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MathGroup Archive 1999

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Re: Out[1]=((5-Sqrt[5])/20 + (5+Sqrt[5])/20 ) == 1/2 ???

  • To: mathgroup at smc.vnet.net
  • Subject: [mg20020] Re: [mg19976] Out[1]=((5-Sqrt[5])/20 + (5+Sqrt[5])/20 ) == 1/2 ???
  • From: "Andrzej Kozlowski" <andrzej at tuins.ac.jp>
  • Date: Sat, 25 Sep 1999 02:40:45 -0400
  • Sender: owner-wri-mathgroup at wolfram.com

Mathematica  tried to compute the numerical values of both sides without 
simplifying them. It did not attempt any simplification because you never
asked it to do so explicitly. It certainly can tell that the two sides are
equal:

In[11]:=
Simplify[( (5 - Sqrt[5])/20 + (5 + Sqrt[5])/20 ) ] == 1/2
Out[11]=
True

It seems to me that as a matter of general strategy Mathematica's behaviour
is reasonable. Since it can't really tell if your expression is complicated
or simple without trying to simplify it first, it does not embark on this
potentially time consuming task unless explicitly asked to do so.

An interesting variant of this occurs in the following situation:

In[12]:=
Simplify[( (5 - Sqrt[5])/20 + (5 + Sqrt[5])/20 ) == 1/2]
\!\($MaxExtraPrecision::"meprec" \(\(:\)\(\ \)\)
    "In increasing internal precision while attempting to evaluate \
\!\(\(\(-\(\(1\/2\)\)\)\) + \(\(1\/20\\ \(\((5 - \ at 5)\)\)\)\) + \(\(1\/20\\
\
\(\((5 + \ at 5)\)\)\)\)\), the limit $MaxExtraPrecision = \!\(50.`\) was \
reached. Increasing the value of $MaxExtraPrecision may help resolve the \
uncertainty."\)
Out[12]=
True

This looks misleadingly as if the output True contradicts the message
Mathematica had just printed. Actually this is simply the result of the fact
that Simplify does not hold its arguments, so Mathematica first tried to
evaluate the expression numerically, failed to do so and then simplified it
correctly. You can avoid getting the message by using:

In[43]:=
Simplify[Unevaluated[( (5 - Sqrt[5])/20 + (5 + Sqrt[5])/20 ) == 1/2]]
Out[43]=
True
--
Andrzej Kozlowski
Toyama International University
JAPAN
http://sigma.tuins.ac.jp
http://eri2.tuins.ac.jp


----------
>From: p_mclean at postoffice.utas.edu.au (Patrick McLean)
To: mathgroup at smc.vnet.net
>To: mathgroup at smc.vnet.net
>Subject: [mg20020] [mg19976] Out[1]=((5-Sqrt[5])/20 + (5+Sqrt[5])/20 ) == 1/2 ???
>Date: Fri, Sep 24, 1999, 12:26 PM
>

> Can any one explain this behaviour???
>
> hilbert% math
> Mathematica 3.0 for Solaris
> Copyright 1988-96 Wolfram Research, Inc.
>  -- Terminal graphics initialized --
>
> In[1]:=   ( (5-Sqrt[5])/20 + (5+Sqrt[5])/20 ) == 1/2
>
> $MaxExtraPrecision::meprec:
>    $MaxExtraPrecision = 50. reached while evaluating
>       1    5 - Sqrt[5]   5 + Sqrt[5]
>     -(-) + ----------- + -----------. Increasing the value of
>       2        20            20
>      $MaxExtraPrecision may help resolve the uncertainty.
>
>         5 - Sqrt[5]   5 + Sqrt[5]    1
> Out[1]= ----------- + ----------- == -
>             20            20         2
>
> In[2]:=
>
> --
> Patrick McLean
>
> No news is good news...
> 


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