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Question re I->-I

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106508] Question re I->-I
  • From: AES <siegman at stanford.edu>
  • Date: Fri, 15 Jan 2010 03:16:48 -0500 (EST)
  • Organization: Stanford University

The extended discussion of I->-I (call this the "iRule") seems to arise 
because 

*  One can write two simple expressions, call them expr1 and expr2, 
containing numbers, symbols (in the Mathematica sense), and the symbol 
I, that anyone with an elementary knowledge of algebra would (and 
should) consider to be exactly equivalent (multiple examples have been 
given).

*  But if one in fact types each of these expressions into a virgin 
Mathematica notebook and kernel and applies the iRule, e.g. in the form 
of three cells containing

         iRule = {I -> -I};

         expr1 /. iRule

         expr2 /. iRule

one gets two totally different answers from the two expressions.  This 
bothers people (me, anyway).

Question (asked for learning purposes):  Does this happen, at this level 
of simplicity, with _any_ other valid Mathematica symbols?

More specifically:  

*  Take any of the pairs of expressions that "fail" in the above manner; 
type them into cells, except replace (by hand!) all the free-standing 
instances of I with any arbitrarily chosen valid Mathematica symbol, 
call it "s" (s can  be x, or xx, or Pi, or E).

*  Do the same as above on these modified expressions except apply the 
"sRule"  {s -> -s}.

Is there any such symbol s that will produce different results for the 
two expressions?

Alternative question:

*  Type in a rational polynomial in a simple variable "s" (format it any 
way you like), and apply this sRule.  Is there any way you can reformat 
this rational polynomial, without changing its meaning, so as to get 
different results when the sRule is applied?


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