MATHEMATICA TUTORIAL

Input Syntax Related Tutorials »| Syntax More About »| Blank BlankNullSequence BlankSequence Optional Out Pattern Slot SlotSequence Functions »

Operator Input Forms

Characters that are not letters, letter-like forms, or structural elements are treated by Mathematica as operators. Mathematica has built-in rules for interpreting all operators. The functions to which these operators correspond may or may not, however, have built-in evaluation or other rules. Cases in which built-in meanings are by default defined are indicated by in the tables below.

Operators that construct two-dimensional boxes—all of which have names beginning with backslash—can only be used inside \(...\). The table below gives the interpretations of these operators within . "Input of Boxes" gives interpretations when no is included.

exprandexpri	any expression
symb	any symbol
patt	any pattern object
stringandstringi	or a sequence of letters, letter-like forms, and digits
filename	like string, but can include additional characters described below
	built-in meanings exist

Objects used in the tables of operator input forms.

Operator Precedence

operator form	full form	grouping
forms representing numbers (see Numbers)
forms representing symbols (see Symbol Names and Contexts)
forms representing character strings (see Character Strings)
e11 e12 ... e21 e22 ... ...	{{e11,e12,...},{e21,e22,...},...}
e11 e12 e21 e22 ...	Piecewise[{{e11,e12},{e21,e22},...}]
expr::string	MessageName[expr,"string"]
expr::string1::string2	MessageName[expr,"string1","string2"]
forms containing (see additional input forms)
forms containing (see additional input forms)
forms containing (see additional input forms)
<<filename	Get["filename"]
	Overscript[expr1,expr2]
expr1\&expr2	Overscript[expr1,expr2]	e\&(e\&e)
	Underscript[expr1,expr2]
expr1\+expr2	Underscript[expr1,expr2]	e\+(e\+e)
	Underoverscript[expr1,expr2,expr3]
expr1\+expr2\%expr3	Underoverscript[expr1,expr2,expr3]
expr1\&expr2\%expr3	Underoverscript[expr1,expr3,expr2]
expr1expr2	Subscript[expr1,expr2]	e(ee)
expr1\_expr2	Subscript[expr1,expr2]	e\_(e\_e)
expr1\_expr2\%expr3	Power[Subscript[expr1,expr2],expr3]
\!boxes	(interpreted version of boxes)
expr1?expr2	PatternTest[expr1,expr2]
expr1[expr2,...]	expr1[expr2,...]	(e[e])[e]
expr1[[expr2,...]]	Part[expr1,expr2,...]	(e[[e]])[[e]]
expr1expr2,...	Part[expr1,expr2,...]	(ee)e
expr1expr2	Part[expr1,expr2,...]	(ee)e
\*expr	(boxes constructed from expr)
expr++	Increment[expr]
expr--	Decrement[expr]
++expr	PreIncrement[expr]
--expr	PreDecrement[expr]
expr1@expr2	expr1[expr2]	e@(e@e)
expr1 expr2	(invisible application, input as Esc @ Esc )
	expr1[expr2]
expr1~expr2~expr3	expr2[expr1,expr3]	(e~e~e)~e~e
expr1/@expr2	Map[expr1,expr2]	e/@(e/@e)
expr1//@expr2	MapAll[expr1,expr2]	e//@(e//@e)
expr1@@expr2	Apply[expr1,expr2]	e@@(e@@e)
expr1@@@expr2	Apply[expr1,expr2,{1}]	e@@@(e@@@e)
expr!	Factorial[expr]
expr!!	Factorial2[expr]
expr*	Conjugate[expr]
expr	Transpose[expr]
expr	ConjugateTranspose[expr]
expr	ConjugateTranspose[expr]
expr'	Derivative[1][expr]
expr''...' (n times)	Derivative[n][expr]
expr1<>expr2<>expr3	StringJoin[expr1,expr2,expr3]	e<>e<>e
expr1^expr2	Power[expr1,expr2]	e^(e^e)
expr1expr2	Power[expr1,expr2]	e(ee)
	Power[Subscript[expr1,expr2],expr3]
expr1\^expr2\%expr3	Power[Subscript[expr1,expr3],expr2]
vertical arrow and vector operators
	Sqrt[expr]
\@ expr	Sqrt[expr]	\@(\@ e)
\@ expr\%n	Power[expr,1/n]
expr1 expr2	Integrate[expr1,expr2]	( e e) e
e3e4	Integrate[e3,{e4,e1,e2}]	( e e) e
other integration operators
expr1expr2	D[expr2,expr1]	e(ee)
expr	Del[expr]	(e)
expr1expr2	DiscreteShift[expr2,expr1]	e(ee)
expr1expr2	DiscreteRatio[expr2,expr1]	e(ee)
expr1expr2	DifferenceDelta[expr2,expr1]	e(ee)
expr	Square[expr]	( e)
expr1 expr2 expr3	SmallCircle[expr1,expr2,expr3]	e e e
expr1 expr2 expr3	CircleDot[expr1,expr2,expr3]	e e e
expr1**expr2**expr3	NonCommutativeMultiply[expr1,expr2,expr3]	e**e**e
expr1expr2expr3	Cross[expr1,expr2,expr3]	eee
expr1.expr2.expr3	Dot[expr1,expr2,expr3]	e.e.e
-expr	Times[-1,expr]
+expr	expr
±expr	PlusMinus[expr]
expr	MinusPlus[expr]
expr1/expr2	expr1(expr2)^-1	(e/e)/e
expr1÷expr2	Divide[expr1,expr2]	(e÷e)÷e
expr1\/expr2	Divide[expr1,expr2]	(e\/e)\/e
expr1\expr2\expr3	Backslash[expr1,expr2,expr3]	e\e\e
expr1expr2expr3	Diamond[expr1,expr2,expr3]	eee
expr1expr2expr3	Wedge[expr1,expr2,expr3]	eee
expr1expr2expr3	Vee[expr1,expr2,expr3]	eee
expr1expr2expr3	CircleTimes[expr1,expr2,expr3]	eee
expr1expr2expr3	CenterDot[expr1,expr2,expr3]	eee
expr1 expr2 expr3	Times[expr1,expr2,expr3]	e e e
expr1*expr2*expr3	Times[expr1,expr2,expr3]	e*e*e
expr1×expr2×expr3	Times[expr1,expr2,expr3]	e×e×e
expr1expr2expr3	Star[expr1,expr2,expr3]	eee
e4	Product[e4,{e1,e2,e3}]	( e)
expr1expr2expr3	VerticalTilde[expr1,expr2,expr3]	eee
expr1expr2expr3	Coproduct[expr1,expr2,expr3]	eee
expr1expr2expr3	Cap[expr1,expr2,expr3]	eee
expr1expr2expr3	Cup[expr1,expr2,expr3]	eee
expr1 expr2 expr3	CirclePlus[expr1,expr2,expr3]	eee
expr1 expr2	CircleMinus[expr1,expr2]	(e e) e
e4	Sum[e4,{e1,e2,e3}]	( e)
expr1+expr2+expr3	Plus[expr1,expr2,expr3]	e+e+e
expr1-expr2	expr1+(-1expr2)	(e-e)-e
expr1±expr2	PlusMinus[expr1,expr2]	(e±e)±e
expr1expr2	MinusPlus[expr1,expr2]	(ee)e
expr1expr2	Intersection[expr1,expr2]	eee
other intersection operators
expr1expr2	Union[expr1,expr2]	eee
other union operators
i;;j;;k	Span[i,j,k]	e;;e;;e
expr1==expr2	Equal[expr1,expr2]	e==e==e
expr1==expr2	Equal[expr1,expr2]	e==e==e
expr1expr2	Equal[expr1,expr2]	eee
expr1!= expr2	Unequal[expr1,expr2]	e!=e!=e
expr1!=expr2	Unequal[expr1,expr2]	e!=e!=e
other equality and similarity operators
expr1>expr2	Greater[expr1,expr2]	e>e>e
expr1>=expr2	GreaterEqual[expr1,expr2]	e>=e>=e
expr1≥expr2	GreaterEqual[expr1,expr2]	e≥e≥e
expr1expr2	GreaterEqual[expr1,expr2]	eee
expr1<expr2	Less[expr1,expr2]	e<e<e
expr1<=expr2	LessEqual[expr1,expr2]	e<=e<=e
expr1≤expr2	LessEqual[expr1,expr2]	e≤e≤e
expr1expr2	LessEqual[expr1,expr2]	eee
other ordering operators
expr1|expr2	VerticalBar[expr1,expr2]	e|e|e
expr1expr2	NotVerticalBar[expr1,expr2]	eee
expr1expr2	DoubleVerticalBar[expr1,expr2]	eee
expr1expr2	NotDoubleVerticalBar[expr1,expr2]	eee
horizontal arrow and vector operators
diagonal arrow operators
expr1===expr2	SameQ[expr1,expr2]	e===e===e
expr1=!=expr2	UnsameQ[expr1,expr2]	e=!=e=!=e
expr1expr2	Element[expr1,expr2]	eee
expr1expr2	NotElement[expr1,expr2]	eee
expr1expr2	Subset[expr1,expr2]	eee
expr1expr2	Superset[expr1,expr2]	eee
other set relation operators
expr1expr2	ForAll[expr1,expr2]	e(ee)
expr1expr2	Exists[expr1,expr2]	e(ee)
expr1expr2	NotExists[expr1,expr2]	e(ee)
!expr	Not[expr]	!(!e)
¬expr	Not[expr]	¬(¬e)
expr1&&expr2&&expr3	And[expr1,expr2,expr3]	e&&e&&e
expr1expr2expr3	And[expr1,expr2,expr3]	eee
expr1expr2expr3	Nand[expr1,expr2,expr3]	eee
expr1expr2expr3	Xor[expr1,expr2,expr3]	eee
expr1expr2expr3	Xnor[expr1,expr2,expr3]	eee
expr1||expr2||expr3	Or[expr1,expr2,expr3]	e||e||e
expr1expr2expr3	Or[expr1,expr2,expr3]	eee
expr1expr2expr3	Nor[expr1,expr2,expr3]	eee
expr1expr2expr3	Equivalent[expr1,expr2,expr3]	eee
expr1expr2	Implies[expr1,expr2]	e(ee)
expr1expr2	Implies[expr1,expr2]	eee
expr1expr2	RightTee[expr1,expr2]	e(ee)
expr1expr2	DoubleRightTee[expr1,expr2]	e(ee)
expr1expr2	LeftTee[expr1,expr2]	(ee)e
expr1expr2	DoubleLeftTee[expr1,expr2]	(ee)e
expr1expr2	SuchThat[expr1,expr2]	e(ee)
expr..	Repeated[expr]
expr...	RepeatedNull[expr]
expr1|expr2	Alternatives[expr1,expr2]	e|e|e
symb:expr	Pattern[symb,expr]
patt:expr	Optional[patt,expr]
expr1~~expr2~~expr3	StringExpression[expr1,expr2,expr3]	e~~e~~e
expr1/;expr2	Condition[expr1,expr2]	(e/;e)/;e
expr1->expr2	Rule[expr1,expr2]	e->(e->e)
expr1→expr2	Rule[expr1,expr2]	e→(e→e)
expr1:>expr2	RuleDelayed[expr1,expr2]	e:>(e:>e)
expr1⧴ expr2	RuleDelayed[expr1,expr2]	e⧴(e⧴e)
expr1/.expr2	ReplaceAll[expr1,expr2]	(e/.e)/.e
expr1//.expr2	ReplaceRepeated[expr1,expr2]	(e//.e)//.e
expr1+=expr2	AddTo[expr1,expr2]	e+=(e+=e)
expr1-=expr2	SubtractFrom[expr1,expr2]	e-=(e-=e)
expr1*=expr2	TimesBy[expr1,expr2]	e*=(e*=e)
expr1/=expr2	DivideBy[expr1,expr2]	e/=(e/=e)
expr&	Function[expr]
expr1:expr2	Colon[expr1:expr2]	e:e:e
expr1//expr2	expr2[expr1]	(e//e)//e
expr1expr2	VerticalSeparator[expr1,expr2]	eee
expr1expr2	Therefore[expr1,expr2]	e(ee)
expr1expr2	Because[expr1,expr2]	(ee)e
expr1=expr2	Set[expr1,expr2]	e=(e=e)
expr1:=expr2	SetDelayed[expr1,expr2]	e:=(e:=e)
expr1^=expr2	UpSet[expr1,expr2]	e^=(e^=e)
expr1^:=expr2	UpSetDelayed[expr1,expr2]	e^:=(e^:=e)
symb/:expr1=expr2	TagSet[symb,expr1,expr2]
symb/:expr1:=expr2	TagSetDelayed[symb,expr1,expr2]
expr=.	Unset[expr]
symb/:expr=.	TagUnset[symb,expr]
expr1expr2	Function[{expr1},expr2]	e(ee)
expr>>filename	Put[expr,"filename"]
expr>>>filename	PutAppend[expr,"filename"]
expr1;expr2;expr3	CompoundExpression[expr1,expr2,expr3]
expr1;expr2;	CompoundExpression[expr1,expr2,Null]
expr1\`expr2	FormBox[expr2,expr1]	e\`(e\`e)

Operator input forms, in order of decreasing precedence.

Additional input forms, in order of decreasing precedence.

Special Characters

Special characters that appear in operators usually have names that correspond to the names of the functions they represent. Thus the character has the name \[CirclePlus] and yields the function CirclePlus. Exceptions are \[GreaterSlantEqual], \[LessSlantEqual] and \[RoundImplies].

The delimiters in matchfix operators have names and .

keyboard characters special character -> \[Rule] :> \[RuleDelayed] == \[Equal] != \[NotEqual]

keyboard characters special character >= \[GreaterEqual] >= \[GreaterSlantEqual] <= \[LessEqual] <= \[LessSlantEqual]

Keyboard and special characters with the same interpretations.

keyboard character special character \[RawColon] : \[Colon] \[RawTilde] ~ \[Tilde] \[RawWedge] ^ \[Wedge] \[RawWedge] ^ \[And] \[RawStar] * \[Star] \[RawBackslash] \ \[Backslash]

keyboard character special character \[RawDot] . \[CenterDot] \[RawVerticalBar] | \[VerticalBar] \[RawVerticalBar] | \[VerticalSeparator] \[RawVerticalBar] | \[LeftBracketingBar] \[RawDash] - \[Dash] ... \[Ellipsis] ...

Some keyboard and special characters with different interpretations.

Precedence and the Ordering of Input Forms

The tables of input forms are arranged in decreasing order of precedence. Input forms in the same box have the same precedence. Each page in the table begins a new box. As discussed in "Special Ways to Input Expressions", precedence determines how Mathematica groups terms in input expressions. The general rule is that if has higher precedence than , then is interpreted as , and is interpreted as .

Grouping of Input Forms

The third columns in the tables show how multiple occurrences of a single input form, or of several input forms with the same precedence, are grouped. For example, is grouped as ("left associative"), while is grouped as ("right associative"). No grouping is needed in an expression like , since Plus is fully associative, as represented by the attribute Flat.

Precedence of Integration Operators

Forms such as have an "outer" precedence just below Power, as indicated in the table above, but an "inner" precedence just above . The outer precedence determines when needs to be parenthesized; the inner precedence determines when needs to be parenthesized.

Spaces and Multiplication

• x y z x*y*z

• 2x 2*x

• 2(x+1) 2*(x+1)

• c(x+1) c*(x+1)

• (x+1)(y+2) (x+1)*(y+2)

• x! y x!*y

• x!y x!*y

Alternative forms for multiplication.

An expression like could potentially mean either or . The first interpretation is chosen because Factorial has higher precedence than Not.

Spaces within single input forms are ignored. Thus, for example, is equivalent to . You will often want to insert spaces around lower precedence operators to improve readability.

• , like , means

• , like , means

• is a single symbol, not

Some cases to be careful about.

Spaces to Avoid

You should avoid inserting any spaces between the different characters in composite operators such as , , and . Although in some cases such spaces are allowed, they are liable to lead to confusion.

Another case where spaces must be avoided is between the characters of the pattern object . If you type , Mathematica will interpret this as , rather than the single named pattern object .

Similarly, you should not insert any spaces inside pattern objects like .

Spacing Characters

• Ordinary keyboard space (\[RawSpace])

• \[VeryThinSpace], \[ThinSpace], ..., \[ThickSpace]

• \[NegativeVeryThinSpace], \[NegativeThinSpace], ..., \[NegativeThickSpace]

• (\[SpaceIndicator])

Spacing characters equivalent to an ordinary keyboard space.

Relational Operators

Relational operators can be mixed. An expression like is converted to Inequality[a, Greater, b, GreaterEqual, c], which effectively evaluates as . (The reason for the intermediate form is that it prevents objects from being evaluated twice when input like is processed.)

File Names

Any file name can be given in quotes after , , and . File names can also be given without quotes if they contain only alphanumeric characters, special characters, and the characters , , , , , , , , , , , and , together with matched pairs of square brackets enclosing any characters other than spaces, tabs, and newlines. Note that file names given without quotes can be followed only by spaces, tabs, or newlines, or by the characters , , or , as well as semicolons and commas.

MORE ABOUT

• Syntax

RELATED TUTORIALS

• Input Syntax

TUTORIAL COLLECTION

• Core Language