Some General Notations and Conventions

Function Names

The names of builtin functions follow some general guidelines.

Function Arguments

The main expression or object on which a builtin function acts is usually given as the first argument to the function. Subsidiary parameters appear as subsequent arguments.

The following are exceptions:


Some builtin functions can take options. Each option has a name, represented as a symbol, or in some cases a string. Options are set by giving rules of the form name->value or name:>value. Such rules must appear after all the other arguments in a function. Rules for different options can be given in any order. If you do not explicitly give a rule for a particular option, a default setting for that option is used.

Options[f]give the default rules for all options associated with f
Options[expr]give the options set in a particular expression
Options[expr,name]give the setting for the option name in an expression
AbsoluteOptions[expr,name]give the absolute setting for name, even if its actual setting is Automatic
SetOptions[f,name->value,]set default rules for options associated with f
CurrentValue[name]give the option setting for the front end option name; can be used on the left-hand side of an assignment to set the option

Operations on options.

Part Numbering

nelement n (starting at 1)
-nelement n from the end

Numbering of parts.

Sequence Specifications

Allall elements
Noneno elements
nelements 1 through n
-nlast n elements
{n}element n only
{m,n}elements m through n (inclusive)
{m,n,s}elements m through n in steps of s

Specifications for sequences of parts.

The sequence specification {m,n,s} corresponds to elements m, m+s, m+2s, , up to the largest element not greater than n.

Sequence specifications are used in the functions Drop, Ordering, StringDrop, StringTake, Take, and Thread.

Level Specifications

nlevels 1 through n
Infinitylevels 1 through Infinity
{n}level n only
{n1,n2}levels n1 through n2
Heads->Trueinclude heads of expressions
Heads->Falsedo not include heads of expressions

Level specifications.

The level in an expression corresponding to a nonnegative integer n is defined to consist of parts specified by n indices. A negative level number -n represents all parts of an expression that have depth n. The depth of an expression, Depth[expr], is the maximum number of indices needed to specify any part, plus one. Levels do not include heads of expressions, except with the option setting Heads->True. Level 0 is the whole expression. Level -1 contains all symbols and other objects that have no subparts.

Ranges of levels specified by {n1,n2} contain all parts that are neither above level n1, nor below level n2 in the tree. The ni need not have the same sign. Thus, for example, {2,-2} specifies subexpressions that occur anywhere below the top level, but above the leaves, of the expression tree.

Level specifications are used by functions such as Apply, Cases, Count, FreeQ, Level, Map, MapIndexed, Position, Replace, and Scan. Note, however, that the default level specifications are not the same for all of these functions.


{imax}iterate imax times
{i,imax}i goes from 1 to imax in steps of 1
{i,imin,imax}i goes from imin to imax in steps of 1
{i,imin,imax,di}i goes from imin to imax in steps of di
{i,list}i takes on the successive values in list
{i,imin,imax},{j,jmin,jmax}, i goes from imin to imax, and for each value of i,j goes from jmin to jmax, etc.

Iterator notation.

Iterators are used in such functions as Sum, Table, Do, and Range.

The iteration parameters imin,imax and di do not need to be integers. The variable i is given a sequence of values starting at imin, and increasing in steps of di, stopping when the next value of i would be greater than imax. The iteration parameters can be arbitrary symbolic expressions, so long as (imax-imin)/di is a number.

When several iteration variables are used, the limits for the later ones can depend on the values of earlier ones.

The variable Null can be any symbolic expression; it need not be a single symbol. The value of Null is automatically set up to be local to the iteration function. This is effectively done by wrapping a Block construct containing Null around the iteration function.

The procedure for evaluating iteration functions is described in "Evaluation".

Scoping Constructs

Function[{x,},body]local parameters
lhs->rhs and lhs:>rhslocal pattern names
lhs=rhs and lhs:=rhslocal pattern names
With[{x=x0,},body]local constants
Module[{x,},body]local variables
Block[{x,},body]local values of global variables
DynamicModule[{x,},body]local variables in a Dynamic interface

Scoping constructs in the Wolfram Language. Functions in the first group scope variables lexically.

Scoping constructs allow the names or values of certain symbols to be local.

Some scoping constructs scope lexically, meaning that literal instances of the specified variables or patterns are replaced with appropriate values. When local variable names are required, symbols with names of the form xxx are generally renamed to xxx$. When nested scoping constructs are evaluated, new symbols are automatically generated in the inner scoping constructs so as to avoid name conflicts with symbols in outer scoping constructs.

When a transformation rule or definition is used, ReplaceAll (/.) is effectively used to replace the pattern names that appear on the righthand side. Nevertheless, new symbols are generated when necessary to represent other objects that appear in scoping constructs on the righthand side.

Each time it is evaluated, Module generates symbols with unique names of the form xxx$nnn as replacements for all local variables that appear in its body.

Block localizes the value of global variables. Any evaluations in the body of a block which rely on the global variable will use the locally specified value even if the variable does not explicitly appear in the body, but is only referenced through subsequent evaluation. The body of the Block may also make changes to the global variable, but any such changes will only persist until the Block has finished executing.

DynamicModule localizes its variables to each instance of the DynamicModule output in a notebook. This means each copy of a DynamicModule output created using copy and paste will use its own localized variables.

Ordering of Expressions

The canonical ordering of expressions used automatically with the attribute Orderless and in functions such as Sort satisfies the following rules:

Mathematical Functions

The mathematical functions such as Log[x] and BesselJ[n,x] that are built into the Wolfram Language have a number of features in common.

Mathematical Constants

Mathematical constants such as E and Pi that are built into the Wolfram Language have the following properties:


The Wolfram Language allows you to make assignments that override the standard operation and meaning of builtin Wolfram Language objects.

To make it difficult to make such assignments by mistake, most builtin Wolfram Language objects have the attribute Protected. If you want to make an assignment for a builtin object, you must first remove this attribute. You can do this by calling the function Unprotect.

There are a few fundamental Wolfram Language objects to which you absolutely cannot assign your own values. These objects carry the attribute Locked, as well as Protected. The Locked attribute prevents you from changing any of the attributes, and thus from removing the Protected attribute.

Abbreviated String Patterns

Functions such as StringMatchQ, Names, and Remove allow you to give abbreviated string patterns, as well as full string patterns specified by StringExpression. Abbreviated string patterns can contain certain metacharacters, which can stand for sequences of ordinary characters.

*zero or more characters
@one or more characters excluding uppercase letters
\\*, etc.literal *, etc.

Metacharacters used in abbreviated string patterns.