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Forum index » Science and Technology » Math » Recreational
Order relation between two radicals....(Ramanujan )
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Virgil
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Joined: 24 Mar 2005
Posts: 5536

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

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

PostPosted: Thu Jun 23, 2005 5:53 am    Post subject: Re: Order relation between two radicals....(Ramanujan ) Reply with 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} .
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Virgil
science forum Guru


Joined: 24 Mar 2005
Posts: 5536

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

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

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

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:

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 Wink
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.
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Henry1
science forum addict


Joined: 15 May 2005
Posts: 58

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

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 Wink

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

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

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

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

"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 Wink
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

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

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

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

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

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