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.
>

```

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