Built into Mathematica
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 Esc inf Esc
. 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, Mathematica
also allows you to use aliases based on TeX names. Thus, for example, you can enter \[Infinity]
using the alias Esc \infty Esc
also supports aliases such as Esc Esc
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
in most fonts. With the notebook front end Mathematica
automatically allows you to use special key combinations when these are available, and with a text-based interface you can get Mathematica
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 Esc inf Esc|
|• Use a TeX alias such as Esc \infty Esc|
|• Use an SGML or HTML alias such as Esc Esc|
|• Click on a button in a palette|
|• Use a special key combination supported by your computer system|
Ways to enter special characters.
In a Mathematica
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 Mathematica
Some special characters are set up to have an immediate meaning to Mathematica
. Thus, for example,
is taken to be the symbol Pi
. Similarly, ≥
is taken to be the operator >=
is equivalent to the function Union
have immediate meanings in Mathematica
Among ordinary characters such as E
, some have an immediate meaning to Mathematica
, but most do not. And the same is true of special characters.
Thus, for example, while
have an immediate meaning to Mathematica
This allows you to set up your own definitions for
has no immediate meaning in Mathematica
This defines a meaning for
just as it would any other function.
Characters such as
are treated by Mathematica
as letters—just like ordinary keyboard letters like a
But characters such as
are treated by Mathematica
. And although these particular characters are not assigned any built-in meaning by Mathematica
, they are nevertheless required to follow a definite syntax
is an infix operator.
The definition assigns a meaning to the
can be evaluated by Mathematica
The details of how input you give to Mathematica
is interpreted depends on whether you are using StandardForm
, and on what additional information you supply in InterpretationBox
and similar constructs.
But unless you explicitly override its built-in rules by giving your own definitions for MakeExpression
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 Mathematica
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.
|Letters||a, E, , , , etc.|
|Letter-like 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
, which looks very similar to
As is typical, however, the operator form
is slightly less elaborate and more stylized than the letter form
. In addition,
is an extensible character which 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]
, both characters are letters. However, Mathematica
treats these characters as different, and in some fonts, for example, they may look quite different.
The result contains four distinct characters.
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 Mathematica
have a precise and consistent syntax, it is necessary at least in StandardForm
to distinguish between an ordinary d
used as a differential operator.
The way Mathematica
does this is to use a special character
as the differential operator. This special character can be entered using the alias Esc dd Esc
uses a special character for the differential operator, so there is no conflict with an ordinary d
When letters and letter-like forms appear in Mathematica
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 Mathematica
does is to create a function whose name is the full name of the special character that appears as the operator.
This constructs an And
function, which happens to have built-in evaluation rules in Mathematica
Following the correspondence between operator names and function names, special characters such as
that represent built-in Mathematica
functions have names that correspond to those functions. Thus, for example, ÷
is named \[Divide]
to correspond to the built-in Mathematica
is named \[Implies]
to correspond to the built-in function Implies
In general, however, special characters in Mathematica
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]
is named \[TildeTilde]
rather than, say, \[ApproximatelyEqual]
Different operator characters that look similar.
There are sometimes characters that look similar but which are used to represent different operators. An example is \[Times]
corresponds to the ordinary Times
function for multiplication; \[Times]
corresponds to the Cross
function for vector cross products. The
is drawn slightly smaller than ×
, corresponding to usual careful usage in mathematical typography.
operator represents ordinary multiplication.
operator represents vector cross products.
The two operators display in a similar way—with \[Times]
slightly larger than \[Cross]
In the example of \[And]
, the \[And]
operator—which happens to be drawn slightly larger—corresponds to the built-in Mathematica
, while the \[Wedge]
operator has a generic name based on the appearance of the character and has no built-in meaning.
You can mix \[Wedge]
operators. Each has a definite precedence.
Some of the special characters commonly used as operators in mathematical notation look similar to ordinary keyboard characters. Thus, for example,
looks similar to the ^
character on a standard keyboard.
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.
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 →
, while Esc && Esc
is the alias for
The most extreme case of characters that look alike but work differently occurs with vertical bars.
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
often gives similar-looking 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.
|Esc nnn Esc||built-in alias for a common character|
|Esc nnn Esc||built-in alias for similar but less common character|
|Esc .nnn Esc||alias globally defined in a Mathematica session|
|Esc ,nnn Esc||alias defined in a specific notebook|
Conventions for special character aliases.
The notebook front end for Mathematica
often allows you to set up your own aliases for special characters. If you want to, you can overwrite the built-in 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.