I'm new in Mathematica software, and I need help with a simple exercise.

Given the list

list = {x, 3.2, y, 1, 2.679, 3, z, 4, -7.9};

I need to make some rules to generate a new list with only three elements.

  • first element is the product of its symbols
  • second element the product of its integers
  • third element the product of its real numbers


newlist = {x y z, 12, -67.72512}  

I hope you can help me.


Using rules for this task may be not very efficient, but here is a rule-based solution:

list /. {
    {List -> Times, _Real   -> 1, _Integer -> 1},
    {List -> Times, _Symbol -> 1, _Real    -> 1}, 
    {List -> Times, _Symbol -> 1, _Integer -> 1}

{x y z, 12, -67.7251}


You can use GroupBy in several ways:

GroupBy[list, Head, Apply[Times]] /@ {Symbol, Integer, Real}

{x y z, 12, -67.7251}

or using a combination of GroupBy andLookup (thanks: CarlWoll):

Lookup[{Symbol, Integer, Real}] @ GroupBy[list, Head, Apply[Times]] 

same result

RotateRight @ Values @ GroupBy[SortBy[Head] @ list, Head, Apply[Times]] 

same result

Alternatively, you can use GatherBy:

RotateRight[Times @@@ GatherBy[SortBy[Head] @ list, Head]]

{x y z, 12, -67.7251}

Also, Replace:

Times @@@ Replace[list, {Except @ Blank @ # -> 1} & /@ {Symbol, Integer, Real}, {1}]

and DeleteCases:

Times @@@ (DeleteCases[list, Except@Blank@#] & /@ {Symbol, Integer, Real})
  • 1
    $\begingroup$ Lookup would be more idiomatic than mapping: Lookup[GroupBy[list, Head, Apply[Times]], {Symbol, Integer, Real}] $\endgroup$ – Carl Woll May 1 at 19:53
  • $\begingroup$ Thank you @CarlWoll; great point. $\endgroup$ – kglr May 1 at 20:00
  • $\begingroup$ Thank you @Shadowray. Hope the updated version works in general. $\endgroup$ – kglr May 2 at 7:30
  • $\begingroup$ Duplicate symbols may change the head of the product. E.g. {x, x, 1, 2.0} $\endgroup$ – Shadowray May 2 at 7:38
  • $\begingroup$ Whoops!! Thank you again @Shadowray. Fixed now. $\endgroup$ – kglr May 2 at 7:47

This doesn't involve any rules, but you could use Cases to select the desired types, using Through and Times as needed:

Times@@@Through@{Cases[_Symbol], Cases[_Integer], Cases[_Real]}@list

{x y z, 12, -67.7251}


A way of doing it with Reap and Sow that would work in versions as old as V5, long before newfangled functions like GroupBy were introduced (V.10).

list = {x, 3.2, y, 1, 2.679, 3, z, 4, -7.9};
     _Symbol, Sow[#, "s"],
     _Integer, Sow[#, "i"],
     _Real, Sow[#, "r"]] & /@ list,
  {"s", "i", "r"}, 
  Times @@ #2&][[-1, All, 1]]
{x y z, 12, -67.7251}
  • $\begingroup$ There's no point in using Switch here: Reap[ Sow[#,Head[#]]& /@ list, {Symbol, Integer, Real}, Times@@ #2&][[-1,All,1]] $\endgroup$ – jjagmath May 3 at 14:02
  • $\begingroup$ @jjagmath. Yes, that is a much better way of doing it. I have updated my answer to include your version. Thanks for bringing it to my attention. $\endgroup$ – m_goldberg May 3 at 21:12

One approach is to select the elements you want, and then mutliply them.

Times @@@ {Select[list, Not[NumberQ[#]] &], 
           Select[list, IntegerQ[#] &], 
           Select[list, NumberQ[#] && Not[IntegerQ[#]] &]}
{x y z, 12, -67.7251}

The first one selects all the symbols (things that are not numbers), the second selects the integers, and the third selects all the reals that are not integers.


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