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Module[{x,y,…},expr]

specifies that occurrences of the symbols x, y, … in expr should be treated as local.

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Module[{x=x₀,…},expr]

defines initial values for x, ….

Details

Module allows you to set up local variables with names that are local to the module.
Module creates new symbols to represent each of its local variables every time it is called.
Module creates a symbol with name xxx$nnn to represent a local variable with name xxx. The number nnn is the current value of $ModuleNumber.
The value of $ModuleNumber is incremented every time any module is used.
Before evaluating expr, Module substitutes new symbols for each of the local variables that appear anywhere in expr except as local variables in scoping constructs.
Symbols created by Module carry the attribute Temporary.
Symbols created by Module can be returned from modules.
You can use Module[{vars},body/;cond] as the right‐hand side of a transformation rule with a condition attached.
Module has attribute HoldAll.
Module constructs can be nested in any way, with inner variables being renamed if necessary.
Module is a scoping construct that implements lexical scoping.

Examples

open allclose all

Basic Examples (1)Summary of the most common use cases

In[1]:=1

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https://wolfram.com/xid/0mq6vq-i2fv4k

In[2]:=2

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https://wolfram.com/xid/0mq6vq-khx9d9

Out[2]=2

Applications (2)Sample problems that can be solved with this function

Dynamic programming with a local function:

In[1]:=1

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https://wolfram.com/xid/0mq6vq-carga

In[2]:=2

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https://wolfram.com/xid/0mq6vq-ilc4w

Out[2]=2

Euclid's algorithm for the GCD using initialized local variables:

In[1]:=1

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https://wolfram.com/xid/0mq6vq-kgez2t

In[2]:=2

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https://wolfram.com/xid/0mq6vq-hefccb

Out[2]=2

Properties & Relations (8)Properties of the function, and connections to other functions

Every time a module is evaluated, a new temporary symbol is created:

In[1]:=1

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https://wolfram.com/xid/0mq6vq-j20eck

Out[1]=1

Module symbols are temporary and are removed if they are no longer referenced:

In[1]:=1

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https://wolfram.com/xid/0mq6vq-g19k9b

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0mq6vq-ux3pk

Out[2]=2

Each use of Module increments $ModuleNumber:

In[1]:=1

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https://wolfram.com/xid/0mq6vq-jjks0b

Out[1]=1

If there is no need to assign to a local variable, a constant should be used instead:

In[1]:=1

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https://wolfram.com/xid/0mq6vq-crvx03

Out[1]=1

With is faster than Module:

In[2]:=2

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https://wolfram.com/xid/0mq6vq-rf4b1

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0mq6vq-elsu9r

Out[3]=3

Block localizes values only; it does not create new symbols:

In[1]:=1

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https://wolfram.com/xid/0mq6vq-kcvf9c

Out[1]=1

Unique creates new variables in a way similar to Module:

In[1]:=1

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https://wolfram.com/xid/0mq6vq-c4vp4u

Out[1]=1

Local variables are not affected by global ones, and vice versa:

In[1]:=1

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https://wolfram.com/xid/0mq6vq-gug7k4

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0mq6vq-pzc8d6

Out[2]=2

A symbol brought into the scope of Module is not affected by naming conflicts:

In[1]:=1

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https://wolfram.com/xid/0mq6vq-3i761

In[2]:=2

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https://wolfram.com/xid/0mq6vq-hwp9ge

Out[2]=2

Possible Issues (3)Common pitfalls and unexpected behavior

Module is a scoping construct; inner local variables shield outer ones:

In[1]:=1

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https://wolfram.com/xid/0mq6vq-cxpmmo

Variables are renamed in nested scopes:

In[1]:=1

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https://wolfram.com/xid/0mq6vq-ur44b

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0mq6vq-e9fdbp

Out[2]=2

Build the function from its parts to avoid the renaming:

In[3]:=3

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https://wolfram.com/xid/0mq6vq-c56zuo

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0mq6vq-cnb70

Out[4]=4

Parallel assignment is not available for Module variables:

In[1]:=1

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https://wolfram.com/xid/0mq6vq-c2pr7

In[2]:=2

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https://wolfram.com/xid/0mq6vq-dlipn6

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0mq6vq-ek10ws

Out[3]=3

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