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Re: Why does Mathematica change the precision of an expression to check equality?

• To: mathgroup at smc.vnet.net
• Subject: [mg69442] Re: Why does Mathematica change the precision of an expression to check equality?
• From: dh <dh at metrohm.ch>
• Date: Wed, 13 Sep 2006 04:00:51 -0400 (EDT)
• References: <ee0spn$b3f$1@smc.vnet.net>

Hi Jean-Marc,
the Help says:
"For exact numeric quantities, Equal internally uses numerical 
approximations to establish inequality. This process can be affected by 
the setting of the global variable $MaxExtraPrecision. "
Obviously the numerical approximation is only done on the expression on 
the LHS, not on the number 1. If 1 is changed into 1. only machine 
precision is used and the test succeed.

Daniel

Jean-Marc Gulliet wrote:
> These thoughts come after answering a similar question in a forum 
> dedicated to anther CAS. Having ran the following code, I am a little 
> perplexed by the behavior of Mathematica.
> 
> y = (Sqrt[5] - 2)*(Sqrt[5] + 2);
> y == 1
> 
> --> N::"meprec" : "Internal precision limit $MaxExtraPrecision = 
> (49.99999999999999) reached while evaluating -1 + (-2 + Sqrt[5])*(2 + 
> Sqrt[5]). More...
> 
> --> (-2 + Sqrt[5])*(2 + Sqrt[5]) == 1
> 
> At least Mathematica returns a warning message in addition to the 
> unevaluated expression
> 
> I used to thought that Mathematica was not attempting to do any 
> algebraic simplifications when testing, say, equality, and that one have 
> to request explicitly such transformations.
> 
> However, it is pretty clear that Mathematica transforms the expression 
> in some way, in this case changing infinite precision -- that is exact 
> numbers -- into arbitrary precision -- that is better precision that 
> hardware but still not exact.
> 
> So the question is, "Why, when an expression is only written with exact 
> numbers, Mathematica would "downgrade" the precision to a lower and 
> inexact one before attempting to answer a boolean question?"
> 
> I do not see the rational behind this design choice...
> 
> Best regards,
> Jean-Marc
> 
> P.S. I know that one can get the correct answer by using Simplify.
>