Special Characters
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.
Mathematica 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 will produce
in most fonts. With the notebook front end
Mathematica automatically allows you to use special key combinations when these are available, and with a textbased 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
>=, while
is equivalent to the function
Union.
and ≥ have immediate meanings in Mathematica.
Out[1]=  

Among ordinary characters such as
E and
i, some have an immediate meaning to
Mathematica, but most do not. And the same is true of special characters.
Thus, for example, while
and
have an immediate meaning to
Mathematica,
and
do not.
This allows you to set up your own definitions for
and
.
has no immediate meaning in Mathematica.
Out[4]=  

This defines a meaning for . 
Now Mathematica evaluates just as it would any other function.
Out[6]=  

Characters such as
and
are treated by
Mathematica as letters—just like ordinary keyboard letters like
a or
b.
But characters such as
and
are treated by
Mathematica as
operators. And although these particular characters are not assigned any builtin meaning by
Mathematica, they are nevertheless required to follow a definite
syntax.
The definition assigns a meaning to the operator. 
Now can be evaluated by Mathematica.
Out[9]=  

The details of how input you give to
Mathematica 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,
Mathematica 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. 
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 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] versus
A, 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.
Out[10]=  

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 and the
used as a differential operator.
The way
Mathematica does this is to use a special character
or
\[DifferentialD] as the differential operator. This special character can be entered using the alias
Esc dd Esc.
Mathematica uses a special character for the differential operator, so there is no conflict with an ordinary d.
Out[11]=  

When letters and letterlike 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 builtin evaluation rules in Mathematica.
Out[13]//FullForm= 
 

Following the correspondence between operator names and function names, special characters such as
that represent builtin
Mathematica functions have names that correspond to those functions. Thus, for example,
÷ is named
\[Divide] to correspond to the builtin
Mathematica function
Divide, and
is named
\[Implies] to correspond to the builtin 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], and
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] and
\[Cross].
\[Times] corresponds to the ordinary
Times function for multiplication;
\[Times] corresponds to the
Cross function for vector cross products. The
for
\[Times] is drawn slightly smaller than
× for
Times, corresponding to usual careful usage in mathematical typography.
In the example of
\[And] and
\[Wedge], the
\[And] operator—which happens to be drawn slightly larger—corresponds to the builtin
Mathematica function
And, while the
\[Wedge] operator has a generic name based on the appearance of the character and has no builtin meaning.
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.
Mathematica 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.
Out[18]=  

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].
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.
Mathematica 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.
Esc nnn Esc  builtin alias for a common character 
Esc nnn Esc  builtin 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 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.