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I would like to apply ListPlot to a set of complex numbers. This is what comes to mind to convert a complex number to a pair: Replace[5 + 2 I, {Complex -> List}, Heads -> True] except it doesn't work. Neither do the other solutions to similar questions I've seen on this site.

I can get around it by defining f[x_]:={Re[x],Im[x]} but I'm curious why the replacement does not work.

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Complex is an atomic type.

AtomQ@Complex[1, 2]
(* True *)

In other words, it is a smallest unit that should be considered indivisible.

For atomic types, you cannot assume any structure. Somewhat confusingly, some of them behave as if they did have a structure in some situations. But this is by no means general.

Notice that this works:

Replace[Complex[1, 2], Complex[a_, b_] :> {a, b}]
(* {1, 2} *)

Even this works:

Replace[Complex[1, 2], head_[a_, b_] -> {head, a, b}]
(* {Complex, 1, 2} *)

So, of course, it is natural to expect this to work too:

Replace[Complex[1, 2], Complex -> List, {1}, Heads -> True]

(Notice that you would need the {1}. This was a mistake in your code, but it's really beside the main point here.)

However, this returns Complex[1,2] back.

This is what I meant when I said some atomic expressions behave as if they had a structure sometimes, but not always. For atoms, all bets are off when trying to decompose them. I suggest you stick to Re and Im, or more conveniently, ReIm.

If you try to decompose other atomic types, you will see different behaviour. Each one has its own quirks.

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  • $\begingroup$ nice answer: +1. was just about to post the solution. Your explanation is very enlightening ! $\endgroup$ – Ali Hashmi May 6 '17 at 11:29
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In:

cs = Table[5 + 2 I, 3];

(*Method 1*)
cs // Map[ReIm]

(*Method 2*)
reim[c_] := ReIm[c]
SetAttributes[reim, Listable];
reim @ cs 

(*Method 3*)
cs // FullForm // ToString // 
  StringReplace[#, "Complex" -> "List"] & // ToExpression

Out:

{{5, 2}, {5, 2}, {5, 2}}
{{5, 2}, {5, 2}, {5, 2}}
{{5, 2}, {5, 2}, {5, 2}}
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  • $\begingroup$ Don't even have to set ReIm to Listable In[232]:= ReIm // Attributes Out[232]= {Listable, Protected} $\endgroup$ – oleflar Jul 10 at 21:30

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