt[3]{2/3} .
I have denoted sqrt[p]{X}:= X^{1/p} , p>= 2 . 

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