# How to | Map a Function over a List

*Mathematica* includes many powerful operations for working with lists. It is often desirable to map a function onto each individual element in a list. While listable functions do this by default, you can use Map to do this with non-listable functions.

First set up a list of the integers from 1 to 5:

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You can map a function over every element of the list using Map; this example uses an undefined function :

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You can use as a shorthand for Map (this is the same command as in the previous example):

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Most mathematical functions have the Listable property, meaning they automatically map over lists:

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If the function is not Listable, you can use Map, instead. Set up a list of five 2×2 matrices:

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Use Map to map MatrixForm over the list to see each of them in mathematical notation:

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Now use Map to calculate the eigenvalues of each of the matrices in the list:

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Map does not just operate on lists. It can be used for any expression:

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Apply is another functional programming operation. It replaces the head of an expression.

You can see how this works using two undefined functions, and :

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Apply has for a shorthand notation (this is the same command as the previous example):

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Common expressions are shown in StandardForm in *Mathematica*, but their underlying FullForm shows how Apply can be used:

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For example, this changes a sum into a product:

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Apply is useful when you want to turn the elements in a list into function arguments.

Create a list of five ordered pairs :

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Mod finds the remainder when dividing the first number of an ordered pair by the second:

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To apply Mod to all of the pairs, you need to work at level 1 of the list (specified by the ):

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You can use as a shorthand to apply at level 1:

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This is another way to do the same thing using a pure function with Function:

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This uses the short form of Function:

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