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From: Philippe 92 on 5 Jan 2010 06:38
mike3 a �crit :
> On Jan 4, 3:09�pm, Aiya-Oba <aaiya...(a)rcc.mass.edu> wrote:
>>
>> ( 44721359550/31622776604)^2 � = � �2
>>
>> = (1.414213562)^2 �= �2
>> � � � � � � � � � � Q E D.
>
> Note that 1.414213562 is NOT equal to sqrt(2).
>
> 1.414213562^2 = 1.999999998944727844 exactly.
>
> 1.999999998944727844 != 2.
>

Hi,

It doesn't matter as
44721359550/31622776604 = 1.414213562269644...
is not equal to 1.414213562

However ( 44721359550/31622776604)^2 = 1.99999999970739665691... != 2

;-)

I mentionned the Babylonian method for extracting the square root.
After verification, this method doesn't give the numbers I gave
(63018038201 / 44560482149 and 26102926097 / 18457556052)
but is even more accurate, as the Babylonian method applied to sqrt(2)
successively gives :
1 / 1 = 1
3 / 2 = 1.5
17 / 12 = 1.416666666666666....
577 / 408 = 1.41421568627450980392156862...
665857 / 470832 = 1.41421356237468991062629557...
886731088897 / 627013566048 = 1.41421356237309504880168962...
(then integer overflow in Javascript)
sqrt(2) = 1.41421356237309504880168872...

The method for obtaining the numbers I gave is the continued fraction
expansion, which gives all the best rationnal approximations of sqrt(2)
Of course the Babylonian method gives numbers which are among the CF
expansion, it just converges much faster (is equivallent to Newton).
The CF expansion is useful when searching a best approximate P/Q with
P,Q in a given range (here in the range "P, Q are 11 digits")

And again the numbers given by Aiya-Oba are much more *in*accurate
that "best approximations"...
44721359550/31622776604 = 1.4142135622696...
the 665857 / 470832 is 100 times more accurate !!!

refs :
<http://en.wikipedia.org/wiki/Methods_of_computing_square_roots>
<http://school.maths.uwa.edu.au/~schultz/3M3/L1Babylonianroot2.html>

Regards.

--
Philippe C., mail : chephip, with domain free.fr
site : http://mathafou.free.fr/ (mathematical recreations)


From: Transfer Principle on 6 Jan 2010 02:10
On Jan 5, 3:38 am, "Philippe 92" <nos...(a)free.invalid> wrote:
> And again the numbers given by Aiya-Oba are much more *in*accurate
> that "best approximations"...
> 44721359550/31622776604 = 1.4142135622696...
> the 665857 / 470832 is 100 times more accurate !!!

Let us point out that 44721359550 and 31622776604 are
approximately the square roots of two sextillion and
one sextillion, respectively. Thus, it becomes obvious
that the OP simply looked at the digits of sqrt(20)
and sqrt(10) to form his fraction. This also explains
why the OP would choose a fraction that obviously is
not even in lowest terms, with the numerator and
denominator both even.

Notice that, to twelve decimal places,

sqrt(20) = 4.472135955000

Indeed, my pocket calculator, which displays nine
decimal places and carries two additional places
internally, claims that:

4.472135955^2 = 20 exactly.

In reality,

4.472135955^2 = 20 + 3.762025(10^-12)

Still, the OP wanted to take advantage of the fact
that sqrt(2(10^19)) is nearly an integer in order to
form his own rational (approximation to) sqrt(2). But
the OP needs a value for sqrt(10), and sqrt(10^21),
unlike sqrt(2(10^21)), is not a near-integer. Indeed,
to ten decimal places:

sqrt(10) = 3.1622776602

so that the OP would have been better off choosing
44721359550/31622776602 instead of 44721359550/31622776604:

44721359550/31622776604 = 1.414213562269644...
44721359550/31622776602 = 1.414213562359087...
sqrt(2) = 1.414213562373095...

bert:
> Idiot. Even the Windows Calculator
> program can show that for those values,
> 2b^2 - a^2 = 292603343132

Replacing b with b-2, the "correct" value, gives:

2b^2 - a^2 = 39621130308

nearly an order of magnitude better.

Of course, there are some "cranks" that even I won't
defend, and the Aiya-Oba is one of them. He has made
several claims, including his current claim that
sqrt(2)eQ, that even an ultrafinitist won't defend. And
since I will not defend any theory more nonstandard
than ultrafinitism, I truly believe that Aiya-Oba does
deserve to be called -- and let me finally remove the
scare quotes here -- a crank.
From: Aiya-Oba on 7 Jan 2010 19:53
On Jan 6, 2:10 am, Transfer Principle <lwal...(a)lausd.net> wrote:
> On Jan 5, 3:38 am, "Philippe 92" <nos...(a)free.invalid> wrote:
>
> > And again the numbers given by Aiya-Oba are much more *in*accurate
> > that "best approximations"...
> > 44721359550/31622776604 = 1.4142135622696...
> > the 665857 / 470832 is 100 times more accurate !!!
>
> Let us point out that 44721359550 and 31622776604 are
> approximately thesquareroots oftwosextillion and
> one sextillion, respectively. Thus, it becomes obvious
> that the OP simply looked at the digits of sqrt(20)
> and sqrt(10) to form his fraction. This also explains
> why the OP would choose a fraction that obviously is
> not even in lowest terms, with the numerator and
> denominator both even.
>
> Notice that, to twelve decimal places,
>
> sqrt(20) = 4.472135955000
>
> Indeed, my pocket calculator, which displays nine
> decimal places and carriestwoadditional places
> internally, claims that:
>
> 4.472135955^2 = 20 exactly.
>
> In reality,
>
> 4.472135955^2 = 20 + 3.762025(10^-12)
>
> Still, the OP wanted to take advantage of the fact
> that sqrt(2(10^19)) is nearly an integer in order to
> form his ownrational(approximation to) sqrt(2). But
> the OP needs a value for sqrt(10), and sqrt(10^21),
> unlike sqrt(2(10^21)), is not a near-integer. Indeed,
> to ten decimal places:
>
> sqrt(10) = 3.1622776602
>
> so that the OP would have been better off choosing
> 44721359550/31622776602 instead of 44721359550/31622776604:
>
> 44721359550/31622776604 = 1.414213562269644...
> 44721359550/31622776602 = 1.414213562359087...
> sqrt(2)                 = 1.414213562373095...
>
> bert:
>
> > Idiot.  Even the Windows Calculator
> > program can show that for those values,
> > 2b^2 - a^2 = 292603343132
>
> Replacing b with b-2, the "correct" value, gives:
>
> 2b^2 - a^2 = 39621130308
>
> nearly an order of magnitude better.
>
> Of course, there are some "cranks" that even I won't
> defend, and the Aiya-Oba is one of them. He has made
> several claims, including his current claim that
> sqrt(2)eQ, that even an ultrafinitist won't defend. And
> since I will not defend any theory more nonstandard
> than ultrafinitism, I truly believe that Aiya-Oba does
> deserve to be called -- and let me finally remove the
> scare quotes here -- a crank.



Thanks everyone, for your interest in this
mathematical revelation, according to which:
Square of the ratio of
44721359550 to 31622776604, is precisely 2.
-Aiya-Oba
From: Aiya-Oba on 16 Jan 2010 17:55
On Jan 7, 7:53 pm, Aiya-Oba <aaiya...(a)rcc.mass.edu> wrote:
> On Jan 6, 2:10 am, Transfer Principle <lwal...(a)lausd.net> wrote:
>
> > On Jan 5, 3:38 am, "Philippe 92" <nos...(a)free.invalid> wrote:
>
> > > And again the numbers given by Aiya-Oba are much more *in*accurate
> > > that "best approximations"...
> > > 44721359550/31622776604 = 1.4142135622696...
> > > the 665857 / 470832 is 100 times more accurate !!!
>
> > Let us point out that 44721359550 and 31622776604 are
> > approximately thesquareroots oftwosextillion and
> > one sextillion, respectively. Thus, it becomes obvious
> > that the OP simply looked at the digits of sqrt(20)
> > and sqrt(10) to form his fraction. This also explains
> > why the OP would choose a fraction that obviously is
> > not even in lowest terms, with the numerator and
> > denominator both even.
>
> > Notice that, to twelve decimal places,
>
> > sqrt(20) = 4.472135955000
>
> > Indeed, my pocket calculator, which displays nine
> > decimal places and carriestwoadditional places
> > internally, claims that:
>
> > 4.472135955^2 = 20 exactly.
>
> > In reality,
>
> > 4.472135955^2 = 20 + 3.762025(10^-12)
>
> > Still, the OP wanted to take advantage of the fact
> > that sqrt(2(10^19)) is nearly an integer in order to
> > form his ownrational(approximation to) sqrt(2). But
> > the OP needs a value for sqrt(10), and sqrt(10^21),
> > unlike sqrt(2(10^21)), is not a near-integer. Indeed,
> > to ten decimal places:
>
> > sqrt(10) = 3.1622776602
>
> > so that the OP would have been better off choosing
> > 44721359550/31622776602 instead of 44721359550/31622776604:
>
> > 44721359550/31622776604 = 1.414213562269644...
> > 44721359550/31622776602 = 1.414213562359087...
> > sqrt(2)                 = 1.414213562373095...
>
> > bert:
>
> > > Idiot.  Even the Windows Calculator
> > > program can show that for those values,
> > > 2b^2 - a^2 = 292603343132
>
> > Replacing b with b-2, the "correct" value, gives:
>
> > 2b^2 - a^2 = 39621130308
>
> > nearly an order of magnitude better.
>
> > Of course, there are some "cranks" that even I won't
> > defend, and the Aiya-Oba is one of them. He has made
> > several claims, including his current claim that
> > sqrt(2)eQ, that even an ultrafinitist won't defend. And
> > since I will not defend any theory more nonstandard
> > than ultrafinitism, I truly believe that Aiya-Oba does
> > deserve to be called -- and let me finally remove the
> > scare quotes here -- a crank.
>
> Thanks everyone, for your interest in this
> mathematical revelation, according to which:
>   Square of the ratio of
>   44721359550 to 31622776604, is precisely 2.
>                             -Aiya-Oba




Please Observe That:
where (a/b)^2 is (44721359550/31622776604)^2,
(a/b)^2 = 2, has the implication of this elusive equation:

(a/b)^2 +\- 0 = 2 (abracadabra).

Such that,

(1.99999999970739665691...) +\- 0 = 2.

Thanks. -Aiya-Oba

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