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Complex Head

  • To: mathgroup at smc.vnet.net
  • Subject: [mg17614] Complex Head
  • From: Jack Goldberg <jackgold at math.lsa.umich.edu>
  • Date: Tue, 18 May 1999 02:45:17 -0400
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Group:

See mg[7605], mg[7548] and mg[7592].  The issue here is 
why does the following occur:

Head[Sqrt[2]*I] -> Times

while 

Head[2*I] -> Complex

At the risk of being shot down and stripped of my medals 
from Mathematica, I offer this explanation.

The answer seems to depend on how far Mathematica goes in determining the 
properties of  x  in an expression like  x*I.  For example,  Mathematica 
will simplify automatically  x = Exp[Pi*I]  to -1  and thus 

Head[x*I] -> Complex

But the kernel will not take the time to decide that Sqrt[2] is real.
Moreover, since  x  could be quite complicated, say some value of the
Bessel function, it is probably impossible to decide on the correct value 
of "Head" for every Mathematica-ly sound numerical choice of x.
Indeed, it may be that  x  has a value for which no known algorithm 
determines whether it is real or complex!  Add to this the requirement 
that Head[...] must always return something, we have an answer to 
Mittermayr's question.   

Jack


  



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