# The Syntax of the *Mathematica* Language

*Mathematica* uses various syntactic rules to interpret input that you give, and to convert strings and boxes into expressions. The version of these rules that is used for StandardForm and InputForm in effect defines the basic *Mathematica* language. The rules used for other forms, such as TraditionalForm, follow the same overall principles, but differ in many details.

a, xyz, | symbols |

"some text", "+" | strings |

123.456, 3*^45 | numbers |

+, ->, ≠ | operators |

(*comment*) | input to be ignored |

Types of tokens in the *Mathematica* language.

When you give text as input to *Mathematica*, the first thing that *Mathematica* does is to break the text into a sequence of *tokens*, with each token representing a separate syntactic unit.

Thus, for example, if you give the input , *Mathematica* will break this into the sequence of tokens , , , and . Here , , and are tokens that correspond to symbols, while and are operators.

Operators are ultimately what determine the structure of the expression formed from a particular piece of input. The *Mathematica* language involves several general classes of operators, distinguished by the different positions in which they appear with respect to their operands.

prefix | !x | Not[x] |

postfix | x! | Factorial[x] |

infix | x+y+z | Plus[x,y,z] |

matchfix | {x,y,z} | List[x,y,z] |

compound | x/:y=z | TagSet[x,y,z] |

overfix | OverHat[x] |

Examples of classes of operators in the *Mathematica* language.

Operators typically work by picking up operands from definite positions around them. But when a string contains more than one operator, the result can in general depend on which operator picks up its operands first.

Thus, for example, could potentially be interpreted either as or as depending on whether or picks up its operands first.

To avoid such ambiguities, *Mathematica* assigns a *precedence* to each operator that can appear. Operators with higher precedence are then taken to pick up their operands first.

Thus, for example, the multiplication operator is assigned higher precedence than , so that it picks up its operands first, and is interpreted as rather than .

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Whatever the precedence of the operators you are using, you can always specify the structure of the expressions you want to form by explicitly inserting appropriate parentheses.

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Extensions of symbol names | x_,#2,e::s, etc. |

Function application variants | e[e],e@@e, etc. |

Power-related operators | √e,e^e, etc. |

Multiplication-related operators
| e,e/e,ee,ee, etc. |

Addition-related operators | ee,e+e,ee, etc. |

Relational operators | e==e,ee,ee,ee,ee, etc. |

Arrow and vector operators | ee,ee,ee,ee, etc. |

Logic operators | _{e}e,e&&e,ee,ee, etc. |

Pattern and rule operators | e..,e|e,e->e,e/.e, etc. |

Pure function operator | e& |

Assignment operators | e=e,e:=e, etc. |

Compound expression | e;e |

Outline of operators in order of decreasing precedence.

The table in "Operator Input Forms" gives the complete ordering by precedence of all operators in *Mathematica*. Much of this ordering, as in the case of and , is determined directly by standard mathematical usage. But in general the ordering is simply set up to make it less likely for explicit parentheses to have to be inserted in typical pieces of input.

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Types of grouping for infix operators.

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The syntax of the *Mathematica* language is defined not only for characters that you can type on a typical keyboard, but also for all the various special characters that *Mathematica* supports.

Letters such as , , and from any alphabet are treated just like ordinary English letters, and can for example appear in the names of symbols. The same is true of letter-like forms such as , , and .

But many other special characters are treated as operators. Thus, for example, and are infix operators, while is a prefix operator, and and are matchfix operators.

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Some special characters form elements of fairly complicated compound operators. Thus, for example, contains the compound operator with elements and .

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Input to *Mathematica* can be given not only in the form of one-dimensional strings, but also in the form of two-dimensional boxes. The syntax of the *Mathematica* language covers not only one-dimensional constructs but also two-dimensional ones.

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