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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 )

"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}
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} .
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 )

"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.
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" 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: TI-66: A = 0.312050634263 B = 0.312050636755 => A < B TI-89: 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 TI-89'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.
Henry1

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.
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
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:

TI-66:
A = 0.312050634263
B = 0.312050636755 => A < B
TI-89:
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)
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?
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?
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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