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Re: Re: Guessing "exact" values

• To: mathgroup at smc.vnet.net
• Subject: [mg76256] Re: [mg76171] Re: [mg76079] Guessing "exact" values
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Fri, 18 May 2007 06:23:50 -0400 (EDT)
• References: <200705160920.FAA29604@smc.vnet.net> <200705170959.FAA01041@smc.vnet.net>

Mathematica can "recognize" algebraic numbers, by using the function 
RootApproximant (or the function Recognize in the Legacy 
NumberTherory package):

ToRadicals[RootApproximant[N[Sqrt[2] + Sqrt[3], 30]]]
Sqrt[5 + 2*Sqrt[6]]

This may not look so impressive  until you check:

FullSimplify[Sqrt[2] + Sqrt[3] - Sqrt[5 + 2*Sqrt[6]]]
0

Anyway, doing this for algebraic numbers has a solid mathematical 
basis: the so called LLL Lattice Reduction) algorithm. I don=92t know, =

however, of any mathematical basis for "recognizing" transcendentals, =

except by means of  stored values or some other "ad hoc" approach 
(essentially "sophisticated guessing")

Andrzej Kozlowski


On 17 May 2007, at 18:59, Murray Eisenberg wrote:

> As I remember, at IMS'06 (Avignon) Stephen Wolfram (via remote link-=

> up)
> played with a function to do just that.  Did I remember correctly?
>
> If so, I cannot recall the name of the function, so I don't know 
> whether
> the function made it into 6.0.  The closest thing I can find is
> RootApproximant, but that doesn't seem to "recognize" an expression
> involving a transcendental number.
>
> Szabolcs wrote:
>> "Another computer algebra system" has a function, identify(), which
>> attempts to guess the exact expression that evaluates to a particular
>> numerical value.
>>
>> Example:
>>
>> In[1]:= N[3Pi+3/2,10]
>>
>> Out[1]= 10.92477796
>>
>>> identify(10.92477796);
>>                                    3
>>                                    - + 3 Pi
>>                                    2
>>
>> Is there a package with similar functionality for Mathematica?
>>
>> Szabolcs
>>
>
> --
> Murray Eisenberg                     murray at math.umass.edu
> Mathematics & Statistics Dept.
> Lederle Graduate Research Tower      phone 413 549-1020 (H)
> University of Massachusetts                413 545-2859 (W)
> 710 North Pleasant Street            fax   413 545-1801
> Amherst, MA 01003-9305
>

• Follow-Ups:
  • Re: Re: Re: Guessing "exact" values
    • From: Daniel Lichtblau <danl@wolfram.com>

• References:
  • Guessing "exact" values
    • From: Szabolcs <szhorvat@gmail.com>
  • Re: Guessing "exact" values
    • From: Murray Eisenberg <murray@math.umass.edu>