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Overriding the definition of Plus so that it does exact arithmetic: Bug or Feature?

• To: mathgroup at smc.vnet.net
• Subject: [mg120227] Overriding the definition of Plus so that it does exact arithmetic: Bug or Feature?
• From: Richard Fateman <fateman at cs.berkeley.edu>
• Date: Thu, 14 Jul 2011 05:21:29 -0400 (EDT)

Consider the definition of the new function P which is supposed to be 
exactly like Plus, except that converts all "Real" floats to exact
rational numbers before adding them up.


P[a_Real,b__]:= Plus[Rationalize[SetAccuracy[a, Infinity]], b];
SetAttributes[P, Attributes[Plus]]

P[c__]:= Plus[c]

This works as expected,  e.g.

P[0.1, -1/10]  returns 1/180143985094819840

which is the difference between the closest double-float to 0.1,
and the exact number 1/10.

Given this, let's put EXACTLY the same definition on Plus

Unprotect[Plus];
Plus[a_Real,b__]:= Plus[Rationalize[SetAccuracy[a, Infinity]], b];
Protect[Plus];

0.1-1/10  gives 0.   (wrong)
0.1+x gives 3602879701896397/36028797018963968 + x  (hm, right).


My guess is that any Rules inserted on Plus by the user are checked 
AFTER the built-in simplifications attached to Plus. I consider this a 
bug. Any workaround come to mind, if you consider this a feature :)


RJF