Special Characters: Mathematical and Other Notation

Built into the Wolfram Language are a large number of special characters intended for use in mathematical and other notation. "Listing of Named Characters" gives a complete listing.

Each special character is assigned a full name such as \[Infinity]. More common special characters are also assigned aliases, such as EscinfEsc. You can set up additional aliases using the InputAliases notebook option discussed in "Options for Expression Input and Output".

For special characters that are supported in standard dialects of TeX, the Wolfram Language also allows you to use aliases based on TeX names. Thus, for example, you can enter \[Infinity] using the alias Esc\inftyEsc. The Wolfram Language also supports aliases such as EscEsc based on names used in SGML and HTML.

Standard system software on many computer systems also supports special key combinations for entering certain special characters. On a Macintosh, for example, Option+5 will produce in most fonts. With the notebook front end the Wolfram System automatically allows you to use special key combinations when these are available, and with a textbased interface you can get the Wolfram System to accept such key combinations if you set an appropriate value for $CharacterEncoding.

Use a full name such as \[Infinity]
Use an alias such as EscinfEsc
Use a TeX alias such as Esc\inftyEsc
Use an SGML or HTML alias such as EscEsc
Click a button in a palette
Use a special key combination supported by your computer system

Ways to enter special characters.

In a Wolfram System notebook, you can use special characters just like you use standard keyboard characters. You can include special characters both in ordinary text and in input that you intend to give to the Wolfram System.

Some special characters are set up to have an immediate meaning to the Wolfram System. Thus, for example, is taken to be the symbol Pi. Similarly, is taken to be the operator >=, while is equivalent to the function Union.

and have immediate meanings in the Wolfram System:
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or [Union] is immediately interpreted as the Union function:
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or [SquareUnion] has no immediate meaning to the Wolfram System:
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Among ordinary characters such as E and i, some have an immediate meaning to the Wolfram System, but most do not. And the same is true of special characters.

Thus, for example, while and have an immediate meaning to the Wolfram System, and do not.

This allows you to set up your own definitions for and .

has no immediate meaning in the Wolfram System:
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This defines a meaning for :
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Now the Wolfram System evaluates just as it would any other function:
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Characters such as and are treated by the Wolfram System as lettersjust like ordinary keyboard letters like a or b.

But characters such as and are treated by the Wolfram System as operators. And although these particular characters are not assigned any builtin meaning by the Wolfram System, they are nevertheless required to follow a definite syntax.

is an infix operator:
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The definition assigns a meaning to the operator:
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Now can be evaluated by the Wolfram System:
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The details of how input you give to the Wolfram System is interpreted depends on whether you are using StandardForm or TraditionalForm, and on what additional information you supply in InterpretationBox and similar constructs.

But unless you explicitly override its builtin rules by giving your own definitions for MakeExpression, the Wolfram System will always assign the same basic syntactic properties to any particular special character.

These properties not only affect the interpretation of the special characters in Wolfram System input, but also determine the structure of expressions built with these special characters. They also affect various aspects of formatting; operators, for example, have extra space left around them, while letters do not.

Lettersa, E, , , , etc.
Letterlike forms, , , , etc.
Operators, , , , etc.

Types of special characters.

In using special characters, it is important to make sure that you have the correct character for a particular purpose. There are quite a few examples of characters that look similar, yet are in fact quite different.

A common issue is operators whose forms are derived from letters. An example is or \[Sum], which looks very similar to or \[CapitalSigma].

As is typical, however, the operator form is slightly less elaborate and more stylized than the letter form . In addition, is an extensible character that grows depending on the summand, while has a size determined only by the current font.

Different characters that look similar.

In cases such as \[CapitalAlpha] versus A, both characters are letters. However, the Wolfram System treats these characters as different, and in some fonts, for example, they may look quite different.

The result contains four distinct characters:
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Traditional mathematical notation occasionally uses ordinary letters as operators. An example is the d in a differential such as dx that appears in an integral.

To make the Wolfram System have a precise and consistent syntax, it is necessary at least in StandardForm to distinguish between an ordinary d and the used as a differential operator.

The way the Wolfram System does this is to use a special character or \[DifferentialD] as the differential operator. This special character can be entered using the alias EscddEsc.

The Wolfram System uses a special character for the differential operator, so there is no conflict with an ordinary d:
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When letters and letterlike forms appear in Wolfram System input, they are typically treated as names of symbols. But when operators appear, functions must be constructed that correspond to these operators. In almost all cases, what the Wolfram System does is to create a function whose name is the full name of the special character that appears as the operator.

The Wolfram System constructs a CirclePlus function to correspond to the operator , whose full name is [CirclePlus]:
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This constructs an And function, which happens to have builtin evaluation rules in the Wolfram System:
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Following the correspondence between operator names and function names, special characters such as that represent builtin Wolfram System functions have names that correspond to those functions. Thus, for example, is named \[Divide] to correspond to the builtin Wolfram System function Divide, and is named \[Implies] to correspond to the builtin function Implies.

In general, however, special characters in the Wolfram Language are given names that are as generic as possible, so as not to prejudice different uses. Most often, characters are thus named mainly according to their appearance. The character is therefore named \[CirclePlus], rather than, say \[DirectSum], and is named \[TildeTilde] rather than, say, \[ApproximatelyEqual].

Different operator characters that look similar.

There are sometimes characters that look similar but that are used to represent different operators. An example is \[Times] and \[Cross]. \[Times] corresponds to the ordinary Times function for multiplication; \[Cross] corresponds to the Cross function for vector cross products. The for \[Cross] is drawn slightly smaller than for \[Times], corresponding to usual careful usage in mathematical typography.

The \[Times] operator represents ordinary multiplication:
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The \[Cross] operator represents vector cross products:
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The two operators display in a similar waywith \[Times] slightly larger than \[Cross]:
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In the example of \[And] and \[Wedge], the \[And] operatorwhich happens to be drawn slightly largercorresponds to the builtin Wolfram System function And, while the \[Wedge] operator has a generic name based on the appearance of the character and has no builtin meaning.

You can mix [Wedge] and [And] operators. Each has a definite precedence:
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Some of the special characters commonly used as operators in mathematical notation look similar to ordinary keyboard characters. Thus, for example, or [Wedge] looks similar to the ^ character on a standard keyboard.

The Wolfram System interprets a raw ^ as a power. But it interprets as a generic Wedge function. In cases such as this where there is a special character that looks similar to an ordinary keyboard character, the convention is to use the ordinary keyboard character as the alias for the special character. Thus, for example, Esc^Esc is the alias for \[Wedge].

The raw ^ is interpreted as a power, but the Esc^Esc is a generic wedge operator:
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A related convention is that when a special character is used to represent an operator that can be typed using ordinary keyboard characters, those characters are used in the alias for the special character. Thus, for example, Esc->Esc is the alias for or \[Rule], while Esc&&Esc is the alias for or \[And].

Esc->Esc is the alias for \[Rule], and Esc&&Esc for \[And]:
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The most extreme case of characters that look alike but work differently occurs with vertical bars.

character name

Different types of vertical bars.

Notice that the alias for \[VerticalBar] is Esc|Esc, while the alias for the somewhat more common \[VerticalSeparator] is Esc|Esc. The Wolfram Language often gives similarlooking characters similar aliases; it is a general convention that the aliases for the less commonly used characters are distinguished by having spaces at the beginning.

EscnnnEscbuiltin alias for a common character
EscnnnEscbuiltin alias for similar but less common character
Esc.nnnEscalias globally defined in a Wolfram System session
Esc,nnnEscalias defined in a specific notebook

Conventions for special character aliases.

The notebook front end for the Wolfram System often allows you to set up your own aliases for special characters. If you want to, you can overwrite the builtin aliases. But the convention is to use aliases that begin with a dot or comma.

Note that whatever aliases you may use to enter special characters, the full names of the characters will always be used when the characters are stored in files.