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MathGroup Archive 2004

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Re: Log[4]==2*Log[2]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg50710] Re: Log[4]==2*Log[2]
  • From: "Richard J. Fateman" <fateman at eecs.berkeley.edu>
  • Date: Fri, 17 Sep 2004 01:17:38 -0400 (EDT)
  • Organization: UC Berkeley
  • References: <200409090917.FAA19334@smc.vnet.net> <200409100805.EAA04777@smc.vnet.net> <chulug$jn1$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Sorry to join the fray so late, and even dare to put
the subject in the Subject line.

There is no reason to suppose that it is impossible
to determine equality by numerical testing.
  One could potentially compute, over a class of
expressions, a bound p that depends on the actual expressions A, B
such that evaluating A within absolute error p, and B within
absolute error p, will show that they are different, if in fact
they are different. And otherwise they are the same.

If the class of expressions is "integers, + * -"  then we know
that evaluation to 0.1 will do it.
There have been papers written on this subject, and even programs.
I am not being coy, just avoiding this newsgroup's censor.

Richard Fateman


DrBob wrote:

>>>Mathematica does not apply any simplification rules but justtries to evaluate the expression numerically and, not
>>>surprisingly, it can't determine if the LHS is zero or not
>>>up to the required precision.
> 
> 
> On the contrary, I think the error message itself clearly indicates the difference IS zero to "the required precision". If 50 digits extra precision isn't enough to determine that the difference ISN'T zero, why doesn't Equal return True?
> 
> Bobby
> 
> On Fri, 10 Sep 2004 04:05:56 -0400 (EDT), Andrzej Kozlowski <andrzej at akikoz.net> wrote:
> 
> 
>>On 9 Sep 2004, at 18:17, Andreas Stahel wrote:
>>
>>
>>>To whom it may concern
>>>
>>>the following answer of Mathematica 5.0 puzzeled me
>>>
>>>Log[4]==2*Log[2]
>>>leads to
>>>
>>>N::meprec: Internal precision limit $MaxExtraPrecision = 50.` reached
>>>while \
>>>evaluating -2\Log[2]+Log[4]
>>>
>>>with the inputs given as answer. But the input
>>>
>>>Log[4.0]==2*Log[2]
>>>
>>>leads to a sound "True"
>>>
>>>Simplify[Log[4]-2*Log[2]]
>>>leads to the correct 0, but
>>>Simplify[Log[4]-2*Log[2]==0]
>>>yields no result
>>>
..........etc etc <snip>


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