are the main type of data in Mathematica
Expressions can be written in the form
. The object h
is known generically as the head
of the expression. The
are termed the elements
of the expression. Both the head and the elements may themselves be expressions.
of an expression can be referred to by numerical indices. The head has index 0; element
has index i
gives the part of expr
with index i
. Negative indices count from the end.
, or Extract
gives the piece of expr
found by successively extracting parts of subexpressions with indices
. If you think of expressions as trees, the indices specify which branch to take at each node as you descend from the root.
The pieces of an expression that are specified by giving a sequence of exactly n
indices are defined to be at level n
in the expression. You can use levels to determine the domain of application of functions like Map
. Level 0 corresponds to the whole expression.
of an expression is defined to be the maximum number of indices needed to specify any part of the expression, plus one. A negative level number -n
refers to all parts of an expression that have depth n
are the basic named objects in Mathematica
The name of a symbol must be a sequence of letters, letter-like forms, and digits, not starting with a digit. Uppercase and lowercase letters are always distinguished in Mathematica
|$Aaaa||global or internal system-defined symbol|
|aaaa$||symbol renamed in a scoping construct|
|aa$nn||unique local symbol generated in a module|
Conventions for symbol names.
Essentially all system-defined symbols have names that contain only ordinary English letters, together with numbers and
. The exceptions are
System-defined symbols conventionally have names that consist of one or more complete English words. The first letter of each word is capitalized, and the words are run together.
Once created, an ordinary symbol in Mathematica
continues to exist unless it is explicitly removed using Remove
. However, symbols created automatically in scoping constructs such as Module
carry the attribute Temporary
, which specifies that they should automatically be removed as soon as they no longer appear in any expression.
When a new symbol is to be created, Mathematica
first applies any value that has been assigned to $NewSymbol
to strings giving the name of the symbol and the context in which the symbol would be created.
If the message General
is switched on, then Mathematica
reports new symbols that are created. This message is switched off by default. Symbols created automatically in scoping constructs are not reported.
The full name of any symbol in Mathematica
consists of two parts: a context
and a short name
. The full name is written in the form
. The context
can contain the same characters as the short name. It may also contain any number of context mark characters
, and must end with a context mark.
At any point in a Mathematica
session, there is a current context $Context
and a context search path $ContextPath
consisting of a list of contexts. Symbols in the current context, or in contexts on the context search path, can be specified by giving only their short names, provided they are not shadowed by another symbol with the same short name.
|name||search $ContextPath, then $Context; create in $Context if necessary|
|`name||search $Context only; create there if necessary|
|context`name||search context only; create there if necessary|
|`context`name||search only; create there if necessary|
Contexts used for various specifications of symbols.
packages, it is conventional to associate contexts whose names correspond to the names of the packages. Packages typically use BeginPackage
to define objects in the appropriate context, and to add the context to the global $ContextPath
prints a warning about any symbols that were created in a package but which are "shadowed" by existing symbols on the context search path.
The context is included in the printed form of a symbol only if it would be needed to specify the symbol at the time of printing
All expressions in Mathematica
are ultimately made up from a small number of basic or atomic types of objects.
These objects have heads that are symbols that can be thought of as "tagging" their types. The objects contain "raw data", which can usually be accessed only by functions specific to the particular type of object. You can extract the head of the object using Head
, but you cannot directly extract any of its other parts.
Atomic objects in Mathematica
are considered to have depth 0 and yield True
when tested with AtomQ
Basic types of numbers.
All numbers in Mathematica
can contain any number of digits. Mathematica
does exact computations when possible with integers and rational numbers, and with complex numbers whose real and imaginary parts are integers or rational numbers.
There are two types of approximate real numbers in Mathematica
: arbitrary precision
and machine precision
. In manipulating arbitrary-precision numbers, Mathematica
tries to modify the precision so as to ensure that all digits actually given are correct.
With machine-precision numbers, all computations are done to the same fixed precision, so some digits given may not be correct.
Unless otherwise specified, Mathematica
treats as machine-precision numbers all approximate real numbers that lie between $MinMachineNumber
and that are input with less than $MachinePrecision
prints machine-precision numbers with $MachinePrecision
digits, except when trailing digits are zero.
In any implementation of Mathematica
, the magnitudes of numbers (except 0) must lie between $MinNumber
. Numbers with magnitudes outside this range are represented by Underflow
Character strings in Mathematica
can contain any sequence of characters. They are input in the form
The individual characters can be printable ASCII (with character codes between 32 and 126), or in general any 8- or 16-bit characters. Mathematica
uses the Unicode character encoding for 16-bit characters.
In input form, 16-bit characters are represented when possible in the form
, and otherwise as
Null bytes can appear at any point within Mathematica