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2.8.15 Operators without Built-in Meanings

When you enter a piece of input such as 2+2, Mathematica first recognizes the + as an operator and constructs the expression Plus[2,2], then uses the built-in rules for Plus to evaluate the expression and get the result 4.
But not all operators recognized by Mathematica are associated with functions that have built-in meanings. Mathematica also supports several hundred additional operators that can be used in constructing expressions, but for which no evaluation rules are initially defined.
You can use these operators as a way to build up your own notation within the Mathematica language.

  • The is recognized as an infix operator, but has no predefined value.
  • In[1]:=


  • In StandardForm, prints as an infix operator.
  • In[2]:=


  • You can define a value for .
  • In[3]:= x_ y_ := Mod[x + y, 2]

  • Now is not only recognized as an operator, but can also be evaluated.
  • In[4]:= 2 3


    A few Mathematica operators corresponding to functions without predefined values.

    Mathematica follows the general convention that the function associated with a particular operator should have the same name as the special character that represents that operator.

  • \[Congruent] is displayed as .
  • In[5]:= x \[Congruent] y


  • It corresponds to the function Congruent.
  • In[6]:= FullForm[%]


    The conventional correspondence in Mathematica between operator names and function names.

    You should realize that even though the functions CirclePlus and CircleTimes do not have built-in evaluation rules, the operators and do have built-in precedences. Section A.2.7 lists all the operators recognized by Mathematica, in order of their precedence.

  • The operators and have definite precedenceswith higher than .
  • In[7]:= x y z // FullForm


    Some two-dimensional forms without built-in meanings.

  • Subscripts have no built-in meaning in Mathematica.
  • In[8]:=


  • Most superscripts are however interpreted as powers by default.
  • In[9]:=


  • A few special superscripts are not interpreted as powers.
  • In[10]:=


  • Bar and hat are interpreted as OverBar and OverHat.
  • In[11]:=